Introduction Basic Description Further Details Advanced Features Concluding Remarks SLEPc: Scalable Library for Eigenvalue Problem Computations Jose E. Roman High Performance Networking and Computing Group (GRyCAP) Universidad Polit´ ecnica de Valencia, Spain August, 2005
Introduction Basic Description Further Details Advanced Features Concluding Remarks Tutorial Outline 1 Introduction 3 Further Details Motivating Examples EPS Options Background on Spectral Eigenproblems Transformation 2 Basic Description 4 Advanced Features Overview of SLEPc 5 Concluding Remarks Basic Usage
Introduction Basic Description Further Details Advanced Features Concluding Remarks Eigenproblems: Motivation Large sparse eigenvalue problems are among the most demanding calculations in scientific computing Example application areas: ◮ Dynamic structural analysis (e.g. civil engineering) ◮ Stability analysis (e.g. control engineering) ◮ Eigenfunction determination (e.g. quantum chemistry) ◮ Bifurcation analysis (e.g. fluid dynamics) ◮ Statistics / information retrieval (e.g. Google’s PageRank)
Introduction Basic Description Further Details Advanced Features Concluding Remarks Motivating Example 1: Nuclear Engineering 10 0.81 Modal analysis of nuclear reactor cores 0.72 0.63 Objectives: 0.54 0.45 ◮ Improve safety 0.36 0.27 0.18 ◮ Reduce operation costs 0.09 0 Lambda Modes Equation 0 0 10 10 L φ = 1 λ M φ 0.6 0.5 0.4 0.3 Target: modes associated to largest λ 0.2 0.1 −1.39e−16 ◮ Criticality (eigenvalues) −0.1 −0.2 −0.3 ◮ Prediction of instabilities and −0.4 −0.5 −0.6 transient analysis (eigenvectors) 0 0 10
Introduction Basic Description Further Details Advanced Features Concluding Remarks Motivating Example 1: Nuclear Engineering (cont’d) Discretized eigenproblem � � ψ 1 � M 11 � � ψ 1 � � � = 1 L 11 0 M 12 − L 21 L 22 ψ 2 0 0 ψ 2 λ Can be restated as N = M 11 + M 12 L − 1 Nψ 1 = λL 11 ψ 1 , 22 L 21 ◮ Generalized eigenvalue problem ◮ Matrix should not be computed explicitly ◮ In some applications, many successive problems are solved
Introduction Basic Description Further Details Advanced Features Concluding Remarks Motivating Example 2: Computational Electromagnetics Objective: Analysis of resonant cavities r ∇ × � ε r � Source-free wave µ − 1 E ) − κ 2 ∇ × (ˆ 0 ˆ E =0 equations r ∇ × � µ r � ε − 1 H ) − κ 2 ∇ × (ˆ 0 ˆ H =0 Target: A few smallest nonzero eigenfrequencies Discretization: 1 st order edge finite elements (tetrahedral) Ax = κ 2 Generalized Eigenvalue Problem 0 Bx ◮ A and B are large and sparse, possibly complex ◮ A is (complex) symmetric and semi-positive definite ◮ B is (complex) symmetric and positive definite
Introduction Basic Description Further Details Advanced Features Concluding Remarks Motivating Example 2: Comp. Electromagnetics (cont’d) Matrix A has a high-dimensional null space, N ( A ) ◮ The problem Ax = κ 2 0 Bx has many zero eigenvalues ◮ These eigenvalues should be avoided during computation λ 1 , λ 2 , . . . , λ k , λ k +1 , λ k +2 , . . . , λ n � �� � � �� � =0 Target Eigenfunctions associated to 0 are irrotational electric fields, � E = −∇ Φ . This allows the computation of a basis of N ( A ) Constrained Eigenvalue Problem where the columns � Ax = κ 2 0 Bx of C span N ( A ) C T Bx = 0
Introduction Basic Description Further Details Advanced Features Concluding Remarks Facts Observed from the Examples ◮ Many formulations ◮ Not all eigenproblems are formulated as simply Ax = λx or Ax = λBx ◮ We have to account for: spectral transformations, block-structured problems, constrained problems, etc. ◮ Wanted solutions ◮ Many ways of specifying which solutions must be sought ◮ We have to account for: different extreme eigenvalues as well as interior ones ◮ Various problem characteristics ◮ Problems can be real/complex, Hermitian/non-Hermitian Goal: provide a uniform, coherent way of addressing these problems
Introduction Basic Description Further Details Advanced Features Concluding Remarks Background on Eigenvalue Problems Consider the following eigenvalue problems Standard Eigenproblem Generalized Eigenproblem Ax = λx Ax = λBx where ◮ λ is a (complex) scalar: eigenvalue ◮ x is a (complex) vector: eigenvector ◮ Matrices A and B can be real or complex ◮ Matrices A and B can be symmetric (Hermitian) or not ◮ Typically, B is symmetric positive (semi-) definite
