Efficient and Reliable Simulation of Quantum Molecular Dynamics

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 970

KATHARINA KORMANN

ACTA UNIVERSITATIS UPSALIENSIS ISSN 1651-6214 ISBN 978-91-554-8466-8 UPPSALA urn:nbn:se:uu:diva-180251 2012 Dissertation presented at Uppsala University to be publicly examined in 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, Friday, October 19, 2012 at 10:00 for the degree of Doctor of Philosophy. The examination will be conducted in English.

Abstract Kormann, K. 2012. Efficient and Reliable Simulation of Quantum Molecular Dynamics. Acta Universitatis Upsaliensis. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 970. 52 pp. Uppsala. ISBN 978-91-554-8466-8.

The time-dependent Schrödinger equation (TDSE) models the quantum nature of molecular processes. Numerical simulations based on the TDSE help in understanding and predicting the outcome of chemical reactions. This thesis is dedicated to the derivation and analysis of efficient and reliable simulation tools for the TDSE, with a particular focus on models for the interaction of molecules with time-dependent electromagnetic fields. Various time propagators are compared for this setting and an efficient fourth-order commutator-free Magnus-Lanczos propagator is derived. For the Lanczos method, several communication-reducing variants are studied for an implementation on clusters of multi-core processors. Global error estimation for the Magnus propagator is devised using a posteriori error estimation theory. In doing so, the self-adjointness of the linear Schrödinger equation is exploited to avoid solving an adjoint equation. Efficiency and effectiveness of the estimate are demonstrated for both bounded and unbounded states. The temporal approximation is combined with adaptive spectral elements in space. Lagrange elements based on Gauss-Lobatto nodes are employed to avoid nondiagonal mass matrices and ill-conditioning at high order. A matrix-free implementation for the evaluation of the spectral element operators is presented. The framework uses hybrid parallelism and enables significant computational speed-up as well as the solution of larger problems compared to traditional implementations relying on sparse matrices. As an alternative to grid-based methods, radial basis functions in a Galerkin setting are proposed and analyzed. It is found that considerably higher accuracy can be obtained with the same number of basis functions compared to the Fourier method. Another direction of research presented in this thesis is a new algorithm for quantum optimal control: The field is optimized in the frequency domain where the dimensionality of the optimization problem can drastically be reduced. In this way, it becomes feasible to use a quasi-Newton method to solve the problem.

Keywords: time-dependent Schrödinger equation, quantum optimal control, exponential integrators, spectral elements, radial basis functions, global error control and adaptivity, high- performance computing implementation

Katharina Kormann, Uppsala University, Department of Information Technology, Division of Scientific Computing, Box 337, SE-751 05 Uppsala, Sweden.

ISSN 1651-6214 ISBN 978-91-554-8466-8 urn:nbn:se:uu:diva-180251 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-180251) List of papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

IK.Kormann,S.Holmgren,H.O.Karlsson.Accuratetimepropagation for the Schrödinger equation with an explicitly time-dependent Hamiltonian. J. Chem. Phys.,128:184101,2008.1 Contributions:Theideasweredevelopedinclosecooperationwiththeco-authors. The author of this thesis implemented the methods and performed analysis as well as computations. She prepared a draft of the manuscript and ﬁnished it with assistance from the co-authors.

II K. Kormann, S. Holmgren, H. O. Karlsson. Global error control of the time-propagation for the Schrödinger equation with a time-dependent Hamiltonian. J. Comput. Sci.,2:178–187,2011.2 Contributions:Theideasweredevelopedinconsultationwiththeco-authors.The author of this thesis implemented the methods and performed analysis as well as computations. She prepared a draft of the manuscript and ﬁnished it with assistance from the co-authors.

III K. Kormann. A time-space adaptive method for the Schrödinger equation. Technical Report 2012-023, Department of Information Technology, Uppsala University, 2012. (Submitted)

Contributions:Theauthorofthisthesisisthesoleauthorofthispaper.

IV K. Kormann, A. Nissen. Error control for simulations of a dissociative quantum system. In G. Kreiss, P. Lötstedt, A. Målqvist, M. Neytcheva, editors, Numerical Mathematics and Advanced Applications 2009, pages 523–531, Springer, Berlin, 2010.3 Contributions:Theideasweredeveloped,themethodsimplemented,andthe manuscript written in close collaboration between the authors of the paper. The author of this thesis had the main responsibility for the parts regarding discretization errors and the error balancing.

1Reprinted with permission. Copyright 2008, American Institute of Physics. 2Reprinted with permission from Elsevier. VM.Gustafsson,K.Kormann,S.Holmgren.Communication-efficient algorithms for numerical quantum dynamics. In K. Jónasson, editor, Applied Parallel and Scientific Computing,volume7134ofLecture Notes in Computer Science, pages 368–378, Springer, Berlin, 2012.3 Contributions:Theideasweredevelopedinclosecollaborationbetweentheauthors of the paper and the manuscript was prepared in cooperation with the first author, in discussions with the third author. The author of this thesis was responsible for the error control. VI K. Kormann, M. Kronbichler. Parallel finite element operator application: Graph partitioning and coloring. In 2011 Seventh IEEE International Conference on eScience,pages332–339,2011.4 Contributions:Theideasandtheimplementationframeworkweredevelopedandthe manuscript drafted in close collaboration between the authors of the paper. The author of this thesis had the main responsibility for the partitioning and coloring algorithm and the application to the Schrödinger equation.

VII K. Kormann, E. Larsson. An RBF-Galerkin approach to the time-dependent Schrödinger equation. Technical Report 2012-024, Department of Information Technology, Uppsala University, 2012. Contributions:Theideasweredevelopedindiscussionbetweentheauthors.The author of this thesis performed the analysis, implemented the method, and carried out the numerical experiments. She prepared a draft of the manuscript and finished it in collaboration with the second author. VIII K. Kormann, S. Holmgren, H. O. Karlsson. A Fourier-coefficient based solution of an optimal control problem in quantum chemistry. J. Optim. Theory Appl.,147:491–506,2010.3 Contributions:Theideasweredevelopedbytheauthorindiscussionwiththeco- authors. The author of this thesis implemented the methods and performed analysis as well as computations. She prepared a draft of the manuscript and finished it in consultation with the co-authors.

3Reprints were made with kind permission from Springer Science and Business Media. 4 c 2011, IEEE. Reprinted with permission. ! Related work

Although not explicitly discussed in the comprehensive summary, the following papers are related to the contents of this thesis.

M. Kronbichler and K. Kormann. A generic interface for parallel • cell-based ﬁnite element operator application. Comp. Fluids, 63:135–147, 2012. K. Kormann and E. Larsson. Radial basis functions for the • time-dependent Schrödinger equation. In Numerical Analysis and Applied Mathematics: ICNAAM 2011,number1389inAIPConference Proceedings, pages 1323–1326, 2011. M. Gustafsson, A. Nissen, and K. Kormann. Stable difference methods • for block-structured adaptive grids. Technical Report 2011-022, Department of Information Technology, Uppsala University, 2011.

The framework for cell-based ﬁnite element operator application has been published as a part of the deal.II library. There are also two tutorial programs published as part of the deal.II project, step-37 (multigrid solver based on the framework) and step-48 (nonlinear example explaining the parallelization of the framework).

Contents

1Introduction...... 9

2Time-DependentQuantumDynamics...... 11 2.1 Born–Oppenheimer Approximation ...... 11 2.2 Interactions with Time-Dependent Fields ...... 13 2.3 Physical and Mathematical Properties ...... 14 2.4 Numerical Challenges ...... 15

3DiscretizationoftheSchrödingerEquation...... 18 3.1 Spatial Discretization ...... 18 3.1.1 Pseudospectral Methods ...... 18 3.1.2 Localized Methods ...... 19 3.1.3 Radial Basis Functions ...... 20 3.2 Time Evolution ...... 21 3.2.1 Suitable Integrators – a Comparison ...... 21 3.2.2 Rotating Wave Approximation ...... 23

4ErrorEstimationandAdaptivity...... 25 4.1 Temporal Adaptivity ...... 25 4.2 Spatial Adaptivity ...... 25 4.3 Residual-Based Error Estimation and Global Error Control ...... 27

5High-PerformanceImplementation...... 30 5.1 Communication-Avoiding Time Stepping ...... 31 5.2 Stencil-Based Implementation ...... 31 5.3 Hybrid Parallelization ...... 33

6QuantumOptimalControl...... 34

7SummaryandOutlook...... 37

8SummaryinSwedish...... 39

References ...... 44

1. Introduction

Quantum mechanics describes how fundamental particles and aggregations thereof interact. While quantum effects are usually not apparent in macro- scopic structures, they govern how chemical products are formed and reac- tants are consumed. In experimental quantum dynamics, scientists measure the effects of the wave nature of particles as, for instance, reaction rates or scattering cross-sections. Such experiments are complemented by a study of the mathematical models. These theoretical solutions reveal the dynamics of the particles and thereby provide understanding of the observables in terms of the underlying molecular properties. For most practical problems, analytical expressions of the solution are unknown, making computer simulations essential. With the aim of controllingreactions,quantumchemists put efforts in prob- ing and manipulating the dynamics of electrons and atomic nuclei. The emer- gence of femtosecond lasers in the late 1980s made it possible to “watch” the motion of nuclei. The development of laser technologies has continued and around the turn of the millennium it even became possible to follow the motion of the electrons, now using attosecond pulses. The possibility to control reactions on the molecular level opens new opportunities for chemical processes, for instance in the semiconductor or catalysis industries [98]. Furthermore, it has potential impact to electronics where scientists try to design miniaturized devices whose structures are made up of very few atoms only [26]. The number of degrees of freedom in the quantum model of a molecule is generally very large. For instance, in the full description of the quantum nature of the comparably small water molecule (H2O), 39 spatial degrees of freedom are involved. Moreover, the motion within a molecule ranges over scales of many orders of magnitude in both time and physical space. The so-called Born–Oppenheimer approximation is a useful simpliﬁcation which allows for splitting electronic and nuclear motion. This way the 30 electronic and the nine nuclear degrees of freedom in the H2Omoleculemodel are separated. Still, the description of the dynamics of the nuclei remains high-dimensional and the corresponding numerical simulation extremely chal- lenging (see Sec. 2.4). This thesis concentrates on such nuclear Schrödinger equations. There are many successful examples of further modeling—mostly involving relations from classical mechanics—that facilitate simulations of larger molecules. It is the aim of this thesis to exploit algorithmic efﬁciency and modern computer power to provide a simulation tool for the dynamics of as large molecules as possible without any classical modeling. Firstly, such a

