on optimal short recurrences for generating orthogonal
play

On optimal short recurrences for generating orthogonal Krylov - PowerPoint PPT Presentation

On optimal short recurrences for generating orthogonal Krylov subspace bases J org Liesen based on joint work with Zden ek Strako s and Petr Tich y (Czech Academy of Sciences), Vance Faber (BD Biosciences, Seattle), Beresford


  1. On optimal short recurrences for generating orthogonal Krylov subspace bases J¨ org Liesen based on joint work with Zdenˇ ek Strakoˇ s and Petr Tich´ y (Czech Academy of Sciences), Vance Faber (BD Biosciences, Seattle), Beresford Parlett (Berkeley), and Paul Saylor (Illinois)

  2. Overview 1. Introduction: Krylov subspace methods 2. Optimal short recurrences 3. Characterization and examples

  3. Introduction: Krylov subspace methods (1) • Methods that are based on ���������� ���� ��� ������ ��������� K � ( A, v � ) ≡ span { v � , Av � , . . . , A � − � v � } , n = 1 , 2 , . . . , where A is a given square matrix and v � is the initial vector. • Must generate bases of K � ( A, v � ), n = 1 , 2 , . . . . • Trivial choice: v � , Av � , . . . , A � − � v � . This is computationally infeasible (recall the Power Method).

  4. Introduction: Krylov subspace methods (1) • Methods that are based on ���������� ���� ��� ������ ��������� K � ( A, v � ) ≡ span { v � , Av � , . . . , A � − � v � } , n = 1 , 2 , . . . , where A is a given square matrix and v � is the initial vector. • Must generate bases of K � ( A, v � ), n = 1 , 2 , . . . . • Trivial choice: v � , Av � , . . . , A � − � v � . This is computationally infeasible (recall the Power Method). • For numerical stability: Well conditioned basis. • For computational efficiency: Short recurrence. • Best of both worlds: ���������� ����� �������� �� ����� ���������� . • First such method for Ax = b : Conjugate gradient (CG) method of Hestenes and Stiefel (1952).

  5. Introduction: Krylov subspace methods (2) The classical CG method of Hestenes and Stiefel (US National Bureau of Standards Preprint No. 1659, March 10, 1952) The residual vectors r � , r � , .. . , r � − � are generated by a short recurrence and form an orthogonal basis of K � ( A, r � ).

  6. Introduction: Krylov subspace methods (3) • CG is for symmetric positive definite A . • (Paige and Saunders, 1975): Short recurrence & orthogonal basis methods for symmetric A .

  7. Introduction: Krylov subspace methods (4) • By the end of the 1970s it was unknown if such methods existed also for general unsymmetric A . • ���� ����� ����� ���� ����������� �������� at Gatlinburg VIII (now Householder VIII) held in Oxford from July 5 to 11, 1981: ���� ���� ���� �����

  8. Introduction: Krylov subspace methods (5) • We want to solve Ax = b iteratively, starting from x � . • Step n = 1 , 2 , . . . : x � = x � − � + α � − � p � − � , direction vector p � − � , scalar α � − � (both to be determined). • Krylov subspace method: span { p � , . . . , p � − � } = K � ( A, v � ) ( v � = r � = b − Ax � ). • CGClike descent method: Error is minimized in some given inner product norm, � � � � = �� , �� � � � � .

  9. Introduction: Krylov subspace methods (5) • We want to solve Ax = b iteratively, starting from x � . • Step n = 1 , 2 , . . . : x � = x � − � + α � − � p � − � , direction vector p � − � , scalar α � − � (both to be determined). • Krylov subspace method: span { p � , . . . , p � − � } = K � ( A, v � ) ( v � = r � = b − Ax � ). • CGClike descent method: Error is minimized in some given inner product norm, � � � � = �� , �� � � � � . • � x − x � � � is minimal iff x − x � ⊥ � span { p � , . . . , p � − � } . • By construction, this is satisfied iff α � − � = � x − x � − � , p � − � � � and � p � − � , p � � � = 0 , j = 0 , . . . , n − 2 , � p � − � , p � − � � � i.e. p � , . . . , p � − � ���� �� � B ����������� ����� �� K � ( A, v � ) �

  10. Introduction: Krylov subspace methods (6) • Faber and Manteuffel answered Golub’s question in 1984: For a general matrix A there exists �� CGClike descent method �������� ������� �� �������� ��� ������������ �� ��� ����� ������� ���� ��� ��� ������� �� ���� �������

  11. Optimal short recurrences (1) Notation: • Matrix A ∈ � � × � , nonsingular. • Matrix B ∈ � � × � , Hermitian positive definite (HPD), defining the B Cinner product, � x, y � � ≡ y ∗ Bx . • Initial vector v � ∈ � � . • d = d ( A, v � ), the grade of v � with respect to A , K � ( A, v � ) ⊂ . . . ⊂ K � ( A, v � ) = K � �� ( A, v � ) = . . . = K � ( A, v � ) .

