Krylov-Subspace Methods - I Lecture 6 Alessandra Nardi Thanks to Prof. Jacob White, Deepak Ramaswamy, Michal Rewienski, and Karen Veroy

Last lecture review Iterative Methods Overview Stationary Non Stationary QR factorization to solve Mx=b Modified Gram-Schmidt Algorithm QR Pivoting Minimization View of QR Basic Minimization approach Orthogonalized Search Directions Pointer to Krylov Subspace Methods

Last lecture reminder QR Factorization – By picture

QR Factorization – Minimization View Minimization Algorithm For i = 1 to N “For each Target Column” For j = 1 to i-1 “For each Source Column left of target” end Orthogonalize Search Direction Normalize

Iterative Methods Solve Mx=b minimizing the residual r=b-Mx Stationary: x(k+1) =Gx(k)+c Jacobi Gauss-Seidel Successive Overrelaxation Non Stationary: x(k+1) =x(k)+akpk CG (Conjugate Gradient)  A symmetric and positive definite GCR (Generalized Conjugate Residual) GMRES, etc etc

Iterative Methods - CG Convergence is related to: Why ? How? Number of distinct eigenvalues Ratio between max and min eigenvalue

Outline General Subspace Minimization Algorithm Review orthogonalization and projection formulas Generalized Conjugate Residual Algorithm Krylov-subspace Simplification in the symmetric case. Convergence properties Eigenvalue and Eigenvector Review Norms and Spectral Radius Spectral Mapping Theorem

Arbitrary Subspace Methods Residual Minimization

Arbitrary Subspace Methods Residual Minimization Use Gram-Schmidt on Mwi’s!

Arbitrary Subspace Methods Orthogonalization

Arbitrary Subspace Solution Algorithm Given M, b and a set of search directions: {w0,…,wk} Make wi’s MMT orthogonal and get new search directions: {p0,…,pk} Minimize the residual:

Arbitrary Subspace Solution Algorithm For i = 0 to k For j = 1 to i-1 end Orthogonalize Search Direction Normalize Update Solution

Krylov Subspace How about the initial set of search directions {w0,…,wk} ? A particular choice that is commonly used is: {w0,…,wk}  {b, Mb, M2b…} Km(A,v)  span{v, Av, A2v, …, Am-1v} is called Krylov Subspace

Krylov Subspace Methods kth order polynomial

Krylov Subspace Methods Subspace Generation The set of residuals also can be used as a representation of the Krylov-Subspace Generalized Conjugate Residual Algorithm Nice because the residuals generate next search directions

Krylov-Subspace Methods Generalized Conjugate Residual Method (k-th step) Determine optimal stepsize in kth search direction Update the solution (trying to minimize residual) and the residual Compute the new orthogonalized search direction (by using the most recent residual)

Krylov-Subspace Methods Generalized Conjugate Residual Method (Computational Complexity for k-th step) Vector inner products, O(n) Matrix-vector product, O(n) if sparse Vector Adds, O(n) O(k) inner products, total cost O(nk) If M is sparse, as k (# of iters) approaches n, Better Converge Fast!

Krylov-Subspace Methods Generalized Conjugate Residual Method (Symmetric Case – Conjugate Gradient Method) An Amazing fact that will not be derived Orthogonalization in one step If k (# of iters )  n, then symmetric, sparse, GCR is O(n2 ) Better Converge Fast!

Summary What is an iterative non stationary method: x(k+1) =x(k)+akpk How search to calculate: Search directions (pk) Step along search directions (ak) Krylov Subspace  GCR GCR is O(k2n) Better converge fast!  Now look at convergence properties of GCR

Krylov Methods Convergence Analysis Basic properties

Krylov Methods Convergence Analysis Optimality of GCR poly GCR optimality property (key property of the algorithm): GCR picks the best (k+1)-th order polynomial minimizing and subject to: 

Krylov Methods Convergence Analysis Optimality of GCR poly GCR Optimality Property Therefore Any polynomial which satisfies the constraints can be used to get an upper bound on

Eigenvalues and eigenvectors review Basic definitions Eigenvalues and eigenvectors of a matrix M satisfy eigenvalue eigenvector

Eigenvalues and eigenvectors review A symplifying assumption Almost all NxN matrices have N linearly independent Eigenvectors The set of all eigenvalues of M is known as the Spectrum of M

Eigenvalues and eigenvectors review A symplifying assumption Almost all NxN matrices have N linearly independent Eigenvectors

Eigenvalues and eigenvectors review Spectral radius The spectral Radius of M is the radius of the smallest circle, centered at the origin, which encloses all of M’s eigenvalues

Eigenvalues and eigenvectors review Vector norms L2 (Euclidean) norm : Unit circle L1 norm : 1 1 L norm : Unit square

Eigenvalues and eigenvectors review Matrix norms Vector induced norm : Induced norm of A is the maximum “magnification” of by = max abs column sum = max abs row sum = (largest eigenvalue of ATA)1/2

Eigenvalues and eigenvectors review Induced norms Theorem: Any induced norm is a bound on the spectral radius Proof:

Useful Eigenproperties Spectral Mapping Theorem Given a polynomial Apply the polynomial to a matrix Then

Krylov Methods Convergence Analysis Overview Matrix norm property GCR optimality property where is any (k+1)-th order polynomial subject to:  may be used to get an upper bound on

Krylov Methods Convergence Analysis Overview Review on eigenvalues and eigenvectors Induced norms: relate matrix eigenvalues to the matrix norms Spectral mapping theorem: relate matrix eigenvalues to matrix polynomials Now ready to relate the convergence properties of Krylov Subspace methods to eigenvalues of M

Summary Generalized Conjugate Residual Algorithm Krylov-subspace Simplification in the symmetric case Convergence properties Eigenvalue and Eigenvector Review Norms and Spectral Radius Spectral Mapping Theorem