9 tool for fully quantum dynamical simulations can be used to validate further approximations and enable error analysis and control. Moreover, there are still effects like tunneling that are very difficult to cover when classical approximations are applied. Quantum effects are also often restricted to a small part of the molecule. In this case, an efficient quantum solver for such a confined part of the molecule could be coupled to a more classical model of the rest of the molecule. The building blocks of an efficient solver as proposed in this thesis are error control to get reliable results and to adapt the discretization to the solution (see Ch. 4)andamemory-leanandparallelimplementation(seeCh.5). Spec- tral elements are suited for this purpose since they are of high order while still being localized enough to enable parallelization. Moreover, they are flexible when it comes to mesh refinement and the dual-weighted residual method provides a theory for global error control. For time-stepping, an exponential Magnus–Lanczos propagator offers good stability and accuracy without the need of solving systems of equations. AGalerkindiscretizationbasedonradialbasisfunctionsasanalyzedinthis thesis has the potential to further improve efficiency and compatibility with semiclassical methods. Finally, it is proposed to consider pulse optimization in Fourier space to solve quantum optimal control problems and to apply a superlinearly converging optimization routine (cf. Ch. 6).

10 2. Time-Dependent Quantum Dynamics

The Schrödinger equation was formulated for the ﬁrst time by Erwin Schrödin- ger in a series of papers [121, 122, 123, 124]. His ﬁrst article [121]considered the case of the hydrogen atom and the Schrödinger eigenvalue problem was devised starting from a classical Hamiltonian differential equation, taking into account several observations on the quantum nature of particles. Schrödinger’s work is still fundamental for contemporary quantum mechanics. The general time-dependent form of the Schrödinger equation, derived in Schrödinger’s fourth article [124], reads ∂ ih¯ Ψ(x,t)=HΨ(x,t), (2.1) ∂t where h¯ is Planck’s constant divided by! 2π and Hˆ is the (quantum) Hamilto- nian consisting of the kinetic and the potential energy operators of the studied system. The spatial coordinates x Rd represent the spatial position of each ∈ particle in the system and t denotes time. The wave function Ψ is found in 1 d 1 d L2(H (R ),(0,t f ]) and has a temporal derivative in L2(H− (R ),(0,t f ]).The square of its modulus, ρ(x,t)= Ψ(x,t) 2,givestheprobabilitydensityofthe | | system. The partial differential equation (2.1)needstobeclosedwithaninitial value, Ψ(x,0)=Ψ0.

2.1 Born–Oppenheimer Approximation When describing the full dynamics of a molecule, the Hamiltonian is established by a sum of the kinetic and the potential energies of each nucleus and each electron as well as the electron-nuclear potential energy. In this section, we will consider the time-independent Schrödinger equation, Hˆ Ψ(x)=EˆΨ(x).

Let us denote by Ri the coordinates of nucleus Ni and by Zi and Mi its charge and mass, respectively. For electron ei,wedenoteitscoordinatesbyri,and by e and m we denote electronic charge and mass, respectively. With this notation, the full molecular Hamiltonian is given by (cf. Refs. [125, 135]),

2 2 2 2 2 h¯ ZiZ je h¯ e Z je H = ∆ + ∆ + −∑ 2M Ni ∑ R R − ∑ 2m ei ∑ r r − ∑ r R i i i> j | i − j| i i> j | i − j| i, j | i − j| ! = TN +VN + Te +Ve +VeN,

! ! ! ! ! 11 where ∆ denotes the Laplacian with respect to the coordinates of particle (*). Since the∗ model includes the position of each particle as (three) degrees of freedom, the full system rapidly becomes extremely high-dimensional. There- fore it is necessary to separate electronic and nuclear coordinates already for small molecules. Since the mass of a nucleus is a factor 103 to 105 larger than the mass of an electron, the nuclei move much slower than the electrons. On the time scale of the vibration of electrons, the nuclei are usually almost stationary. In the Born–Oppenheimer approximation, it is assumed that the wave function Ψ(r,R) can be split into an electronic and a nuclear part as Ψ(r,R)=φ(r;R)ψ(R). Then the electronic Schrödinger equation is solved separately for fixed nuclear coordinates, el Heφ(r;R)=Eˆ (R)φ(r;R), with the electronic Hamiltonian! being He = Te +Ve + VeN . Substituting this ansatz and the electronic Schrödinger equation into the full equation yields ! ! ! ! h¯ 2 H (φ(r;R)ψ(R)) = ∑ (φ(r;R)∆Niψ(R)+2∇N iφ(r;R) ∇Niψ(R) − i 2Mi − · ˆ el ˆ ! +∆Ni φ(r;R)ψ(R))+ (E (R)+VN)φ(r;R)ψ(R). (2.2) The terms 2∇N iφ(r;R) ∇N ψ(R) and ∆N φ(r;R)ψ(R) in (2.2)hampera − · i i separation of the nuclear part. For most configurations, they are however of the same magnitude as the mass ratio between electrons and nuclei to some power. Therefore, they are dismissed in the Born–Oppenheimer approximation. However, these terms become relevant at nuclear configurations where several electronic states have the same potential energy. To improve the Born–Oppenheimer approximation it is common to couple several nuclear Schrödinger equations for different electronic states. The nuclear wave functions for each electronic state are collected in a vector ψ(R)= T (ψ 1(R),...,ψ n(R)) .Usually,oneappliesaso-calleddiabatic transformation and models the coupling of the various electronic states by potential terms. For a two-state system, for instance, the Hamiltonian becomes the block matrix Tˆ +Vˆ Vˆ Hˆ = 1 c , Vˆ H Tˆ +Vˆ " c 2 # where Vˆ1/2 is the diabatic potential energy surface (PES) of state 1 or 2, respectively, and Vˆc models the coupling of the two PES in the diabatic framework. A PES for a certain state is the sum of the internuclear repulsion and the eigenenergy.

12 AtypicalfeatureofaPESisthatitgoestoinﬁnityastheinternucleardis- tance approaches zero due to nuclear-nuclear repulsion. Moreover, as the distance becomes large and the nuclei become too far from each other to interact, the PES approaches an asymptotic value. Many PES have a minimum value and there exist one or more vibrational eigenstates that are bound, that is the molecular bond is stable, whereas others are monotonically decreasing for increasing distance and all eigenstates are unstable so that dissociation even- tually occurs.

2.2 Interactions with Time-Dependent Fields Given precomputed values for the electronic spectrum Eˆ el(R) in the diabatic representation, we can form the molecular Hamiltonian describing the dynamics of the nuclei under the Born–Oppenheimer approximation. In addition, we often do not only want to consider the dynamics of an isolated molecule but its interaction with a time-dependent field. Applications we have in mind are, for instance, multiple pulse optical spectroscopy or control of molecular dynamics by laser fields (cf. Ref. [135]). Optical spectroscopy can be applied to analyze the structure of complex molecules like biopolymers, and the manipulation of molecular dynamics can initiate chemical reactions. The period of nuclear vibrations in a molecule is on the time scale of tens or hundreds of femtoseconds. It was thus with the introduction of femtosecond lasers in the mid-1980s that it became experimentally possible to track and even manipulate the dynamics of nuclei in molecules [101]. Zewail and his group probed various molecules with femtosecond pulse pairs [151], and later also demonstrated how molecular states could be controlled with the help of time-delayed pulses [110].1 Fleming and co-workers [120]usedtime-delayed pulses with controlled phase for manipulating molecules. The groups of Crim [130]andZare[22]attemptedtousetunablelaser-pulseswherethewave- length could be varied for steering chemical reactions. Advances in pulse shaping techniques based on grating and filtering by, amongst others, Weiner and co-workers [63, 141], spurred the development of femtosecond chemistry. The recent development of attosecond lasers also facilitated physicists to probe the motion of electrons, taking place on the atto-scale, cf. Ref. [80]. To model interactions of a molecule with a femtosecond laser field, the molecular Hamiltonian (denoted H0 in the following) is augmented by an additional term representing the interaction with the electromagnetic field. A weak field can be modeled classically! using the dipole approximation. Given

1A. H. Zewail was rewarded the 1999 Nobel price in Chemistry “for his pioneer- ing investigations of chemical reactions on the time-scale they really occur.” (quote from the extended version of the press release of the Royal Swedish Academy of Sciences, see www.nobelprize.org/nobel_prizes/chemistry/laureates/1999/advanced- chemistryprize1999.pdf, September 4, 2012.)

