A rational Krylov method for non-linear matrix problems Karl - PowerPoint PPT Presentation

A rational Krylov method for non-linear matrix problems Karl Meerbergen (Joint work with Roel Van Beeumen and Wim Michiels) KU Leuven SIMAX – Valencia

Motivation Solution of linear eigenvalue problem Ax = λ Bx is pretty well understood ◮ Appealing method: shift-and-invert Arnoldi ◮ Reliable for computing several eigenvalues near a shift point. ◮ Rational Krylov: more than one interpolation points = poles. Polynomial eigenvalue problems ◮ We can use Krylov methods (when we ‘linearize’) K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 2 / 24

Motivation Solution of linear eigenvalue problem Ax = λ Bx is pretty well understood ◮ Appealing method: shift-and-invert Arnoldi ◮ Reliable for computing several eigenvalues near a shift point. ◮ Rational Krylov: more than one interpolation points = poles. Polynomial eigenvalue problems ◮ We can use Krylov methods (when we ‘linearize’) Solution of non-linear eigenvalue problem A ( λ ) x = 0 ◮ Residual inverse iteration and Newton iterations are locally convergent methods ◮ Globally convergent methods by using polynomial approximation of A ( λ ) K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 2 / 24

Outline Motivation 1 Polynomial eigenvalue problem 2 Nonlinear eigenvalue problem 3 Rational Krylov 4 A statement on moment matching and projection 5 K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 3 / 24

Polynomial eigenvalue problem Polynomial eigenvalue problem: A 0 x + λ A 1 x + · · · + λ N A N x = 0 Transformed to ( A λ B ) x = 0 −     0     A 0 A 1 A p − 1 − A N x · · · · · · · · · 0 I I λ x         = 0 λ . .   ...   ...     − . .         . .         λ p − 1 x 0 I I Is a linear problem, so we can use Arnoldi’s method Is a well known technique. K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 4 / 24

Linearization for polynomial eigenvalue problems (Shift and) invert transform A − 1 B :       x S 1 · · · · · · S p x λ x I 0 λ x       λ − 1  =  .   .   .  ... . . .       . . .      λ p − 1 x λ p − 1 x I 0 Krylov sequence:   v 0    .  . . K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 5 / 24

Linearization for polynomial eigenvalue problems (Shift and) invert transform A − 1 B :       x S 1 · · · · · · S p x λ x I 0 λ x       λ − 1  =  .   .   .  ... . . .       . . .      λ p − 1 x λ p − 1 x I 0 Krylov sequence:     w = S 1 v v v 0        .  0 .   . .   . . K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 5 / 24

Linearization for polynomial eigenvalue problems (Shift and) invert transform A − 1 B :       x S 1 · · · · · · S p x λ x I 0 λ x       λ − 1  =  .   .   .  ... . . .       . . .      λ p − 1 x λ p − 1 x I 0 Krylov sequence:       w t = S 1 w + S 2 v v w v 0            .  v 0 .     . .     0 .   .  .  . . K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 5 / 24

Linearization for polynomial eigenvalue problems (Shift and) invert transform A − 1 B :       x S 1 · · · · · · S p x λ x I 0 λ x       λ − 1  =  .   .   .  ... . . .       . . .      λ p − 1 x λ p − 1 x I 0 Krylov sequence:       u = S 1 t + S 2 w + S 3 v   w t v t w v 0                .  w v 0 .       .   .     v 0 .     .    .  0 .   .  .  . . K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 5 / 24

Linearization for polynomial eigenvalue problems Shift-and-invert transform:       x S 1 · · · · · · S p x λ x I 0 λ x       λ − 1  =  .   .   .  ... . . .       . . .      λ p − 1 x λ p − 1 x I 0 Krylov sequence:   v 1 v 2 v 3 v 4 · · · v k 0 v 1 v 2 v 3 v k − 1    .  .   . 0 v 1 v 2 v k − 2    . .  . .   . 0 v 1 .   . .   . .   . 0 .   .   .  . v 1    0 K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 6 / 24

Nonlinear eigenvalue problem Straightforward idea: choose an approximation of a fixed degree and solve the corresponding polynomial eigenvalue problem: ◮ Polynomial expansion, e.g. Taylor, Chebyshev, . . . A ( λ ) ≈ A 0 p 0 ( λ ) + · · · + A N p N ( λ ) [Amiraslani, Corless, Lancaster, 2009], [Effenberger, Kressner, 2011] ◮ Rational expansion, e.g. (Chebyshev) Pad´ e α 0 α N A ( λ ) ≈ A 0 + · · · + A N λ − σ 0 λ − σ N [Su & Bai, 2011] ◮ Spectral discretization A ( λ ) ≈ A 0 + λ A 0 p 0 ( λ ) + · · · + A N p N ( λ ) α 0 p 0 ( λ ) + · · · + α N p N ( λ ) [Trefethen 2000], [Michiels, Niculescu 2007], [Breda, Maset, Vermiglio 2005] In this talk, we explore a ‘flexible’ choice of N and p j . K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 7 / 24

Taylor-Arnoldi [Jarlebring, Meerbergen, Michiels, 2010] Power sequence on Taylor series A ( λ ) ≈ A 0 + λ A 1 + λ 2 A 2 + · · · For k iterations of Arnoldi, use Companion linearization with N > k Arnoldi:   v 0   .   . . Breakthrough for solving the delay (equation) eigenvalue problem K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 8 / 24

Taylor-Arnoldi [Jarlebring, Meerbergen, Michiels, 2010] Power sequence on Taylor series A ( λ ) ≈ A 0 + λ A 1 + λ 2 A 2 + · · · For k iterations of Arnoldi, use Companion linearization with N > k Arnoldi: w = − A − 1     0 A 1 v v 0 v     .     0 .   .  .  . . Breakthrough for solving the delay (equation) eigenvalue problem K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 8 / 24

Taylor-Arnoldi [Jarlebring, Meerbergen, Michiels, 2010] Power sequence on Taylor series A ( λ ) ≈ A 0 + λ A 1 + λ 2 A 2 + · · · For k iterations of Arnoldi, use Companion linearization with N > k Arnoldi: t = − A − 1     0 ( A 1 w + A 2 ) v   w v w 0 v         .     0 v .     .    .  0 .   .  .  . . Breakthrough for solving the delay (equation) eigenvalue problem K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 8 / 24

Taylor-Arnoldi [Jarlebring, Meerbergen, Michiels, 2010] Power sequence on Taylor series A ( λ ) ≈ A 0 + λ A 1 + λ 2 A 2 + · · · For k iterations of Arnoldi, use Companion linearization with N > k Arnoldi: u = − A − 1       t 0 ( A 1 t + A 2 w + A 3 ) v   w v t w 0 v             .     v w 0 .       .      .  v 0 .     .    .  0 .   .   . . . Breakthrough for solving the delay (equation) eigenvalue problem K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 8 / 24

Other polynomials Use polynomials in order to ◮ avoid derivatives or ◮ interpolation in more than one point. K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 9 / 24

Other polynomials Use polynomials in order to ◮ avoid derivatives or ◮ interpolation in more than one point. Newton polynomial Lagrange polynomial Chebyshev polynomial K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 9 / 24

Other polynomials Use polynomials in order to ◮ avoid derivatives or ◮ interpolation in more than one point. Newton polynomial Lagrange polynomial Chebyshev polynomial A (partly) successful generalization of Taylor-Arnoldi to rational Krylov K. Meerbergen (KU Leuven) Nonlinear Rational Krylov SIMAX – Valencia 9 / 24