Adaptive Multiresolution 3D Hartree-Fock-Bogoliubov Solver for Nuclear Structure
G. Fann1, W. Nazarewicz2, J. Pei3, Y. Shi2, W. S. Thornton4 and R. J. Harrison4,5
1) Oak Ridge National Laboratory, Oak Ridge, TN 37831 2) Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI 48824, USA 3) State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China 4) Stony Brook University, Stony Brook, NY 11794, USA 5) Brookhaven National Laboratory, Upton, NY 11973, USA
A solver for the Hartree-Fock-Bogoliubov equations is a key element of any density functional theory framework to enable a quantitative understanding of complex nuclear structures and its extensions to complex nuclei. This work highlights NUCLEI's close collaborations between domain scientists, applied mathematicians, and computer scientists to develop a new adaptive multiscale solver to enable scalable and high accuracy simulations of complex many-body open systems on leadership-class systems.
Physics Problems Computational Problems MADNESS Multiple scales and complex topologies arise in Solutions for the needed 3D solutions in DFT, within a large and MADNESS is a general purpose numerical environment for complex many-body open quantum systems such as asymmetric domain without symmetry, require orders of deploying advanced scientific algorithms on petascale and magnitude more computational resource than 2D simulations. beyond – emphasizing productivity and performance. ● triaxial and reflection-asymmetric nuclei, MADNESS-HFB introduced an adaptive and variational ● weakly bound halo states, pseudo-spectral framework for solving self-consistent equations A fast prototyping, high level coding – a nearly symbolic ● cluster configurations, of nuclear DFT, without symmetry restrictions, on massively programming object-oriented C++ environment: ● nuclear fragments from heavy-ion fusion reactions, parallel computers. The new framework solves multi-scale − ψ(r ) ψ(r ) ● cold Fermi gases, and physics challenges in nuclear and atomic physics problems 1 1 2 E=∫( Δ+V ) ψ(r)ψ(r)dr +∫ dr 1 dr 2 ● pasta phases in a neutron star crust. involving many-particles systems, by combining the 2 |r1−r2|
eigensolution of an HFB equation and a corresponding set of real_function_3d Vnuc = real_factory_3d(world).f(V); Systems, with symmetry breaking and complex and Lippman-Schwinger equations (with k 2 = − 2 E ), with improved real_function_3d−1 psi = real_factory_3d(world).f(guess); ψ(r1) ψ(r2) changing topologies, can be described within asymptotic behavior. real_convolution_3d op = CoulombOperator(world, rlo, thresh); | | real_function_3d rho = square(psi).truncate(); nuclear density functional theory (DFT) framework. e−k r− s E= ( double two_electron_energyΔ =+ inner(op(rho),rho);V ) ψ(r)ψ(r)dr + dr dr −1 ∫ double nuclear_attraction_energy = 2.0*inner(Vnuc,rho); ∫ 1 2 ( Δ+V ) ψ= E ψ → (Δ +2 E)ψ=V ψ → ψ=−2 V ψ dr See Sagert's poster for an example of MADNESS in a ∫ | | 2 4 π r−s double kinetic_energy2 = 0.0; |r −r | periodic crust simulation. for (int axis=0; axis<3; axis++) { 1 2 real_derivative_3d D = free_space_derivative
Self-consistent solutions of Hartree-Fock- double total_energy = kinetic_energy + two_electron_energy + nuclear_attraction_energy + nuclear_repulsion_energy; Bogoliubov (HFB or Bogoliubov-de Gennes) The colors represent the nodes holding non- trivial blocks of coefficients of a function equations are a key element of any DFT 00 approximated in Alpert's multiwavelet basis. framework. The HFB equations define “quasi- AA parallel runtime library is responsible for managing task particles” which approximate ground-states of scheduling and placement, as well as dependencies (futures) many-body