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Preconditioned Techniques for Large Eigenvalue Problems

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Preconditioned Techniques for Large Eigenvalue Problems Preconditioned Techniques For Large Eigenvalue Problems Kesheng Wu June Abstract This research fo cuses on nding a large number of eigenvalues and eigenvectors of a sparse symmetric or Hermitian matrix for example nding eigenpairs of a matrix These eigenvalue problems are challenging b ecause the matrix size is to o large for traditional QR based algorithms and the number of desired eigenpairs is to o large for most common sparse eigenvalue algorithms In this thesis we approach this problem in two steps First we identify a sound preconditioned eigenvalue pro cedure for computing multiple eigenpairs Second we improve the basic algorithm through new preconditioning schemes and sp ectrum transformations Through careful analysis we see that b oth the Arnoldi and Davidson metho ds have an appropriate structure for computing a large number of eigenpairs with preconditioning We also study three variations of these two basic algorithms Without preconditioning these metho ds are mathematically equivalent but they dier in numerical stability and complexity However the Davidson metho d is much more successful when preconditioned Despite its success the preconditioning scheme in the Davidson metho d is seen as awed b ecause the preconditioner b ecomes illconditioned near convergence After comparison with other metho ds we nd that the eectiveness of the Davidson metho d is due to its preconditioning step b eing an inexact Newton metho d We pro ceed to explore other Newton metho ds for eigenvalue problems to develop preconditioning schemes without the same aws We found that the simplest and most eective preconditioner is to use the Conjugate Gradient metho d to approximately solve equations generated by the Newton metho ds Also a dierent strategy of enhancing the p erformance of the Davidson metho d is to alternate b etween the regular Davidson iteration and a p olynomial metho d for eigenvalue problems To use these p olynomials the user must decide which intervals of the sp ectrum the p olynomial should suppress We studied dierent schemes of selecting these intervals and found that these hybrid metho ds with p olynomials can b e eective as well Overall the Davidson metho d with the CG preconditioner was the most successful metho d the eigenvalue problems we tested Chapter Electronic Structure Simulation Before entering into the main topic of this thesis we use this chapter to introduces the context of our research which is also serve as motivation for our research In this chapter we fo cus on one application that is the main driving force b ehind this research in eigensystem solvers the electronic structure simulation pro ject at the University of Minnesota We will give some background information ab out electronic structure simulation and characteristics of the eigenvalue problems generated from the simulation The eigenvalue algorithm developed in later chapters are designed to solve the eigenvalue problem of the same characteristics as these matrices Introduction Many scientic and engineering applications give rise to eigenvalue problems that is they require the solution of the following equation Ax x nn n where A C is n n matrix C is an eigenvalue and x C is an eigenvector The particular eigenvalue problem we are interested in is generated from a materials science research pro ject The aim of the research pro ject is to understand the dynamics of microscopic particles sp ecically electronic structures of complex systems Using quantum physics it is p ossible to explain and predict certain material prop erties at a microscopic scale A key step to the simulation of electronic structure is to nd a steady state or a quasisteady state Intuitively the pro cess of nding this steady state is adjusting the electron distribution to minimize the total energy of the system The total energy is a nonlinear function of the electron distribution Finding the minimal energy and the corresp onding electron distribution can b e viewed as solving for the smallest eigenvalue and the corresp onding eigenvector of a nonlinear eigenvalue problem The SelfConsistent Field SCF iteration is the primary scheme of solving this nonlinear eigenvalue problem At each step of the SCF iteration a linear eigenvalue problem is generated This is source of our matrix eigenvalue problem The Schrodinger equation is an nonlinear eigenvalue problem At each step of the SCF iteration a matrix eigenvalue problem is generated to contrast with the overall nonlinear eigenvalue problem we will refer to this matrix eigenvalue problem as the linear eigenvalue problem in this chapter However after this chapter we will b e concentrating on this linear eigenvalue problem the eigenvalue problem will refer the linear eigenvalue problem In quantum physics the electron distribution is represented by a wavefunction The governing equation is the Schrodinger equation H E where H is the Hamiltonian op erator for the system and E the total energy is the wavefunction The Hamiltonian op erator describ es the motion and interaction of the particles of the system The wavefunction describ es where the particles are The Schrodinger equation for any nontrivial system is complex nonlinear Partial Dierential Equation PDE There are many numerical metho ds for solving a PDE One of the most common numerical approaches to solve the Schrodinger equation is to discretize it in a planewave basis which is similar to sp ectral techniques for solving partial dierential equations This discretization scheme turns the Hamiltonian into a large dense matrix that is often to o large to store in a computers main memory Usually the matrix is only used in the linear eigenvalue problem one workaround to this diculty is to use a Lanczostype eigenvalue routine which only need to access the matrix through matrixvector multiplications Because the Fast Fourier Transformation FFT can b e used to accomplish the most computation intensive op eration in the matrixvector multiplication the matrixvector multiplication is relatively inexp ensive with this alternative With the planewave basis the problem is solved in the Fourier Space A dierent approach is to solve the problem in real space by discretizing the Schrodinger equation with the nite dierence scheme For lo calized systems such as a cluster of atoms highorder nite dierence scheme has shown to b e more ecient than planewave techniques in nding solutions of same accuracy The matrices resulting from b oth nite dierence metho ds and planewave techniques are large if the number of particles in the system is large In addition the size of the matrix is also aected by the desired accuracy of the solution characteristics of the atoms involved and the physical quantities to b e computed For complex systems involving hundreds of atoms the matrix size could b e on the order of millions Complex systems may also require one to solve many more linear eigenvalue problems b efore an acceptable solution for the nonlinear system is reached If we solve the Schrodinger equation directly only the smallest eigenvalue is needed However if we try to do so the p otential function V would b e to o complex to compute In next section we will show a scheme of simplifying this V This scheme makes V tractable at the same time it requires computation of a large number of eigenvalues from the linear eigenvalue problem The number of eigenvalues required is prop ortional to the number of atoms in the system Most of the eigenvalue metho ds traditionally used for this computation are not very eective for nding a large number of eigenvalues This is an additional challenge for an eective simulation Ab Initio pseudop otential simulation The electronic structure of a condensed matter system eg cluster liquid or solid is describ ed by a quantum wavefunction which can b e obtained by solving the Schrodinger equation This equation is very complex b ecause the Hamiltonian op erator H describ es motions of all particles in the system and the interactions among all of them Complete analytical solution is only p ossible for the simplest atoms Signicant sim plication is required to compute any large system Most theories of condensed matter systems make the following three fundamental approximations to make them manageable BornOpp enheimer approximation BornOpp enheimer approximation neglects the kinetic energy of the nuclei This is a go o d approximation b ecause of two main reasons First the mass of nuclei is much larger than the mass of electrons in the system typically more times larger nuclei move very slowly compared to electrons Second we are not interested in the average motion of the whole system in other word we will solve the system in the nuclei frame of reference Because of this approximation the wave function we use only involves the electrons Thus the original electronnuclear problem now b ecomes a pure electron problem Under this circumstance the wavefunction describ es the distribution of electrons only Let r denote a p oint in space the density of electron distribution at r is dened by H r r r where the sup erscript H denotes complex conjugate Lo cal density approximation To explain the concept of Lo cal Density Approximation LDA we will rst describ e a more general theory the density functional theory The density functional theory transforms a manyelectron problem into an oneelectron problem The simplied Schrodinger equation in this case is called the KohnSham equation r V r r r tot i i i where r is the usual Laplacian op erator V r denotes
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