Harmonic Rayleigh-Ritz for the multiparameter eigenvalue problem Bor - PowerPoint PPT Presentation

Harmonic Rayleigh-Ritz for the multiparameter eigenvalue problem Bor Plestenjak Department of Mathematics University of Ljubljana This is joint work with Michiel Hochstenbach Harrachov, 2007 1/21

Outline • Multiparameter eigenvalue problem (MEP) • Jacobi–Davidson type methods for MEP • Harmonic Rayleigh–Ritz for GEP and MEP • Numerical examples • Conclusions Harrachov, 2007 2/21

Two-parameter eigenvalue problem • Two-parameter eigenvalue problem: A 1 x = λB 1 x + µC 1 x ( MEP ) A 2 y = λB 2 y + µC 2 y, where A i , B i , C i are n × n matrices, λ, µ ∈ C , x, y ∈ C n • Eigenvalue: a pair ( λ, µ ) that satisfies (MEP) for nonzero x and y . • Eigenvector: the tensor product x ⊗ y . • Goal: compute eigenvalues ( λ, µ ) close to a target ( σ, τ ) and eigenvectors x ⊗ y . Harrachov, 2007 3/21

Tensor product approach A 1 x = λB 1 x + µC 1 x ( MEP ) A 2 y = λB 2 y + µC 2 y • On C n ⊗ C n of the dimension n 2 we define ∆ 0 = B 1 ⊗ C 2 − C 1 ⊗ B 2 ∆ 1 = A 1 ⊗ C 2 − C 1 ⊗ A 2 ∆ 2 = B 1 ⊗ A 2 − A 1 ⊗ B 2 . • MEP is equivalent to a coupled GEP ∆ 1 z = λ ∆ 0 z ( ∆ ) ∆ 2 z = µ ∆ 0 z, where z = x ⊗ y . • MEP is nonsingular ⇐ ⇒ ∆ 0 is nonsingular. • ∆ − 1 0 ∆ 1 and ∆ − 1 0 ∆ 2 commute. Harrachov, 2007 4/21

Right definite problem ∆ 0 = B 1 ⊗ C 2 − C 1 ⊗ B 2 A 1 x = λB 1 x + µC 1 x ∆ 1 z = λ ∆ 0 z ( MEP ) ∆ 1 = A 1 ⊗ C 2 − C 1 ⊗ A 2 ( ∆ ) A 2 y = λB 2 y + µC 2 y ∆ 2 z = µ ∆ 0 z ∆ 2 = B 1 ⊗ A 2 − A 1 ⊗ B 2 MEP is right definite when A i , B i , C i are Hermitian and ∆ 0 is positive definite. Atkinson (1972): ☞ ☞ ☞ x ∗ B 1 x x ∗ C 1 x ☞ ⇒ ( x ⊗ y ) ∗ ∆ 0 ( x ⊗ y ) = ☞ ☞ ∆ 0 positive definite ⇐ ☞ > 0 for x, y � = 0 . ☞ y ∗ B 2 y y ∗ C 2 y Harrachov, 2007 5/21

Right definite problem ∆ 0 = B 1 ⊗ C 2 − C 1 ⊗ B 2 A 1 x = λB 1 x + µC 1 x ∆ 1 z = λ ∆ 0 z ( MEP ) ∆ 1 = A 1 ⊗ C 2 − C 1 ⊗ A 2 ( ∆ ) A 2 y = λB 2 y + µC 2 y ∆ 2 z = µ ∆ 0 z ∆ 2 = B 1 ⊗ A 2 − A 1 ⊗ B 2 MEP is right definite when A i , B i , C i are Hermitian and ∆ 0 is positive definite. Atkinson (1972): ☞ ☞ ☞ x ∗ B 1 x x ∗ C 1 x ☞ ⇒ ( x ⊗ y ) ∗ ∆ 0 ( x ⊗ y ) = ☞ ☞ ∆ 0 positive definite ⇐ ☞ > 0 for x, y � = 0 . ☞ y ∗ B 2 y y ∗ C 2 y If MEP is right definite then • eigenpairs are real • there exist n 2 linearly independent eigenvectors • eigenvectors of distinct eigenvalues are ∆ 0 -orthogonal, i.e., ( x 1 ⊗ y 1 ) T ∆ 0 ( x 2 ⊗ y 2 )=0 Harrachov, 2007 5/21