13 target state

pulse

initial state

internuclear distance

Figure 2.1. Schematic configuration of the interaction of a molecule with a time- dependent field ε(t).Thegroundstate(dashedline)iscoupledtoanexcitedstate (dash-dot line) with a laser field. The second excited state (dotted line) is coupled to the first excited state by a static crossing of the PES. the laser field ε(t) and the transition dipole moment µˆ ,the(nuclear)time- dependent Schrödinger equation (TDSE) reads ∂ ih¯ ψ(R,t)= Hˆ (R)+µˆ (R)ε(t) ψ(R,t). (2.3) ∂t 0 $ % Fig. 2.1 shows a typical configuration where we have a molecule in its ground state and want to excite it to a target state with the help of a femtosecond pulse.

2.3 Physical and Mathematical Properties In this section, we discuss some fundamental properties of the physical model that also play an important role for the design of numerical methods. First of all, total probability in a closed quantum mechanical system is conserved. This means physically that no matter is destroyed or created. Probability conservation can be expressed as

ψ(R,t) 2 dR = 1forallt, & | | i.e., the L2 norm of the wave function is conserved. An important property of the evolution of the TDSE is time reversibility. No information is destroyed, as it would be if, e.g., diffusion were involved. Hence, if the Hamiltonian is known for the studied time interval, we can re- construct the initial state from the ﬁnal state. In the mathematical formulation

14 of the TDSE, the self-adjointness of the Hamiltonian (together with the imag- inary unit) assures norm conservation and time reversibility. Since classical Hamiltonian systems are well-studied, it is of interest that the TDSE can be rewritten as a classical system by defining the configuration variable q(R,t)=√2h¯Re(ψ(R,t)) and the canonically conjugate momentum p(R,t)=√2h¯Im(ψ(R,t)) (cf. Ref. [53]). An important conservation law for classical Hamiltonian systems is area conservation for the flow. Such a flow is called symplectic.ThispropertyintheclassicalversionoftheTDSEassures norm conservation of ψ (cf. Ref. [89]).

2.4 Numerical Challenges The simulation of molecular processes based on the chemical model discussed in the previous chapter entails several challenges. Most standard numerical methods for discretization of the spatial variables of a partial differential equation are grid-based, i.e., one distributes nodes in a more or less regular pattern over the computational domain and evaluates the solution in those points. One intricacy with such a discretization is the fact that the number of independent variables increases linearly with the number N of atoms in the system. More precisely, the total number of degrees of freedom is 3N,i.e.,threecoordinates for each nucleus. Six degrees of freedom are external2 (three translational and three rotational), giving 3N 6internaldegreesoffreedom[135,Sec.12.4]. − Only the internal degrees of freedom are of importance for a quantum description of an isolated molecule. For a grid based model, the number of grid points grows exponentially with the dimensionality: When n grid points per dimen- 3N 6 sion are used, the number of grid points becomes n − .Hence,onlyvery small molecules can be handled using such numerical schemes. Adaptive mesh refinement (cf. [33], Paper III), sparse grids (cf. [54, 50]), and parallelization (cf. [15, 57], Papers V and VI)areoptionstoincreasethe problem size that can be simulated slightly. Also, using radial basis functions as proposed in Paper VII is an option to reduce the size of the discrete system. An alternative to full quantum simulations is to use semiclassical methods that contain further modeling. As opposed to purely classical models, the particles are still represented by a wave function. The basis functions are not defined by some abstract approximation space but their parameters are determined by (classical properties of) the problem. Heller [62]introducedtheconceptof frozen Gaussians where the initial wave packet is represented by several complex Gaussians with fixed width, so-called coherent states, whose centers and phases are propagated according to classical trajectories. This method has been refined in many ways. Within the MP/SOFT framework [149]moreco- herent states are successively added and their parameters are optimized. The

2Note that the number of internal degrees of freedom is 3N 5forlinearmoleculeswherethe − rotation along the molecular axis collapses.

15 group of Martínez has refined semiclassical methods to also cover more complicated quantum effects [11]byamultiplespawningtechnique.In[37]semiclassical propagation with Hagedorn wave packets was proposed where the propagation according to the harmonic oscillator is corrected by a perturba- tion term. The method of coupled coherent states [126]usesMonte-Carlo sampling of trajectories. A different ansatz is provided by the multi configuration time-dependent Hartree (MCTDH) method [100]whichisbasedona decomposition of the wave packet into aproductofsingleparticle functions and the high-dimensional linear PDE is reduced to a number of low dimensional nonlinear PDEs. In this way, larger molecules can be studied if they exhibit a certain structure. Despite much progress in this area, difficulties can still arise when e.g. tunneling occurs or static or dynamic couplings are con- tained in the model. The wave function does not only depend on the spatial coordinates but also on time. Since multiple time scales are usually present in the model, a huge number of time steps is often necessary. High-frequency oscillations have to be resolved or modeled. High order and adaptive integrators or multi-scale algorithms can help in reducing the number of time steps. There are various sources for high-frequency oscillations depending on the particular type of application. In the semiclassical scaling, oscillations occur due to high frequencies in the spatial variables. For simulations of interaction with a time- dependent field, the oscillations of the laser field pose limitations to the step size. A simple multi-scale model, allowing for larger time steps, is the so- called rotating wave approximation (see Sec. 3.2.2). A review of numerical integrators for highly oscillatory Hamiltonian systems was provided by Co- hen et al. [28]. Athirdpotentialdifficultywhenconstructingnumericalmethodsistheun- boundedness of the spatial domain in which the TDSE is posed. For a wide class of problems the wave function stays within a certain domain, i.e., the molecule stays intact. In such cases, the computational domain can be truncated and the PDE is usually closed by periodic or homogeneous Dirichlet boundary conditions. However, the modeling of dissociative processes poses difficulties when the domain is truncated and parts of the wave packet leaves the computational domain. Standard methods to handle dissociation from the computational domain are complex absorbing potentials [95, 55]andperfectly matched layers [12, 1, 103]. In both cases, artificial damping is introduced to the system and physical properties like norm conservation and time reversibility are no longer valid in the computational domain. Therefore, numerical methods designed for bound states can be difficult to generalize to dissociative problems. On the other hand, particular pseudospectral methods have been designed for unbounded domains (see e.g. [140, 102]). A similar alternative discussed in Paper VII is to use globally defined radial basis functions without any boundary conditions. However,thesemethodsstillneedtoresolve the whole wave packet to some extent.

16 The challenges discussed so far concern solving one single TDSE problem. When this equation appears within an optimization loop, e.g., for ﬁnding the optimal shape of an interacting laser pulse, additional complexity is introduced. The Schrödinger equation has to be solved many times,andmoreover an adjoint equation might have to be computed to calculate optimality conditions. This introduces extra demands on both computing power and memory size.

17 3. Discretization of the Schrödinger Equation

For a computer simulation of the TDSE, we have to discretize spatial and temporal variables. Preferably, the discretization should satisfy the same conservation properties as the continuous problem (cf. Sec. 2.3). Moreover, it is desirable that the method can easily and efﬁciently be generalized to multiple dimensions. In this chapter, we review common methods from the literature with a special emphasis on the methods applied in the attached papers.

3.1 Spatial Discretization Since the solutions of the time-dependent Schrödinger equation are generally of class C∞,high-ordermethodsaremoreefficientconsideringbothcomput- ing time and memory requirements. Therefore, pseudospectral methods are commonly used in the field. While these methods work very well in a standard setting, stencil-based methods and radial basis functions are more flexible alternatives.

3.1.1 Pseudospectral Methods The idea of spectral methods is to represent the approximate solution as a weighted sum of basis functions with global support. When a collocation approach based on this weighted sum is used, one is talking about a pseudospectral method [48, 44]. Pseudospectral methods have been introduced as discrete variable representation (DVR) to quantum dynamics [87, 86]. Colbert and Miller [29]introducedthesinc-DVRandFeit&Fleck[41]andKosloff& Kosloff [77]proposedthedynamicFouriermethodwheretheHamiltonianis not computed explicitly but the action of the kinetic energy operator is evaluated via the Fast Fourier Transform. While the Fourier method is limited to bound-state computations where the computational domain can be truncated, pseudospectral discretizations based on Hermite functions [14, 36]operatein infinite domains. Especially, the dynamic Fourier method is very popular in the quantum dynamics community due to its high accuracy and fast evaluation. Moreover, it can nicely be combined with the so-called split operator time-evolution method (see Sec. 3.2.1)yieldinganeasy-to-implementfulldis- cretization. Pseudospectral methods are generalized to higher dimensions by tensor products which makes adaptive refinement difficult. However, one does not

18 need to use the full grid. Instead one can use sparse representations based on the hyperbolic cross. Hallatschek [61]devisedasparsegridalgorithmfor the Fourier method which was applied to the Schrödinger equation by Gradi- naru [49]. Also, Griebel and Hamaekers [54]consideredsparsegridsforthe Schrödinger equation. Attempts have been made to use the Fourier method more efﬁciently by transforming the coordinates according to the speciﬁc potential. This procedure is called the mapped Fourier method [39]andwas extended to time-dependent problems by Kleinekathöfer and Tannor [75].