  12. Optimal short recurrences (1) Notation: • Matrix A ∈ � � × � , nonsingular. • Matrix B ∈ � � × � , Hermitian positive definite (HPD), defining the B Cinner product, � x, y � � ≡ y ∗ Bx . • Initial vector v � ∈ � � . • d = d ( A, v � ), the grade of v � with respect to A , K � ( A, v � ) ⊂ . . . ⊂ K � ( A, v � ) = K � �� ( A, v � ) = . . . = K � ( A, v � ) . Our goal: Generate a B Corthogonal basis v � , . . . , v � of K � ( A, v � ). 1. span { v � , . . . , v � } = K � ( A, v � ) , for n = 1 , . . . , d , 2. � v � , v � � � = 0 , for j � = k , j, k = 1 , . . . , d .

  13. Optimal short recurrences (2) • Standard way for generating the B Corthogonal basis: Arnoldi’s method. � v � �� = Av � − h ��� v � , n = 1 , . . . , d − 1 , � �� � Av � , v � � � h ��� = , d = dim K � ( A, v � ) . � v � , v � � � (No normalization for convenience.)

  14. Optimal short recurrences (2) • Standard way for generating the B Corthogonal basis: Arnoldi’s method. � v � �� = Av � − h ��� v � , n = 1 , . . . , d − 1 , � �� � Av � , v � � � h ��� = , d = dim K � ( A, v � ) . � v � , v � � � (No normalization for convenience.) matrix of size d × ( d − 1) • Rewritten in matrix notation: AV � − � = V � H ��� − � , where h � � � � � � h � �� − � . ... . 1 . ... V � ≡ [ v � , . . . , v � ] , H ��� − � ≡ h � − � �� − � 1 V ∗ � BV � is diagonal , d = dim K � ( A, v � ) .

  15. Optimal short recurrences (3) • The ���� ���������� in Arnoldi’s method, � v � �� = Av � − h ��� v � , n = 1 , . . . , d − 1 , � �� is an ������� � s + 2 ������ ���������� when � v � �� = Av � − h ��� v � , n = 1 , . . . , d − 1 . � � � − � • For s = 1: Optimal 3Cterm recurrence, v � �� = Av � − h ��� v � − h � − � �� v � − � • Why ������� ? �� ���� ��� �������������� ���� � �� ���������� � ���� ��� �������� � ! � ������� ��� ��"������

  16. Optimal short recurrences (4) s d − s − 2 � �� � � �� � largest upper triangle that is zero   ∗ ∗ � � � ∗ 0 � � � 0 .   ... .   ∗ ∗ ∗ � � � ∗ .     ... ... ... ...   0   Optimal ( s + 2)Cterm recurrence:   ... ... ...   ∗   H ��� − � �� ( s + 2) ����� ���������   . ... ... ... .   .     ... ...   ����� ������ ���������� � ������������ ∗     ...   ∗ ∗

  17. Optimal short recurrences (4) s d − s − 2 � �� � � �� � largest upper triangle that is zero   ∗ ∗ � � � ∗ 0 � � � 0 .   ... .   ∗ ∗ ∗ � � � ∗ .     ... ... ... ...   0   Optimal ( s + 2)Cterm recurrence:   ... ... ...   ∗   H ��� − � �� ( s + 2) ����� ���������   . ... ... ... .   .     ... ...   ����� ������ ���������� � ������������ ∗     ...   ∗ ∗ ��������� (L. and Strakoˇ s, 2007) Given A , B as above and a nonnegative integer s with s + 2 ≤ d ��� ( A ). ( d ��� ( A ) = degree of A ’s minimal polynomial.) Then A ������ ��� B �� ������� ( s + 2) ����� ���������� , if • for any v � the matrix H ��� − � is at most ( s + 2)Cband Hessenberg, and • for at least one v � the matrix H ��� − � is ( s + 2)Cband Hessenberg.

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend


More recommend