Numerical methods First option: standard algorithms for explicitly computed matrices ∆ : ∆ 0 = B 1 ⊗ C 2 − C 1 ⊗ B 2 A 1 x = λB 1 x + µC 1 x ∆ 1 z = λ ∆ 0 z ( MEP ) ∆ 1 = A 1 ⊗ C 2 − C 1 ⊗ A 2 ( ∆ ) A 2 y = λB 2 y + µC 2 y ∆ 2 z = µ ∆ 0 z ∆ 2 = B 1 ⊗ A 2 − A 1 ⊗ B 2 Algorithms that work with matrices A i , B i , C i : • Blum, Curtis, Geltner (1978), and Browne, Sleeman (1982): gradient method, • Bohte (1980): Newton’s method for eigenvalues, • Ji, Jiang, Lee (1992): Generalized Rayleigh Quotient Iteration. • Continuation method: – Shimasaki (1995): for a special class of RD problems, – P. (1999): for RD problems, Tensor Rayleigh Quotient Iteration, – P. (2000): for weakly elliptic problems. • Jacobi-Davidson type methods. – Hochstenbach, P. (2002): for RD problems, – Hochstenbach, Koˇ sir, P. (2005): for general nonsingular MEP, – Hochstenbach, P. (2007): JD with harmonic extraction. Harrachov, 2007 6/21

Jacobi–Davidson method Subspace methods compute eigenpairs from low dimensional subspaces. They work as follows: • Extraction: We compute an approximation to an eigenpair from a given search subspace. Usually, we solve the same type of eigenvalue problem as the original one, but of a smaller dimension. • Expansion: After each step we expand the subspace by a new direction. As the search subspace grows the eigenpair approximations should converge to an eigenpair. Jacobi–Davidson method is a subspace method where: • a new direction to the subspace is orthogonal or oblique to the last chosen Ritz vector, • approximate solutions of certain correction equations are used for expansion. JD method can be efficiently generalized for two-parameter eigenvalue problems, while this is not clear for subspace methods based on Krylov subspaces. Harrachov, 2007 7/21

JD-like method for the right definite case: extraction Ritz–Galerkin conditions: search spaces = test spaces: u ∈ U k , v ∈ V k ( A 1 − σB 1 − τC 1 ) u ⊥ U k ( A 2 − σB 2 − τC 2 ) v ⊥ V k ⇒ projected right definite two-parameter eigenvalue problem U T k A 1 U k c = σU T k B 1 U k c + τU T k C 1 U k c V T k A 2 V k d = σV T k B 2 V k d + τV T k C 2 V k d Ritz vectors: u = U k c , v = V k d for c, d ∈ R k Ritz value: ( σ, τ ) , Ritz pair: (( σ, τ ) , u ⊗ v ) Will not discuss the correction equation and the deflation. Works well for exterior eigenvalues. Harrachov, 2007 8/21

Two-sided JD-like method for a general problem: extraction Petrov–Galerkin conditions: search spaces u i ∈ U ik , test spaces v i ∈ V ik ( A 1 − σB 1 − τC 1 ) u 1 ⊥ V 1 k ( A 2 − σB 2 − τC 2 ) u 2 ⊥ V 2 k , ⇒ projected two-parameter eigenvalue problem V ∗ σV ∗ 1 k B 1 U 1 k c 1 + τV ∗ 1 k A 1 U 1 k c 1 = 1 k C 1 U 1 k c 1 V ∗ σV ∗ 2 k B 2 U 2 k c 2 + τV ∗ 2 k A 2 U 2 k c 2 = 2 k C 2 U 2 k c 2 , where u i = U ik c i � = 0 for i = 1 , 2 and σ, τ ∈ C . Petrov vectors: u i = U ik c i , v i = V ik d i , c i , d i ∈ C k Petrov value: ( σ, τ ) , Petrov triple: (( σ, τ ) , u 1 ⊗ u 2 , v 1 ⊗ v 2 ) Usually performs better than the one-sided method. Works well for exterior eigenvalues, is less favorable for interior ones. Harrachov, 2007 9/21