3.1.2 Localized Methods Localized stencils as in finite differences or finite elements allow for more general grids with refinement adapted to the solution (see Ch. 4). Moreover, localized methods reduce the data dependencies for derivative computations to anumberofneighboringpointscomparedtotheglobaldependencyinpseu- dospectral derivative approximation. For a given number of grid points, localized methods are, of course, less accurate, but their superior properties when it comes to parallelization and adaptivity can nonetheless make high order stencils competitive. Particularly finite differences have attracted attention for the solution of the TDSE. A drawback of finite difference methods compared to the Fourier method is the fact that the dispersion relation is only approxi- mately recovered. Gray and Goldfield [51]havethereforedeviseddispersion fitted finite differences. The convenience of finite differences is their ease of implementation and the good conditioning of high-order stencils [56]. Finite element methods, on the other hand, are mostly used with comparably low order. However, there is a variant—referred to as spectral elements—that is suited for use with high orders [73]. When using high order elements, the nodes have to be clustered towards the element boundary in order toachieveawell-conditionedapprox- imations and avoid effects from the Runge phenomenon. One such type of elements is the Lagrange element with support points distributed according to the nodes of the Gauss–Lobatto quadrature rule. This type of element also comes with a second advantage: One can use Gauss–Lobatto quadrature with nodes corresponding to the interpolation points when evaluating the mass and stiffness integrals which gives a diagonal mass matrix. In this way, the integrals are only approximated but the approximation order of the quadrature rule is equal to or higher than the approximation order of the finite element. This type of elements was used for the solution of quantum scattering problems by Manolopoulos & Wyatt [96]andRescigno&McCurdy[112]. With the efficient implementation framework proposed in Paper VI and [82], high-order finite elements can be almost as narrow as finite differences on equi-spaced grids while at the same time allowing for mesh refinement without loss in the approximation order (cf. Chs 4 and 5).

19 3.1.3 Radial Basis Functions Approximation using radial basis functions (RBF) [144, 38, 24]isanalter- native to grid-based methods. In this method, the basis functions are radial functions centered around a set of points that can be arbitrarily distributed over the computational domain. The solution is approximated by a weighted sum of the basis functions and the partial differential equation can be solved by either choosing a collocation or a Galerkin ansatz. There are various types of radial functions that are commonly used for RBF- based approximations. Choosing a globally supported and globally smooth function, one gets a spectrally convergent method [144, 114]. Radial basis function discretization based on a Galerkin formulation of the partial differential equation was analyzed in [143]. Compared to RBF-collocation [72], however, RBF-Galerkin has received much less attention so far. This can primarily be attributed to the fact that a Galerkin formulation requires the evaluation of integrals. In Galerkin formulations for finite elements this is usually done by numerical quadrature. For spectral methods, however, the use of numerical quadrature would spoil the approximation order. In our special setting for the TDSE, however, analytical evaluation is a viable alternative: When using Gaussian basis functions the computational domain does not have to be truncated and there are closed-form expressions for the integrals involved in these settings—at least as long as the potentials are polynomials or Gaussians. Collocation, on the other hand, is easier to implement, but finding a stable discretization is more intricate. Problems with eigenvalue stability related to boundary treatment were pointed out by Platte and Driscoll [108]. For the TDSE, we do not have to bother with boundary conditions. Nevertheless, we have encountered stability problems due to the combination of second derivatives and the potential term. Fornberg and Lehto [45]describedastabilization procedure based on artificial diffusion. The problem with this remedy, however, is that damping is introduced to the whole spectrum. Hence the eigenval- ues still do not represent the correct physics and we prefer an RBF-Galerkin ansatz for the TDSE. In Paper VII,weanalyzetheconvergencepropertiesof our Galerkin ansatz without explicit boundary treatment. When reducing the fill distance between the nodes in a fixed domain, exponential convergence is shown up to a point where the error flattens out due to the systematic error from the part of the domain where no basis functions are centered. Numer- ical experiments show that RBF discretizations are more accurate than the dynamic Fourier method already for an equidistant node distribution. On the other hand, dense matrices have to be handled for RBF computations, and the efficiency must be improved to actually outperform the dynamic Fourier method. Conceivable approaches are truncation or some other type of local- ization or the application of fast transforms.

20 3.2 Time Evolution As discussed in Sec. 2.4,thetemporaldimensionhastobetreatedwithextra care since the number of time steps needed is usually much larger than the number of grid points per dimension. Throughout this section, we consider the semi-discretized TDSE, d i u(t)= H(t)u(t), (3.1) dt −h¯ where H denotes a matrix that represents the Hamiltonian in the approximation space and u =(ui) is the coefﬁcient vector representing the wave function. There is a large spectrum of methods designed for time-evolution of ODE systems. For our purpose, an explicit method1 is preferable since solving a linear system can become very costly and memory consuming when high- order spatial methods in high dimensions are involved. The monographs [59, 85, 117]presentmethodsspeciallysuitedforHamiltoniansystems.Thereare two families of methods that are most often used for the TDSE, exponential integrators and partitioned Runge–Kutta (PRK) methods [91]whichweboth will discuss in more detail below. Note that, in this thesis, we only consider the case where short time steps are required because we assume that the Hamiltonian is explicitly time-dependent. The situation with a time-independent Hamiltonian has been studied exten- sively, see the comparative study by Leforestier et al. [84].

3.2.1 Suitable Integrators – a Comparison Exponential integrators For a linear ODE with a time-independent right-hand side, i.e., for the semi- discretized TDSE with time-independent Hamiltonian, the evolution operator is analytically given as the exponential of the Hamiltonian matrix times the time span. The situation becomes more complicated for time-dependent right- hand sides but, for sufﬁciently small time intervals, the Magnus expansion [18, 94]providesanexpressionoftheevolutionoperatorU(t + ∆t,t),deﬁned by u(t + ∆t)=U(t + ∆t,t)u(t),intheformoftheexponentialofaseriesex- pansion,

U(t + ∆t,t)=exp ∑ θl . 'l 0 ( ≥ Each term θl contains l + 1integralsoverl commutators of the Hamiltonian 2 l +3 at different points in time. Since the lth (l > 0) term decays as (∆t) & 2 ' for sufﬁciently small time steps, a truncated version of this expansion provides a natural starting point for numerical evolution methods.

1Amethodthatdoesnotinvolvethesolutionoflargelinearequationsystems,e.g.,through factorization or iterative solvers.

21 After truncating the Magnus expansion, the matrix exponential has to be evaluated. The simplest approach is the split operator [40]. The method is asecond-orderaccurateexponentialStrangsplittingwheretheHamiltonian is split into potential and kinetic energy operators and the time-dependence is evaluated at the mid-point. This form is appealing if we have a fast transformation between the diagonal representations of the potential and the kinetic energy operators. This is the case for a Fourier spectral approximation in space where derivative computations are diagonal operations in Fourier space and where the Fast Fourier Transform represents a computationally cheap transformation from coordinate to Fourier space. AmoreflexiblealternativeistouseaKrylovsubspacemethodforcomputa- tions of the action of the matrix exponential. For a symmetric Hamiltonian matrix, coming from an arbitrary spatial discretization method, the Lanczos algorithm can be used and for non-symmetric Hamiltonians, that can for instance arise if absorbing boundaries are applied, the computationally more expensive Arnoldi method. Krylov methods can also be combined with a higher-order truncation of the Magnus expansion to get more accurate integrators. Evaluat- ing higher Magnus terms is quite costly, though, since an increasing number of commutators has to be computed. Several attempts have been made to simplify the terms. Blanes et al. [19]rewritetheexpansionreducingthetotalnumber of commutators for truncation up to a certain order and Blanes & Moan [21] split the matrix exponential to completely avoid commutators (with the price of several exponentials to be computed per time step). However, one can use the special structure of the Hamiltonian in the case of a matter-field-interaction problem where the spatial dependence in the transition dipole moment is usually rather weak. In this case the second order Magnus term simplifies to a block-diagonal matrix. Hence, increasing the order of the truncation from order two to four can be achieved with very little additional computational effort. This is proposed in Paper I.

Runge–Kutta methods AsecondclassofintegratorsreliesonthefactthatthecomplexHamilto- nian system of the TDSE can be transformed into a real Hamiltonian system as pointed out in Sec. 2.3.So-calledpartitionedRunge–Kuttamethodsare symplectic methods of Runge–Kutta type that are designed to mimic the conservation laws exhibited by classical Hamiltonian systems. However, most PRK methods are implicit. Explicit PRK methods are available for separable Hamiltonian functions where Hˆ (p,q)=Hˆ (p)+Hˆ (q).IftheHamiltonianis explicitly time-dependent, this separation cannot be made. However, Gray and Verosky [53]exploitedtheMagnusexpansiontokeepthemethodexplicitfor time-dependent Hamiltonians. Sanz-Serna and Portillo [118]proposedamore elegant procedure, by introducing an additional conjugate pair of variables to represent the time dimension. The separation also fails in case the Hamilto-

22 nian is complex-valued. This is, e.g., the case when modeling dissociation by adding complex terms to the potential or the kinetic energy. By varying the number of stages, it is possible to design optimal Runge– Kutta coefficients for special problems in terms of accuracy and stability. Suzuki [132, 133, 134]andYoshida[150]demonstratedhowtoconstructhigh order symplectic methods. Later McLachlan [99]cameupwiththeideato construct methods with an optimal error constant for given order and Blanes &Moan[20]refinedthisconcept.InRefs.[17]and[52], optimal coefficients for TDSE problems have been devised, even though both articles only provide examples with time-independent Hamiltonians. In Paper I, the numerical propagation for a model of the rubidium diatom (Rb2)isstudied.Inthesecalculations,themethodsproposedinRef.[20] perform better than those tailored to the TDSE [17, 52]. However, exponential integrators are even more efficient for low to moderate accuracy requirements.

3.2.2 Rotating Wave Approximation The simulation of matter-field interaction requires the resolution of the oscillations of the laser pulse. This necessitates small time steps. For computations with a low accuracy requirement, this shortage can be overcome with the help of the rotating wave approximation (RWA) [111]. Consider a two state system with Hamiltonian Tˆ +Vˆ (R) f (t)cos(ωt) Hˆ = g , f (t)cos(ωt) Tˆ +Vˆ (R) " e # where ω is the frequency of the laser field and f ( ) aslowlyvaryingenvelope · function. Define the transformation I 0 Wˆ := , 0eiωt I " − · # of the wave packet. Then, the TDSE for the transformed wave packet ϕ := 1 Wˆ − ψ reads

∂ϕ Tˆ +Vˆ (R) 1 f (t) 1 + e i2ωt ih¯ = g 2 − ϕ. ∂t 1 f (t) 1 + ei2ωt Tˆ +Vˆ (R) h¯ω " 2 $e − % # If the laser frequency is very$ high compared% to the variations of the envelope, i2ωt the time-averaged influence of the terms e± is insignificant. In the RWA, this term is dismissed. When propagating the solution of the RWA-TDSE, the oscillatory frequency of the laser field need not be resolved. This corresponds to a separation of scales where effects on the scale of π/ω and below are neglected. A detailed discussion of the modeling error introduced by the RWA can be found in Paper I.

