Extrapolation of operator moments, with problems C. Brezinski, - PowerPoint PPT Presentation

Extrapolation of operator moments, with applications to linear algebra Extrapolation of operator moments, with problems C. Brezinski, ∗ applications to linear algebra problems P. Fika, + and M. Mitrouli + Introduction C. Brezinski, ∗ P. Fika , + and M. Mitrouli + Motivation of the problem Mathematical ∗ University of Lille - France, + University of Athens - Greece tools Estimates Applications The trace The error norm Numerical results Conclusions

Extrapolation of operator MAIN TOPICS moments, with applications to linear algebra Motivation of the problem problems C. Brezinski, ∗ The mathematical landscape P. Fika, + and M. Mitrouli + Extrapolation procedures and estimates Introduction Motivation of Estimation of the the problem Tr ( A q ) , q ∈ Q Mathematical { error of the solution of the linear system Ax = f tools Estimates Applications Numerical results The trace The error norm Numerical results Conclusions

Introduction Let A ∈ R p × p symmetric positive definite (spd) matrix. Extrapolation of operator moments, with We are interested in obtaining estimations of applications to linear algebra problems Tr ( A q ) , q ∈ Q C. Brezinski, ∗ P. Fika, + and ( x − y , x − y ) , x is the exact solution of Ax = f , M. Mitrouli + Introduction y is any approximation of x , e = ∣∣ x − y ∣∣ is the error . Motivation of the problem Mathematical tools ↪ These estimates will be obtained by Estimates Applications extrapolation of the moments of A . The trace The error norm Numerical results Conclusions

Motivation of the problem Extrapolation Estimates for the error have applications in the of operator moments, with applications choice of the best parameter in Tikhonov regularization . to linear algebra The computation of Tr ( A q ) , have applications in problems C. Brezinski, ∗ P. Fika, + and M. Mitrouli + Statistics : specification of classical optimality criteria. Introduction Matrix theory : computation of the characteristic polynomial. Motivation of the problem Dynamical Systems : determination of their invariants. Mathematical tools Differential Equations : solution of Lyapunov matrix equation. Estimates Applications Crystals : for the selection of measurement directions in elastic The trace The error norm strain determination of single crystals. Numerical results Conclusions

The mathematical landscape Extrapolation The singular value decomposition of operator moments, The singular value decomposition of an spd matrix A ∈ R p × p is with applications to linear algebra A = UΣU T , problems with UU T = I p , Σ = diag ( σ 1 ,...,σ p ) with σ 1 ≥ ⋯ ≥ σ p > 0 . C. Brezinski, ∗ P. Fika, + and M. Mitrouli + A q = UΣ q U T , Introduction q ∈ Q . Motivation of the problem Mathematical Let x be an arbitrary nonzero vector in R p and U = [ u 1 ,...,u p ] tools Estimates It holds Applications p A q x = ∑ The trace σ q k ( u k , x ) u k . The error norm k = 1 Numerical results Conclusions

The mathematical landscape Extrapolation The moments of operator moments, with The moments of A with respect to a vector z are defined by applications to linear algebra c ν ( z ) = ( z , A ν z ) = ∑ k ( z ) , k α 2 problems σ ν C. Brezinski, ∗ k P. Fika, + and where α k ( z ) = ( z , u k ) . M. Mitrouli + Introduction Motivation of Extrapolation of moments was first introduced in the problem Mathematical C. Brezinski, Error estimates for the solution of linear systems, tools SIAM J. Sci. Comput. , 21 (1999) 764–781. Estimates Applications The trace The error norm Numerical results Conclusions

Extrapolation procedures and estimates Extrapolation of operator moments, Using some moments with a non–negative integer index , we estimate the moments c q ( z ) for any fixed index q ∈ Q . with applications to linear algebra The estimates are based on the integer moments of A with problems ν = n ∈ N . C. Brezinski, ∗ P. Fika, + and M. Mitrouli + For this purpose, we will approximate the moments c q ( z ) Introduction by interpolating or extrapolating the c n ( z ) ’s, for different Motivation of the problem values of the non–negative integer index n , at the points q , Mathematical tools by a conveniently chosen function obtained by Estimates keeping only one or two terms in the summations . Applications The trace The error norm Numerical results Conclusions

One–term estimates Knowing c 0 ( z ) and c 1 ( z ) , we will look for s , and α ( z ) Extrapolation of operator moments, satisfying the interpolation conditions with applications c 0 ( z ) = α 2 ( z ) to linear algebra c 1 ( z ) = s α 2 ( z ) problems C. Brezinski, ∗ P. Fika, + and and, then, c q ( z ) will be estimated by M. Mitrouli + c q ( z ) ≃ e q ( z ) = s q α 2 ( z ) . Introduction Motivation of the problem Mathematical Proposition 1 tools 1 ( z ) c q c q ( z ) ≃ e q ( z ) = Estimates ( z ) . Applications c q − 1 The trace 0 The error norm Numerical e q ( z ) ∈ R , q ∈ Q, since c 1 ( z ) > 0 . results Conclusions