Rayleigh–Ritz for GEP For a GEP Ax = λBx we want an approximate eigenpair ( θ, u ) , where u is in a given search subspace U k and θ is close to the given target τ ∈ C . Standard Ritz–Galerkin condition Au − θBu ⊥ U k leads to U ∗ k AU k c = θ U ∗ k BU k c, where the columns of U k form an orthonormal basis for U k and c ∈ C k . Harrachov, 2007 10/21

Rayleigh–Ritz for GEP For a GEP Ax = λBx we want an approximate eigenpair ( θ, u ) , where u is in a given search subspace U k and θ is close to the given target τ ∈ C . Standard Ritz–Galerkin condition r := Au − θBu ⊥ U k leads to U ∗ k AU k c = θ U ∗ k BU k c, where the columns of U k form an orthonormal basis for U k and c ∈ C k . For interior eigenvalues, even for a Ritz value θ ≈ τ , � r � can be large and the approximate eigenvector may be poor. As a remedy, the harmonic Rayleigh–Ritz was proposed: • standard eigenproblem: Morgan (1991), Paige, Parlett, Van der Vorst (1995), • GEP: Fokkema, Sleijpen, Van der Vorst (1998), Stewart (2001). Assuming A − τB is nonsingular we consider a spectral transformation ( A − τB ) − 1 Bx = ( λ − τ ) − 1 x. The interior eigenvalues λ ≈ τ are exterior eigenvalues of ( A − τB ) − 1 B . Harrachov, 2007 10/21

Harmonic Rayleigh–Ritz for GEP To avoid working with ( A − τB ) − 1 we impose a Petrov–Galerkin condition ( A − τB ) − 1 Bu − ( θ − τ ) − 1 u ⊥ ( A − τB ) ∗ ( A − τB ) U k , or, equivalently, Au − θBu = ( A − τB ) u − ( θ − τ ) Bu ⊥ ( A − τB ) U k , leading to the projected eigenproblem U ∗ k ( A − τB ) ∗ ( A − τB ) U k c = ( θ − τ ) U ∗ k ( A − τB ) ∗ BU k c. This approach has two motivations: • it retrieves exact eigenvectors in the search space; • a harmonic Ritz pair ( θ, u ) satisfies (Stewart (2001)) � Au − τBu � ≤ | θ − τ | · � Bu � ≤ | θ − τ | · � BU k � . Harrachov, 2007 11/21

Harmonic Rayleigh–Ritz for two-parameter eigenvalue problem GEP: Ax = λBx subspace is U k , target is τ A 1 x = λB 1 x + µC 1 x MEP: A 2 y = λB 2 y + µC 2 y subspace is U k ⊗ V k , target is ( σ, τ ) Harrachov, 2007 12/21

Harmonic Rayleigh–Ritz for two-parameter eigenvalue problem GEP: Ax = λBx subspace is U k , target is τ Rayleigh–Ritz: Au − θBu ⊥ U k A 1 x = λB 1 x + µC 1 x MEP: A 2 y = λB 2 y + µC 2 y subspace is U k ⊗ V k , target is ( σ, τ ) Harrachov, 2007 12/21

Harmonic Rayleigh–Ritz for two-parameter eigenvalue problem GEP: Ax = λBx subspace is U k , target is τ Rayleigh–Ritz: Au − θBu ⊥ U k A 1 x = λB 1 x + µC 1 x MEP: A 2 y = λB 2 y + µC 2 y subspace is U k ⊗ V k , target is ( σ, τ ) ( A 1 − θB 1 − ηC 1 ) u ⊥ U k Rayleigh–Ritz: ( A 2 − θB 2 − ηC 2 ) v ⊥ V k Harrachov, 2007 12/21

Harmonic Rayleigh–Ritz for two-parameter eigenvalue problem GEP: Ax = λBx subspace is U k , target is τ Rayleigh–Ritz: Au − θBu ⊥ U k ( A − τB ) − 1 Bx = ( λ − τ ) − 1 x Spectral transformation: A 1 x = λB 1 x + µC 1 x MEP: A 2 y = λB 2 y + µC 2 y subspace is U k ⊗ V k , target is ( σ, τ ) ( A 1 − θB 1 − ηC 1 ) u ⊥ U k Rayleigh–Ritz: ( A 2 − θB 2 − ηC 2 ) v ⊥ V k Harrachov, 2007 12/21