23 5 10

0 10 error

ï5 10 Gauss Filon RWA FilonïRWT ï10 10 ï2 0 2 4 10 10 10 10 step size [fs] Figure 3.1. Matter-field interaction for the rubidium diatom solved with Magnus– Lanczos propagator of order four. The integrals in the Magnus expansion are inte- grated with Gauss or Filon-type quadrature, respectively. The methods start converging once ∆t is below the period of the oscillation of the laser field (verticle line). The RWA gives good results first but flattens out. Filon-type integration applied after the rotating wave transformation (Filon-RWT) yields slightly improved results compared to the RWA.

The RWA is suitable for low accuracy computations where we do not need to resolve the oscillations of the laser pulse with the time step. Clearly a straight-forward discretization with a standard method for numerical quadrature fails to give a meaningful approximation if the oscillations are underre- solved. On the other hand, there are so-called Filon-type methods for highly oscillatory quadrature [69]thataremoresuitabletouseinthisregime.Still, the RWA is considerably more accurate but Filon-type integration can slightly improve the result when applied after transformation with W (see Figure 3.1).

24 4. Error Estimation and Adaptivity

In the previous chapter, we have discussed several methods that are suitable for the numerical solution of the time-dependent Schrödinger equation. One major goal of the research discussed in this thesis is to provide an efﬁcient solver that adapts the discretization to the shape of the solution and provides the solution with a certain prescribed accuracy. In order to design such a solver one needs to have an error estimate at hand and a way to discretize the Hamiltonian for unevenly distributed resolution.

4.1 Temporal Adaptivity In order to design an algorithm that automates the choice of the time step and is capable of meeting a given error tolerance, adaptive step size control is desirable. A theoretical study of accuracy and convergence rates for several standard time-marching methods for the TDSE was provided by Lubich [92]. However, those estimates are not easily computable within a step size control algorithm. Within the framework of Runge–Kutta methods, efﬁcient techniques for local error control have been designed based on embedded methods of two consecutive orders. Admittedly, a straight-forward implementation of PRK methods with variable step size suffers from the fact that symplecticity is lost (cf. Ref. [117]). Sophisticated techniques have been developed to circumvent this problem (cf., e.g., Ref. [16, 60]) which are, however, strongly dependent on a good choice of a parameter function. For the Magnus expansion, on the other hand, an easy-to-compute error estimate based on extrapolation is available [19]andhasbeensuccessfully applied to the TDSE (cf. Ref. [145]andPaperII). This can be combined with aLanczosalgorithmthatchoosesthesizeoftheKrylovsubspacetomeetthe same tolerance. The Lanczos error can be estimated according to Ref. [67], similar to what is used to stop iterative solvers like conjugate gradients [115].

4.2 Spatial Adaptivity While pseudospectral methods can only be adapted to a specific problem by transforming the coordinates (cf. Sec. 3.1.1), finite elements and finite differences allow for more general grid adaptation. Although finite elements with

25 tetrahedral meshes allow for arbitrary mesh refinement on unstructured grids without hanging nodes, we prefer to use structured refinement to benefit from the reduced memory consumption. Also, the additional degrees of freedom in- side each tetrahedron for high order exhibit some structure anyway. The mesh is split into patches of different refinement level. Interfaces between these patches can either be treated continuously with constraints on degrees of freedom that are only active on one side of the interface [113]ordiscontinuously with penalty terms. The latter procedure is usually referred to as discontinuous Galerkin [65]andusedinconjunctionwithpenalizationoverallfaces. However, this comes with computational and memory overhead compared to the methods used in Papers III and VI. Finite differences, on the other hand, are very easy to implement on equidistant grids, but adaptive mesh refinement is more intricate to achieve in a stable manner. For the treatment of domain boundaries, the so-called summation- by-parts technique was proposed by Kreiss and Scherer [81]toachievestable discretizations. This construction can also be used to treat interfaces between blocks with different refinement levels both in a continuous [79]andadis- continuous setting [97, 104]. However, order reduction around interfaces and especially corner points is problematic. While adaptivity calls for elaborate treatment of irregularities in the mesh when using grid-based methods, radial basis functions are mesh tolerant. In principle, one can distribute the centering points in an arbitrary pattern and the theory is naturally formulated for scattered data (cf. [24,Ch.5]and[38,Ch. 1]). However, ill-conditioning can appear when the nodes are clustered too closely. This calls for scaling of the shape parameters which in turn deterio- rates the convergence. Optimal scattering of nodes in radial basis function discretization is an active area of research and many questions are still open. An idea in one dimension is to equidistribute the points according to the arclength of the solution [119]. This is, however, difficult to generalize to higher dimensions. A mathematically rigorous but computationally costly framework is the greedy algorithm by Ling and Schaback [88]whichphrasestheproblemof scattering the nodes as an optimization problem. Driscoll and Heryudono [34] propose a subsampling method where the shape parameter is proportional to the distance to the nearest neighbor despite the loss of spectral convergence. In the setting of the Schrödinger eigenvalue problem, Degani [31]proposes an alternative strategy that places the points according to the potential in the Hamiltonian, i.e., relying on properties of the equation instead of the solution. On the sphere, where no boundaries have to be taken into account, it has been proposed to redistribute the points based on an electro-static repulsion model of the shape of the solution [43].

26 4.3 Residual-Based Error Estimation and Global Error Control For both temporal and spatial discretizations, much research has been on designing error estimates. For spatial discretization, various techniques for error estimation appear in the literature. There are strategies that are only based on the value of the solution, some on derivative information of the solution, and others on the residual. Multiresolution analysis provides a tool to analyze the properties of the solution and can be used to identify regions where the solution contains more information and regions where it contains less. Such wavelet analysis was applied by Jameson[70]toadaptfinitedifferencestencils to the solution. Error estimation based on derivatives in an a priori estimate was analyzed by Eriksson & Johnson [35]. All of these strategies examine errors at a given point in time only, but for time-dependent problems one wants to know how these errors influence the result at the final simulation time. A posteriori error estimates that are based not only on the residuals but also incorporate information from a dual problem fill this gap. A duality argument was first proposed by Babuška and Miller [3]andrefinedbyBeckerandRannacher[9, 10]. While an a posteriori error estimation can in principle be derived for any discretization technique, the built-in Galerkin orthogonality of the finite element method makes it a powerful tool. Since the error is perpendicular to the approximation, it is possible to estimate the error with which each cell contributes to the global error. For finite differences, for instance, such a posteriori estimates can only make a meaningful prediction of the global error. Paper IV employs the global estimation. Duality-based error control can also be formulated for ordinary differential equations, see [27]. Papers II and III are devoted to the design of a reliable simulation method for the TDSE. For the numerical solution u of the TDSE at the final time t f , we consider a functional, represented by φ,ofthetotalerror,e(t ) := ψ(t ) f f − u(t f ).DefiningthedualTDSE ∂ ih¯ χ(t)=Hˆ χ(t), χ(t )=φ, 0 t < t , ∂t f ≤ f we can express the error functional as

t f ∂ ∂ φ,e(t ) = χ(t), e(t) + χ(t),e(t) dt, (4.1) ) f * ) ∂t * )∂t * &0 " # where , denotes the L inner product. For a ﬁnite element semi-discretiza- )· ·* 2 tion satisfying a Galerkin orthogonality relation, equation (4.1)becomes φ,e(t ) = (4.2) ) f * t f i h¯ 2 i ∂ ∇χ (t), ∇u(t) + χ ,Vˆ (t)u(t) + χ (t), u(t) dt, 0 h¯ ) ⊥ 2M * h¯ ) ⊥ * ) ⊥ ∂t * & " #

27 Figure 4.1. Final solution on an adaptively refined mesh. Simulation of matter-field interaction in OClO with Gauss–Lobatto elements of order 7. where χ denotes the part of the dual solution in the orthogonal complement of the approximation⊥ space. As mentioned in Sec. 3.1.2,weuseaspecial type of elements, Gauss–Lobatto elements, in Paper III,andadiscretization based on these elements does not satisfy the weak equation exactly in the approximation space. However, the truncation due to inexact quadrature is not of leading order (for elements of order greater than three) and can therefore be neglected in a first-order error estimate. When discretizing in time as well, we get an additional term in the error expression (4.2)accountingfortheperturbationduetonumericaltimeprop- agation. Since there are very efficient methods for temporal propagation of the TDSE and we expect the number of time steps to exceed the number of grid points per dimension for the application we have in mind, we do not use finite elements for discretization of the time variable. Instead, we choose a Magnus–Lanczos propagator. When combining temporal and spatial adaptivity in such a mixed discretization method, care has to be taken because refinement in space can influence the temporal error (cf. [66]). Therefore, we propose an adaptive algorithm that handles the temporal adaptivity independently. We consider the spatially semi-discretized TDSE as a system of ordinary differential equations and apply the theory developed by Cao and Petzold [27]. Exploiting the linearity and the norm conservation of the TDSE, we derive an aposterioriestimatethatisindependentofthesolutionofthedualproblem, see Paper II.Figure4.1 shows the final solution and the adapted mesh of a simulation of the OClO molecule with the algorithm described in Paper III. When designing an adaptive algorithm that should control errors from various sources, it is important to balance the different error terms. Paper IV

28 considers such error balancing for the error due to numerical approximation in time and space and an additional error from domain truncation that becomes important when dissociative configurations are considered. The dissociative part of the wave packet is handled by a perfectly matched layer as described in [103]. An argumentation as in Paper II is used to dismiss the dual problem even though the norm of the dual is no longer conserved due to the artificial dissipation in the layer. However, it is argued that the error estimate is still efficient if one makes sure that the dual problem and the computational domain are chosen such that the interesting features of the problem take place within the computational domain. While residual-based error estimation has proved to be quite effective for localized methods, it is more intricate in a setting with spectral methods: Since local changes exhibit a global effect, it is difficult to predict the effect of a basis change. Also boundary effects have to be taken into account. We have tested straight-forward residual-based error estimation for radial basis function approximation with little success.