systems beyond HF + BCS theory. For with user enabled load and memory balancing. All ↑ ↓ a two-component system ( and ) with communication are asynchronously. All intra-node corresponding HF Hamiltonian h , chemical communication by performed by parallel threads. Interface λ δ potentials and a pairing potential , the HFB to Intel TBB available. equation is h −λ δ u u Fast “mathematical O(N)” computation with guaranteed ↑ ↑ i =E i [ δ¯t −h +λ ] v i v precision through the use of MRA in conjunction with compact ↓ ↓ [ i ] [ i ] Examples of adaptive representations of wave-functions, for a nonlinear multiscale approximations for high-order (up to 32+) and accurate solutions of coupled 1-6D systems of time- Benchmarking given precision, are illustrated. Each wave-function and dependent integral and differential equations with functional operator has its own adaptive spectral representation. For Comparisons were made between the MADNESS- and a variety boundary conditions – solving systems with example, the modulus squared of the single-neutron HFB solver against 2D coordinate-space solver several trillions of variables. corresponding to -5.214 MeV calculations for 110Mo (a) with its using B-splines (HFB-AX) and a 3D solver based on corresponding spectral refinement structure (b). Examples of the harmonic-oscillator basis expansion (HFODD). • Controlled truncation for accuracy and sparsity, hierarchy of other wave-functions are pictured above. Single-particle density z equations by scale (a la renormalization)
) z n • Compact non-linear approximation of high-dimensional functions 110 110 ( distributions for Mo 0 Mo 1 An important example is the simulation of a Fermi gas and the g and integral operators (Green's functions) o illustrates (see semilog l min(M ) d x n ' superfluid phase in spin-imbalanced systems in an elongated k1 n 1 2 1 k 1 f (x …,x )= σ f (l )(x )+O(ϵ) inset) the superior x n0 0 n n 1, n ∑ l ∏ i i ) f()()() x si0 i 0 x d il il x l=1 i=1 3 - trap, such as the Fulde-Ferrell-Larkin-Ovchinnikov phase that i0 n ' 0 l 0 i 0 (l) asymptotic properties of m = σ > f ‖f i ‖2 1 l 0 (
y n y exhibits oscillating pairing gap using an asymmetric local the MADNESS-HFB Runs from Linux and Apple laptops to clusters and leadership solutions to DFT over density approximation. The transversal pairing field are indicative of the Larkin-Ovchinnikov phase. The simulation class computers. Accelerators versions (GPU and Intel Phi) conventional techniques x, y, z (fm) under development. using 1140 and 1540 HFODD(1140) used a 320 trap units. HFODD(1540) References: harmonic-oscillator basis MADNESS ● R.J. Harrison, G. I. Fann, T. Yanai, Z. Gan, G. Beylkin, “Multiresolution quantum chemistry: Basic theory and initial applications,” J. Chem. Phys. 121 11587 (2004) functions. x, y, z (fm) ● G. I. Fann, J. Pei, R.J. Harrison, J. Jia, J. Hill, M. Ou, W. Nazarewicz, W. A. Shelton, N. Schnuck, “Fast multiresolution methods for density functional theory in nuclear physics,” JPG, Phys. Conf. Ser. 180 012080 (2004) Funding: This material is based on work supported by the U.S. Department of Energy, Office of ● J.C. Pei, G.I. Fann, R.J. Harrison, W. Nazarewicz, Y. Shi, S. Thornton, “Adaptive multiresolution 3D Hartree-Fock-Bogoliubov Science, SciDAC NUCLEI, Office of Science, Divisions of Advanced Scientific Computing solver for nuclear structure,” Phys. Rev. C 90 024317 (2014) Research, under contract DE-AC05-00OR22725 with Oak Ridge National Laboratory. This is a ● R. Harrison, et. al., “MADNESS: A multiresolution adaptive numerical environment for scientific simulation,” arXiv:1507.01888v1, part of the SciDAC NUCLEI Project (PI: J. Carlson, LANL). July 5, 1025.