pymor.algorithms.krylov

Module for computing (rational) Krylov subspaces’ bases.

Module Contents

pymor.algorithms.krylov.arnoldi(A, E, b, r)

Block Arnoldi algorithm.

Computes an orthonormal basis for the Krylov subspace

$$\mathrm{span}\{E^{-1}b,E^{-1}A{E}^{-1}b,{\left(E^{-1}A\right)}^2{E}^{-1}b,\dots ,{\left(E^{-1}A\right)}^{r-1}{E}^{-1}b,\}.$$

Parameters

A

The Operator A.

E

The Operator E. If None, the identity operator is assumed.

b

The VectorArray b.

r

Order of the Krylov subspace (positive integer).

Returns

V

Orthonormal basis for the Krylov subspace as a VectorArray with len(V) <= r*len(b) (strict inequality in case of deflation).

pymor.algorithms.krylov.rational_arnoldi(A, E, b, sigma, trans=False)

Rational Arnoldi algorithm.

If trans == False, using Arnoldi process, computes a real orthonormal basis for the rational Krylov subspace

$$\mathrm{span}\{({\sigma }_{1}E-A{)}^{-1}b,({\sigma }_{2}E-A{)}^{-1}b,\dots ,({\sigma }_{r}E-A{)}^{-1}b\},$$

otherwise, computes the same for

$$\mathrm{span}\{({\sigma }_{1}E-A{)}^{-T}{b}^T,({\sigma }_{2}E-A{)}^{-T}{b}^T,\dots ,({\sigma }_{r}E-A{)}^{-T}{b}^T\}.$$

Interpolation points in sigma are allowed to repeat (in any order). Then, in the above expression,

$$\underset{m\text{ times}}{\underbrace{({\sigma }_{i}E-A{)}^{-1}b,\dots ,({\sigma }_{i}E-A{)}^{-1}b}}$$

is replaced by

$$({\sigma }_{i}E-A{)}^{-1}b,({\sigma }_{i}E-A{)}^{-1}E({\sigma }_{i}E-A{)}^{-1}b,\dots ,{(({\sigma }_{i}E-A{)}^{-1}E)}^{m-1}({\sigma }_{i}E-A{)}^{-1}b.$$

Analogously for the trans == True case.

Parameters

A

Real Operator A.

E

Real Operator E.

b

Real vector-like operator (if trans is False) or functional (if trans is True).

sigma

Sequence of interpolation points (closed under conjugation).

trans

Boolean, see above.

Returns

V

Orthonormal basis for the Krylov subspace as a VectorArray.

pymor.algorithms.krylov.tangential_rational_krylov(A, E, B, b, sigma, trans=False, orth=True)

Tangential Rational Krylov subspace.

If trans == False, computes a real basis for the rational Krylov subspace

$$\mathrm{span}\{({\sigma }_{1}E-A{)}^{-1}B{b}_1,({\sigma }_{2}E-A{)}^{-1}B{b}_2,\dots ,({\sigma }_{r}E-A{)}^{-1}B{b}_r\},$$

otherwise, computes the same for

$$\mathrm{span}\{({\sigma }_{1}E-A{)}^{-T}{B}^T{b}_1,({\sigma }_{2}E-A{)}^{-T}{B}^T{b}_2,\dots ,({\sigma }_{r}E-A{)}^{-T}{B}^T{b}_r\}.$$

Interpolation points in sigma are assumed to be pairwise distinct.

Parameters

A

Real Operator A.

E

Real Operator E.

B

Real Operator B.

b

VectorArray from B.source, if trans == False, or

B.range, if trans == True.

sigma

Sequence of interpolation points (closed under conjugation), of the same length as b.

trans

Boolean, see above.

orth

If True, orthonormalizes the basis using pymor.algorithms.gram_schmidt.gram_schmidt.

Returns

V

Optionally orthonormal basis for the Krylov subspace as a VectorArray.