29 5. High-Performance Implementation

Since applications in quantum dynamics are large-scale problems, an efficient and parallel implementation is essential. In most cases, memory is the limit- ing factor. A common strategy is therefore to decompose the spatial domain into partitions, distributed between nodes in a computer cluster and to let each node perform the simulation on its share. However, the simulation on the different parts of the domain are not independent since derivative computations use information from neighboring cells. Therefore, the nodes have to com- municate and exchange information. Since communication is generally more expensive than computation on contemporary computer architectures with in- terconnected nodes (clusters), care has to be taken when implementing data exchange. It is preferable to overlap data exchange with computation when- ever possible. In order to optimally exploit parallelism on all levels, hybrid parallelization has to be considered in addition. The implementations considered in this thesis include three levels of parallelism: Distributed-memory parallelization with the Message Passing Interface (MPI), shared-memory parallelization with Intel Threading Building Blocks (TBB) or OpenMP, and explicit vectorization. Two examples of software projects that are targeted to computing the solution of the full time-dependent Schrödinger equation in a parallel setting are PyProp [15]andHAParaNDA[57]. The former was developed by Birkeland and is maintained at the Bergen Computational Quantum Mechanics Group. The code is based on a tensor product discretization which allows for combi- nations of different types of approximations—spectral and localized—for the individual independent variables of the problem. The HAParaNDA project by Gustafsson, on the other hand, is focussed on finite difference approximations. Currently, it supports time-stepping based on the Lanczos propagation methods including the step-size control devised in Paper II.Thespatialdiscretiza- tion is built on equally sized blocks with ghost layers. The parallelization is a hybrid of OpenMP and MPI. Paper V relies on the HAParaNDA framework. It analyzes the impact of communication in the Lanczos time-stepping algorithm. Paper VI discusses a stencil-based implementation of finite elements (published in the deal.II library [7]) that is similar to finite difference stencils. The focus of the paper is on strategies for shared-memory parallelization. The experiments reported in Paper III are based on this implementation.

30 5.1 Communication-Avoiding Time Stepping One way to reduce the overhead due to communication is to use a temporal propagation method where several matrix-vector products can be computed independently from each other. Most common numerical propagators, however, construct the approximation of the new time step with a number of stages that are constructed in an iteration where each stage depends on the previous ones. In the Lanczos algorithm, we moreover have to compute inner products that require all-to-all communication which is considerably more expensive than the nearest-neighbor communication needed in sparse matrix-vector products. Each stage of the Lanczos algorithm in its standard formulation requires one matrix-vector product as well as two inner products that require communication. Paige [106]hasconsideredrearrangementsofthealgorithminaway that is less efficient for serial implementations (since one extra vector update is necessary) but avoids one global synchronization point per stage by clustering the two inner products. The so-called s-step Lanczos [74]methodgoesone step further: It really modifies the algorithm with the aim of clustering blocks of s stages of the Lanczos algorithm between synchronization points. This reduction in communication comes with the price of one extra matrix-vector product as well as the fact that the resulting algorithm is unstable in floating- point arithmetics when the Krylov subspace becomes too large. It is therefore essential to control the error and to choose the Krylov subspace adaptively. In Paper V,wehavetestedthesecommunication-reducingLanczosvari- ants in the setting of a temporal propagator with adaptively chosen size of the Krylov subspace. However, we do not see a clear advantage of the communication reduction in our experiments. The result was rather similar run-times for all three Lanczos variants with a slight advantage for the few-synchronization Lanczos. An enhancement of the s-step Lanczos method is the so-called communication-avoiding Lanczos method proposed by Hoemmen [68]. The numerical features of this algorithm were further investigated by Gustafsson et al. [58].

5.2 Stencil-Based Implementation High-order methods are preferable for the TDSE since they keep the number of variables needed low. Also, the number of arithmetic operations involving each degree of freedom (DoF) increases with the order which reduces memory transfer in relation to the number of computation. This comes, however, with the price that each DoF couples to a larger number of other DoFs for derivative approximations. Therefore, a sparse-matrix representation of the Hamiltonian will contain many entries per row. Storing this matrix requires much more memory than a vector storing the solution data. Hence, memory consumption would be dominated by the matrix, and it is essential to rely on an implementation that does not form the matrix explicitly. While it is relatively common

31 4 1 10 10 sparse matrixïfree 3 0 10 10

sparse 2 ï1 10 10 matrixïfree flops per DoF wall clock time

1 ï2 10 10 1 2 3 4 5 6 8 10 1 2 3 4 5 6 8 10 degree of finite element degree of finite element (a) Computational complexity. (b) Wall clock time on Nehalem EP. Figure 5.1. Comparison of complexity per degree of freedom (DoF) and wall time in seconds per matrix-vector product for a 3D Laplacian (problem size ﬁxed at 1.77 million DoFs) for Gauss–Lobatto elements with matrix-free and sparse-matrix implementations.

to use stencil-based implementations for finite difference methods, finite elements are mostly implemented by assembling a matrix in the beginning of the computation and then solving the problem using this matrix. This formulation is chosen in most common open-source projects for generic finite element programming like, e.g., deal.II [7]orFEniCS[90]. Specialized software targeted at spectral elements, on the other hand, choose a matrix-free implementation. Examples are SPECFEM 3D [76, 8]andNektar++[137, 138]. However, these packages do not support refined meshes with hanging nodes. In [82], data structures for matrix-free implementation of finite elements within the deal.II framework were proposed. These structures provide the framework for the experiments with adaptive finite elements reported in this thesis. For high order finite elements in high dimensions, a stencil-based implementation does not only reduce memory consumption but the computational complexity, too. This is specially relevant for Gauss–Lobatto elements where the complexity per degree of freedom is reduced from O(pd) in a sparse- matrix setting to O(dp) for d dimensions and convergence order p.The speedup in practical computations is even higher: Sparse matrix-vector products are memory bandwidth bound so that higher arithmetic intensity can be achieved for a matrix-free implementation where data is reused. Figure 5.1 il- lustrates complexity and run time for the evaluation of the action of the Lapla- cian in three dimensions. In these simulations, the matrix-free version reaches about 50% of the peak performance for order ten while the corresponding number for the sparse matrix is only about 10%.

32 5.3 Hybrid Parallelization When solving partial differential equations on a distributed-memory system, there are two issues that have to be dealt with: the parallelization of the grid structure and data types that enable communication of vector entries at the boundary of the processor’s domain. The more general the grid structure, the more complicated and memory-consuming becomes the data structure to hold the grid information. To simplify the grid structure in adaptive mesh refinement, Berger and Colella proposed a structured version [13]. For instance, the software projects Chombo [30], SAMRAI [147], and AMROC [32]provide solvers with finite differences on structured adaptively refined meshes. The p4est package [25]handlesparallelmeshesbyaglobalcoarsemeshand processor-local hierarchically refined cells. It can be combined with generic (finite element) PDE solver software and there is an interface [6]tothedeal.II package. When the grid structure of the problem is set up, computations boil down to the computation of matrix-vector products or the solution of linear equation systems. It is very common to rely on linear algebra packages like PETSc [5, 4]orTrilinos[64]tohandlecommunicationbetweenremotenodes.While this generally works very well for implementations relying on matrices, the corresponding data structures suffer from a relatively large overhead in case the computations are broken down into cell-wise evaluations of the differential operator. In this case, it is favorable to explicitly implement communication using MPI commands. In an operator evaluation, computations on the various cells are not independent since more than one cell contributes to the results for DoFs at element boundaries. For shared memory computations, we therefore have to prevent data loss which can arise when processes try to write into the same data field simultaneously. In order to avoid this, coloring strategies [115,Sec.12.4]are commonly used. Simple coloring suffers from explicit synchronization points and poor data reuse in processor caches. In Paper VI,wethereforepropose astrategyofpartitioningandcoloringontwolevelswhichavoidsexplicit synchronization. The dependencies form a graph of tasks that are dynamically scheduled using Intel TBB. Compared to pure coloring, the cache performance is also improved. On a single processing unit, instruction-level parallelism is possible through vectorized data types. Explicit vectorization is heavily used in the BLAS packages, but is more rare in user code. Matrix-vector products that are evaluated cell-wise can be vectorized by clustering the instructions for two (or more, depending on the processor’s instruction set) cells [82].