Introduction Basic Description Further Details Advanced Features Concluding Remarks Solution of the Eigenvalue Problem There are n eigenvalues (counted with their multiplicities) nev = number of Partial eigensolution: nev solutions eigenvalues / λ 0 , λ 1 , . . . , λ nev − 1 ∈ C eigenvectors x 0 , x 1 , . . . , x nev − 1 ∈ C n (eigenpairs) Different requirements: ◮ Compute a few of the dominant eigenvalues (largest magnitude) ◮ Compute a few λ i ’s with smallest or largest real parts ◮ Compute all λ i ’s in a certain region of the complex plane
Introduction Basic Description Further Details Advanced Features Concluding Remarks Single-Vector Methods The following algorithm converges to the dominant eigenpair ( λ 1 , x 1 ) , where | λ 1 | > | λ 2 | ≥ · · · ≥ | λ n | Notes: Power Method ◮ Only needs two vectors Set y = v 0 For k = 1 , 2 , . . . ◮ Deflation schemes to find v = y/ � y � 2 subsequent eigenpairs y = Av ◮ Slow convergence θ = v ∗ y (proportional to | λ 1 /λ 2 | ) Check convergence ◮ Fails if | λ 1 | = | λ 2 | end
Introduction Basic Description Further Details Advanced Features Concluding Remarks Variants of the Power Method Shifted Power Method ◮ Example: Markov chain problem has two dominant eigenvalues λ 1 = 1 , λ 2 = − 1 = ⇒ Power Method fails! ◮ Solution: Apply the Power Method to matrix A + σI Inverse Iteration ◮ Observation: The eigenvectors of A and A − 1 are identical ◮ The Power Method on ( A − σI ) − 1 will compute the eigenvalues closest to σ Rayleigh Quotient Iteration (RQI) ◮ Similar to Inverse Iteration but updating σ in each iteration
Introduction Basic Description Further Details Advanced Features Concluding Remarks Spectral Transformation A general technique that can be used in many methods = ⇒ Ax = λx Tx = θx In the transformed problem ◮ The eigenvectors are not altered ◮ The eigenvalues are modified by a simple relation ◮ Convergence is usually improved (better separation) Shift-and-invert Shift of Origin Example transformations: T SI = ( A − σI ) − 1 T S = A + σI Drawback: T not computed explicitly, linear solves instead
Introduction Basic Description Further Details Advanced Features Concluding Remarks Invariant Subspace A subspace S is called an invariant subspace of A if A S ⊂ S ◮ If A ∈ C n × n , V ∈ C n × k , and H ∈ C k × k satisfy A V = V H then S ≡ C ( V ) is an invariant subspace of A Objective: build an invariant subspace to extract the eigensolutions Partial Schur Decomposition AQ = QR ◮ Q has nev columns which are orthonormal ◮ R is a nev × nev upper (quasi-) triangular matrix
Introduction Basic Description Further Details Advanced Features Concluding Remarks Projection Methods The general scheme of a projection method: 1. Build an orthonormal basis of a certain subspace 2. Project the original problem onto this subspace 3. Use the solution of the projected problem to compute an approximate invariant subspace ◮ Different methods use different subspaces ◮ Subspace Iteration: A k X ◮ Arnoldi, Lanczos: K m ( A, v 1 ) = span { v 1 , Av 1 , . . . , A m − 1 v 1 } ◮ Dimension of the subspace: ncv (number of column vectors) ◮ Restart & deflation necessary until nev solutions converged
Introduction Basic Description Further Details Advanced Features Concluding Remarks Summary Observations to be added to the previous ones ◮ The solver computes only nev eigenpairs ◮ Internally, it works with ncv vectors ◮ Single-vector methods are very limited ◮ Projection methods are preferred ◮ Internally, solvers can be quite complex (deflation, restart, ...) ◮ Spectral transformations can be used irrespective of the solver ◮ Repeated linear solves may be required Goal: hide eigensolver complexity and separate spectral transform
Introduction Basic Description Further Details Advanced Features Concluding Remarks Executive Summary SLEPc : Scalable Library for Eigenvalue Problem Computations A general library for solving large-scale sparse eigenproblems on parallel computers ◮ For standard and generalized eigenproblems ◮ For real and complex arithmetic ◮ For Hermitian or non-Hermitian problems Current version: 2.3.0 (released July 2005) http://www.grycap.upv.es/slepc
Introduction Basic Description Further Details Advanced Features Concluding Remarks SLEPc and PETSc SLEPc extends PETSc for solving eigenvalue problems PETSc: Portable, Extensible Toolkit for Scientific Computation ◮ Software for the solution of PDE’s in parallel computers ◮ A freely available and supported research code ◮ Usable from C, C++, Fortran77/90 ◮ Focus on abstraction, portability, interoperability, ... ◮ Object-oriented design (encapsulation, inheritance and polymorphism) ◮ Current: 2.3.0 http://www.mcs.anl.gov/petsc SLEPc inherits all good properties of PETSc
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