One–term estimates Assume that A − 1 exists, and let κ = ∥ A ∥ ⋅ ∥ A − 1 ∥ . Extrapolation of operator moments, with applications to linear Theorem algebra problems If A is symmetric positive definite, then, for any vector z , the C. Brezinski, ∗ one–term estimate e n ( z ) satisfies the following inequalities for P. Fika, + and M. Mitrouli + n ∈ Z , n ≠ 0 , Introduction e n ( z ) ≤ c n ( z ) ≤ (( 1 + κ ) 2 Motivation of 2 d − 1 ) e n ( z ) , the problem Mathematical 4 κ tools Estimates where n > 1 Applications d = { n − 1 , The trace ∣ n ∣ , n < 0 , n = 1 The error norm Numerical results Conclusions

Two–term estimates Estimate c q ( z ) , q ∈ Q , by keeping two terms Extrapolation of operator moments, c q ( z ) ≃ e q ( z ) = s q 1 ( z ) + s q 2 ( z ) . with 1 a 2 2 a 2 applications (1) to linear algebra problems 1 ( z ) and a 2 2 ( z ) will be computed by The unknowns s 1 , s 2 , a 2 C. Brezinski, ∗ P. Fika, + and imposing the interpolation conditions , M. Mitrouli + c n ( z ) = e n ( z ) = s n 1 ( z ) + s n 2 ( z ) , Introduction 1 a 2 2 a 2 (2) Motivation of the problem for different integer values of the integer n . Mathematical c n ( z ) ’s satisfy the difference equation of order 2 tools Estimates c n + 2 ( z ) − sc n + 1 ( z ) + pc n ( z ) = 0 , Applications (3) The trace The error norm where s = s 1 + s 2 and p = s 1 s 2 . Numerical results Conclusions

Two–term estimates Using this relation for n = 0 and 1 gives s and p . Extrapolation of operator moments, s = c 0 ( z ) c 3 ( z ) − c 1 ( z ) c 2 ( z ) p = c 1 ( z ) c 3 ( z ) − c 2 2 ( z ) with applications c 0 ( z ) c 2 ( z ) − c 2 1 ( z ) c 0 ( z ) c 2 ( z ) − c 2 1 ( z ) , to linear algebra √ problems (4) s 2 − 4p )/ 2 and e q ( z ) follows with s 1 , 2 = ( s ± C. Brezinski, ∗ P. Fika, + and M. Mitrouli + 1 ( z ) = c 0 ( z ) s 2 − c 1 ( z ) 2 ( z ) = c 1 ( z ) − c 0 ( z ) s 1 Introduction a 2 s 2 − s 1 a 2 s 2 − s 1 (5) , , Motivation of the problem Mathematical Proposition 2 tools The moment c q ( z ) can be estimated by the two–term formula Estimates Applications c q ( z ) ≃ e q ( z ) = s q 1 ( z ) + s q 2 ( z ) , q ∈ Q , The trace 1 a 2 2 a 2 (6) The error norm Numerical e q ( z ) ∈ R , if q ∈ Q. results Conclusions

The trace Extrapolation of operator Theorem moments, with M. Hutchinson, A stochastic estimator of the trace of the applications to linear influence matrix for Laplacian smoothing splines, algebra problems Commun. Statist. Simula., 18 (1989) 1059–1076. C. Brezinski, ∗ Let P. Fika, + and A ∈ R p × p symmetric, Tr ( A ) ≠ 0 , M. Mitrouli + X a discrete random variable with values 1 , − 1 with Introduction equal probability 0 . 5 , Motivation of the problem x a vector of p independent samples from X . Mathematical Then ( x , Ax ) is an unbiased estimator of Tr ( A ) . tools Estimates E (( x , Ax )) = Tr ( A ) , Applications The trace The error norm Var (( x , Ax )) = 2 ∑ a 2 Numerical ij , results i ≠ j Conclusions

Extrapolation This Theorem tells us that of operator moments, Tr ( A q ) = E (( x , A q x )) = E ( c q ( x )) , x ∈ X p with applications to linear algebra . problems Thus, estimates of Tr ( A q ) could be obtained by C. Brezinski, ∗ P. Fika, + and M. Mitrouli + Introduction extrapolating the moments at the point q , Motivation of the problem computing the expectation E ( e q ( x )) of e q ( x ) , x ∈ X p . Mathematical tools Estimates Applications The trace The error norm Numerical results Conclusions

For the one–term estimates , for x ∈ X p and q = n ∈ Z , we Extrapolation of operator moments, have with applications Proposition 3 to linear algebra problems If the matrix A is symmetric positive definite, then, for the C. Brezinski, ∗ one–term estimates, we have the bounds P. Fika, + and M. Mitrouli + E ( e n ( x )) ≤ Tr ( A n ) ≤ (( 1 + κ ) 2 2 d − 1 ) E ( e n ( x )) , Introduction 4 κ Motivation of the problem Mathematical where d = { n − 1 , n > 1 tools ∣ n ∣ , n < 0 , n = 1 Estimates Applications The trace The error norm ↪ If A is orthogonal , then κ ( A ) = 1 → Tr ( A n ) = E ( e n ( x )) . Numerical results Conclusions