33 6. Quantum Optimal Control

In the laboratory, physicists use laser pulses to manipulate the energetic state of molecules and thereby initiate chemical reactions. They design the laser fields that yield the desired outcome. This chapter is devoted to the question of how to use simulation and numerical optimization to find a suitably shaped laser field for a given purpose. Agenericobjectivewouldbetofindapulseε that minimizes the function 1 t f J1(ε,ψ)= ψ(R,t) O ψ(R,t) dt, (6.1) t t t ) | | * f − 0 & 0 where O = O δ(t t )+O (t) and ψ is the! wave packet solving the TDSE 1 · − f 2 (2.3)fortheinteractionwiththelaserfieldε.TheoperatorO1 defines some target state! at! final time t f and! O2(t) allows us to include a time-dependent objective, such as the penalization of a special molecular state! (that is unde- sired). ! In this form, the problem is ill-posed since there is no restriction on the strength of the pulse. A strategy to resolve this issue is to introduce a so-called Tikhonov regularization, i.e., to add the term

t f 2 J2(ε)=λ ε (t)dt, λ > 0, &t0 to the objective function. The constant λ indirectly controls the total energy of the optimal pulse and has to be chosen in a way so the resulting pulse has a strength that can be achieved with the experimental equipment. The complete optimization problem now reads

min J (ε,ψ)=J1(ε,ψ)+J2(ε), ε L2([t0,t f ]) ∈ (6.2) ∂ subject to ih¯ ψ = H + µε(t) ψ, ψ(R,t )=ψ . ∂t 0 0 0 ) * The optimality system for (6.2)isgivenby! ∂ ih¯ ψ = H + µˆ ε(t) ψ, ψ(R,0)=ψ , (state equation) ∂t 0 0 ∂ )i * 1 χ = !Hˆ0 + µˆ ε(t) χ O1(t)ψ(r,t), ∂t −h¯ − t f t0 (6.3) − $ % χ(x,t f )=O2ψ(r,t f ), ! (adjoint equation) 1 Im( χ µˆ ψ )+ηε = 0, (complementarity) h¯ ) | |! * 34 where χ is the adjoint variable. It was shown by Peirce et al. [107]that(6.2) has a solution in the special case where

J (ε,ψ)= ψ(t ) φ,ψ(t ) φ . 1 ) f − f − *

The essential property of J1 in the proof is weak lower semi-continuity, and hence the reasoning in [107]canalsobeadoptedtomoregeneralobjectives. The first studies of quantum optimal control problems included only final time objectives. In the late 1980s, the pulse shaping problem was formulated as an optimal control problem and solved with the conjugate gradient (CG) method [78, 107, 128]. The CG algorithm is a standard first order method which may suffer from slow convergence [46]. The efficiency of the numerical optimization was improved in a paper by Somlói and co-workers [131]who adopted the Krotov method [83]forquantumcontrol.TheKrotoviteration was refined by Zhu and co-workers [152, 153]andbyMadayandTurinici [93]. The idea of this method is to solve the first-order optimality condition (6.3)forε,yielding 1 ε(t)= Im( χ µˆ ψ ). −h¯η ) | | * Since χ and ψ depend on ε,afixedpointiterationisappliedtofindtheoptimal field. Within the quantum control literature, the method is referred to as the monotonic algorithm. Ohtsuki et al. [105]showedhowtoincludetime-dependenttargetsofthe general form (6.1)totheoptimalcontrolformulation.Thistime-dependent objective problem was also tackled with the monotonic algorithm (cf. also the tutorial [146]). All of these methods are based on the first order optimality condition. Usu- ally, convergence can be improved by using a quasi-Newton method that includes information of an approximate Hessian. Among those methods, the BFGS algorithm [23, 42, 47, 127]isthemostsuccessfulone.Adisadvantage of a quasi-Newton method for the quantum optimal control problem is that the number of control parameters is the number of time steps, which in general is very large. Therefore, memory shortage will make it impossible to store the full approximate Hessian. Hence, only low-memory variants are possible. Judson and Rabitz [71]proposedanalternativewaytodesignlaserpulses: feedback control. This offers a possibility to design laser pulses also for systems where the Hamiltonian is unknown or for molecules that are too large to be simulated in a computer. The computer only successively generates pulses (usually based on some global optimization strategy) and then the fitness of the pulse is determined by an experiment. In the experiment, the incoming laser pulse is grated to spatially disperse light of different frequencies which then can be delayed independently from each other through a spatial filter [142, 148]. This technique is also-called Fourier transform femtosecond pulse shaping.Assionetal.[2]pointoutthattheoptimizationprocedureshould

35 consider the spectral phase as the experimental shaper does. In Ref. [129], a special global optimization strategy is applied to find optimal Fourier coefficients with computer simulation. Using the Fourier coefficients as control variables like in feedback control algorithms, one can drastically reduce the dimensionality of the optimization problem as demonstrated in Paper VIII.Inthisway,itbecomesfeasibleto use quasi-Newton methods even for long-time quantum optimal control problems. Moreover, one can make sure that the theoretically found pulses can be realized in practice. Note that a quasi-Newton method was also applied to quantum optimal control of Bose–Einstein condensate in the context of von Winckel and Borzì’s study on a suitable norm for minimization [136].

36 7. Summary and Outlook

This thesis focuses on the numerical solution of the time-dependent Schrödin- ger equation modeling matter-field interaction on the femtosecond time scale. This phenomenon is fundamentally quantum mechanical in nature. Therefore, semiclassical methods have difficulties to fully describe the action and there is aneedtodevelopefficientdirectmethodsforthesolutionoftheTDSE.The underlying partial differential equation is characterized by a highly-oscillatory temporal dependence of the Hamiltonian operator. In Paper I,aMagnus– Lanczos propagator of order four is adapted for this special problem. In a comparison of various propagation methods its efficiency has been underlined. This propagator is put into an adaptive setting in Paper II.Theadaptivityis enhanced in Paper III where mesh refinement in a spectral element setting is included. Paper VI describes an efficient implementation framework for spectral elements that uses the data types for matrix-free implementation introduced in [82]anddescribestheparallelizationstrategieswithfocusonanovel coloring-type algorithm for shared-memory architectures. Another aspect of parallelization is considered in Paper V which analyzes communication in the Lanczos algorithm. Paper IV considers error estimation for a dissociative system and combines the results for the time evolution in Paper II with errors from spatial discretization and domain truncation. With the proposed implementation and adaptivity, three-particle problems in three dimensions can be solved on a multi-core processor. An implementation of the framework that can handle high-dimensional grids would enable the study of larger molecules on computer clusters. Supposedly the algorithm developed for the linear Schrödinger equation could be applied to simulate quantum fluids and quantum turbulence. However, this requires the models and the theory to be augmented to include nonlinearities. Paper VII proposes a radial basis function approximation of the TDSE in aGalerkinsetting.Weshowconvergenceandstabilityforthismethodand report experiments that demonstrate the high accuracy of the method. There are two features of this ansatz that are promising: the high accuracy combined with more locality in the method compared to pseudospectral methods as well as the resemblance to coherent states in semiclassical methods. On the other hand, the present implementation of the method is both memory and CPU-time intense. Therefore, the inherent locality of the method needs to be exploited in order to facilitate parallelization and to save memory. Another direction of further research is to devise a strategy to scatter the nodes of the radial basis functions. It would be interesting to analyze how the

37 error estimation theory proposed for more traditional discretization methods in this thesis should be modified to fit in the RBF setting. One challenge in this research is how to model boundary effects: At the boundary, nodes need to be clustered to countervail ill-conditioning of the interpolation [109], an effect that cannot directly be measured by the size of the solution or the residual in this region. The resemblance of the radial basis functions to wave packets used in semiclassical methods suggests a combination of the two for simulations where a small number of independent variables models truly quantum dynamical processes while a larger number can be treated almost classically. Finally, a new algorithm to solve the quantum optimal control problem is devised in Paper VIII.ThepulseisoptimizedinFourierspaceandaquasi- Newton method is applied. The optimal pulse can be found in fewer iterations compared to the well-established monotonic algorithm and the control variables in Fourier space give the possibility to restrict the pulse to a certain band width. The latter fact could be explored further in combination with physical requirements. For instance, the resolution of the discrete Fourier components could be coupled to the resolution of the pulse shaper. In the present implementation of the algorithm, the envelope of the field is also modeled with the Fourier components. This might be reasonable to change by apply- ing more local transforms [116]. Another direction of research is to consider more complicated objectives like revival patterns [139]orproblemsincluding dissociative processes.

38 8. Effektiva och tillförlitliga simuleringar av kvantmolekyldynamik

Kvantmekaniken beskriver hur materiens minsta byggstenar växelverkar. Se- dan laser, med frekvenser i samma storleksordning som vibrationerna i atom- kärnorna, uppfanns har en hel disciplin utvecklats som försöker förstå – och även manipulera – molekylernas struktur med hjälp av laserns elektromagnetiska fält. Kvanteffekterna uppenbarar sig inte i de flesta makroskopiska struk- turer, men blir avgörande när man vill designa så kallade nanokomponenter som består av enbart ett fåtal molekyler. I experimentell kvantmekanik mäter man effekter av partikelrörelser vilka ger indikation på hur partiklarnas våg- funktion ser ut. Därför bidrar det till förståelsen av mätningarna att lösa Schrö- dingerekvationen som är den matematiska beskrivningen av vågfunktionerna. Oftast är man tvungen att lösa ekvationerna med hjälp av datorer då analytis- ka lösningar inte är kända. När det gäller lösning av Schrödingerekvationen med hjälp av datorer är huvudutmaningen dimensionalitetens förbannelse: eftersom antalet rumsdimensioner i ekvationen är proportionerligt mot antalet partiklar växer antalet noder i en nätbaserad beräkningsmetod exponentiellt med antalet partiklar. Denna doktorsavhandling beaktar frågan hur man kan utnyttja moderna nu- meriska metoder och datorkraft för att simulera hur molekyler beter sig i väx- elverkan med elektromagnetiska fält på ett effektivt och tillförlitligt sätt. Hu- vudfokus ligger på utveckling av en lösare som beräknar resultatet med efter- strävad noggrannhet. Lösaren utnyttjar adaptivitet i rum såväl som i tid och är parallelliserad och implementerad på ett minneseffektivt sätt. Spektrala metoder ger mycket effektiv rumsdiskretisering i enkla lösare för Schrödingerekvationen, men de är mycket svåra att parallellisera och anpas- sa till lösningens utseende. Denna avhandling föreslår istället användandet av spektralelement, finita element av hög noggrannhetsordning. Fördelen med spektralelement är tvåfaldig: De kombinerar förhållandevis hög noggrannhet med relativt lokaliserade stenciler som går att parallellisera över datorkluster. Därutöver finns det genom metoden för dualviktade residualer en teori för hur de lokala diskretiseringsfelen påverkar slutsresultatets noggrannhet. Efter diskretiseringen i rummet uppstår ett system av ordinära differentialekvationer som löses med en numerisk tidsstegningsmetod. Idealiskt ska metoden bevara lösningens norm på samma sätt som den är bevarad i den kontinuerliga modellen. Dessutom vill man använda sig av en explicit metod som undviker lösning av linjära system vilket är mycket kostsamt och minneskrä- vande i högre dimensioner. Det finns två typer av tidsstegare som har dessa

39 egenskaper: exponentialintegratorer och partitionerade Runge-Kutta-metoder. Iavhandlingenjämförsolikautformningaravdessametoderförsimulering av växelverkan mellan ljus och materia. För denna tillämpning härleds en fjärde ordningens exponentialintegrator baserad på Magnusutvecklingen vilken undviker evaluering av matriskommutatorer. Denna är annars nödvändigt ihögreordningenstermeriMagnusutvecklingen.Matrisexponentialenberäk- nas slutligen approximativt med hjälp av Lanczosalgoritmen. Denna metod visar sig vara mycket effektivt och vidareutvecklas till att inkludera felkontroll. Eftersom Schrödingerekvationen är självadjungerande blir det lokala trunke- ringsfelet varken dämpat eller förstärkt vilket utnyttjas i den globala felupp- skattningen. Det visar sig dock att man kan applicera teorin även för dissocia- tiva system där modellen inte är självadjungerande. Kombinerar man en rumsdiskretisering baserad på spektralelement med en exponentialintegrator, är massmatrisen som vanligen uppstår i finita element- diskretiseringar besvärande eftersom den innebär att man är tvungen att lösa ett linjärt ekvationssystem trots att tidsstegningsmetoden är explicit. Därför används Lagrangepolynom som centreras i noderna hos Gauss-Lobatto-kvadraturformeln, och sedan approximeras integralerna i mass- och styvhetsmatriserna med just denna kvadraturformel. På så sätt blir massmatrisen diagonal och kan enkelt inverteras. Eftersom Gauss-Lobatto-kvadraturformeln fördelar noderna så att de ligger tätare närmare randen kringgår man också Rungefeno- menet för högre ordningens polynomapproximation. Därför är dessa element mycket lämpade som spektralelement. Ivanligafallimplementerasfinitaelementbaseratpåenglesmatrissomin- nehåller all information om differentialoperatorn projicerat på finita element- rummet. När man använder basfunktioner av högre ordning är dock matrisen inte speciellt gles och en sådan implementation kräver mycket minne. Som en del av denna avhandling presenteras ett ramverk inom biblioteket deal.II för att applicera finita element-operatorer cellbaserat. Idén är att för varje cell lagra hur den kan transformeras till en enhetscell. Funktionen transformeras sedan elementvis till enhetscellen där differentialoperatorn appliceras. Denna implementation gör det möjligt att utnyttja tensorproduktstrukturen som finns in Gauss-Lobatto-elementen för att minska beräkningskomplexiteten. Dess- utom reduceras minnensanvändningen – och därigenom minnesflödet under beräkningarna – för högre ordningar. Ramverket är parallelliserat på tre ni- våer: MPI kan användas på kluster med distribuerat minne, Intel Threading Building Blocks används för att hantera parallellism på ett system med delat minne, och operationer på två eller flera celler klumpas ihop för att utnyttja beräkningskärnornas vektorenheter. Denna avhandling fokuserar på parallel- lisering på system med delat minne. När man parallelliserar matris-vektor- multiplikation och vektorerna sparas på delat minne måste man undvika att olika processorer räknar på celler som delar frihetsgrader. Detta görs vanligen genom att dela upp domänen i ”olika färger” så att celler som tillhör samma färg inte delar frihetsgrader. Sedan bearbetar man en färg i taget. Problemet är

40 att man får en synkroniseringspunkt efter varje färg och att man kan få ganska många färger på adaptiva nåt. Därför föreslås en uppdelning på två nivåer som undviker explicita synkroniseringspunkter. Istället läggs beräkningarna ut dy- namiskt enligt denna uppdelning så att två beräkningsenheter aldrig samtidigt skriver i samma element. En annan aspekt av parallelliseringen som behandlas är kommunikationen mellan olika datorer inom ett kluster. Medan matris-vektor-multiplikation vanligen bara kräver utbyte av data mellan grannar kräver skalärprodukter att al- la beräkningsnoder kommunicerar med varandra. Detta försvårar parallelliseringen. I en vanlig implemention av Lanczosalgoritmen finns det två ställen ivarjeiterationdärenskalärproduktberäknas.Idennaavhandlingundersöks olika varianter av Lanczosalgoritmen som är designade för att minska kommunikationen till priset av lite fler beräkningar. Det visar sig att effekterna av minskad kommunikation och ökat beräkningsarbete ungefär tar ut varandra på de system som det har räknats på. Som ett sidospår undersöks också möjligheten att använda radiella basfunktioner (RBF) som diskretiseringsmetod för Schrödingerekvationen. Olika basfunktioner omnämns i litteraturen, men Gaussianer visar sig vara mest lämp- liga för Schrödingerekvationen. RBF är en modern metod som är mycket lo- vande av två anledningar: metoden ger spektral noggrannhet men till skillnad från vanliga spektralmetoder kan man sprida basfunktionernas centrum god- tyckligt. Dessutom liknar basfunktionerna de funktioner som används i vissa semiklassiska metoder. Det finns alltså goda förutsättningar för att kombinera en RBF-diskretisering med semiklassika metoder. Eftersom basfunktionerna är definierade på hela rummet och avtar mot oändligheten på samma sätt som lösningen av Schrödingerekvationen är man inte tvungen att applicera några artificiella randvillkor utan kan betrakta problemet på hela rummet på samma sätt som i det kontinuerliga problemet. Inom RBF-litteraturen är det vanligast med kollokation vid lösning av differentialekvationer, men denna avhandling föreslår istället en Galerkinansats för Schrödingerekvationen. Detta eftersom det uppstår stabilitetsproblem med en kollokationsansats samtidigt som integralerna i mass- och styvhetsmatriserna kan beräknas analytiskt på ett relativt enkelt sätt. Både stabilitet och konvergens visas. Metoden konvergerar exponentiellt upp till en viss nivå, vilken beror på storleken av domänen som basfunktionerna är fördelade på. Metoden visar sig vara mycket noggrann, men det finns behov att öka effektiviteten när det gäller lösningen av det linjära systemet som uppstår vid diskretiseringen. Slutligen behandlar avhandlingen även problemet att optimera ett elektro- magnetiskt fält som manipulerar en molekyl, t. ex. med målet att initiera en kemiskt reaktion. Trots att laserpulserna tas fram i experiment genom att för- ändra parametrarna i frekvensrummet, är det vanligt att hitta det optimala fäl- tet inom tidsrummet. I denna avhandling föreslås istället att betrakta pulsen i frekvensrummet även i optimeringsproblemet. På detta sätt blir det lättare att säkerställa att pulsen som hittas teoretiskt också kan framställas i praktiken

41 samtidigt som man oftast kan koncentrera sig på färre parametrar. Därigenom blir det också möjligt att använda en kvasi-Newtonmetod (som kräver att det lagras en approximativ Hessian) istället för den monotona algoritmen, en linjär algoritm som vanligen används i detta sammanhang. Effekten är en avsevärd uppsnabbning.

42 Acknowledgments

First of all, I want to express my gratitude to my advisor Sverker Holmgren for introducing me to this exciting subject and countless stimulating discussions. I have appreciated the freedom and responsibility he gave me to develop my own ideas and interests. Invaluable for this thesis was the help that Hans O. Karlsson offered me to understand the chemistry behind the computations and to acquire knowledge on new challenges in chemistry. Many thanks to Elisabeth Larsson for sharing her broad knowledge on radial basis functions and for our great collaboration. IhaveverymuchenjoyeddoingresearchtogetherwithMagnusGustafsson, Martin Kronbichler, and Anna Nissen. I want to thank Martin Berggren and Eddie Wadbro for sharing their expertise on numerical optimization and Axel Målqvist for sharing his on ﬁnite elements. Also, I gratefully acknowledge discussions on various topics of programming with Wolfgang Bangerth, Mag- nus Gustafsson, Martin Kronbichler, Emanuel Rubensson, and Elias Rudberg. IamthankfultoAriehIserlesaswellasGabrielTurinici,ClaudeLeBries,and Julien Salomon for inviting me to discuss my research. Many thanks to Vasile Gradinaru for discussing my research as an opponent at the occasion of my licentiate defense. Finally, I appreciated Emil Kieri’s and Martin Kronbich- ler’s detailed comments to this comprehensive summary and other parts of my work. Over the years the colloquia on numerical quantum dynamics, radial basis functions, high dimensional problems, and high performance computing at the Division of Scientiﬁc Computing have been excellent platforms for exchang- ing ideas. This work was supported by the Graduate School in Mathematics and Com- puting (FMB).

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