A class of nonlinear acceleration techniques based on Krylov subspaces

• Yousef Saad

Room: Sophie Germain There has been a surge of interest in recent years in general-purpose `acceleration' methods that take a sequence of vectors converging to the limit of a fixed point iteration, and produce from it a faster converging sequence. A prototype of these methods that attracted much attention recently is the Anderson Acceleration (AA) procedure. We introduce the nonlinear Truncated Generalized Conjugate Residual (nlTGCR) algorithm, an alternative to AA which is designed from a careful adaptation of the Conjugate Residual method for solving linear systems of equations to the nonlinear context. The various links between nlTGCR and inexact Newton, quasi-Newton, and multisecant methods are exploited to build a method that has strong global convergence properties and that can also exploit symmetry when applicable. Taking this algorithm as a starting point we explore a number of other acceleration procedures including a short-term (`symmetric') version of Anderson Acceleration which we call Anderson Acceleration with Truncated Gram-Schmidt.

Convergence of GMRES accelerated by deflation and preconditioning

• Nicole Spillane

Room: Sophie Germain This is joint work with Daniel Szyld (Temple University) We present new convergence bounds for weighted, preconditioned, and deflated GMRES applied to non-Hermitian linear systems. These bounds are given for the case when the Hermitian part of the coefficient matrix is positive definite, the preconditioner is Hermitian positive definite, and the weight is equal to the preconditioner. The decrease in residual is bounded with respect to: - the condition number of the preconditioned Hermitian part of the problem matrix, - certain measure of how non-Hermitian the problem is. This indicates how to choose the preconditioner and the deflation space in order to accelerate convergence. One such choice of deflation space is proposed, and numerical experiments illustrate the effectiveness of such space.

Perturbations of functions of matrices and applications to density functional theory

• Antoine Levitt

Room: Sophie Germain

Enhanced stabilization strategies for some linearized PDEs

• Youcef Mammeri

Room: Sophie Germain Stabilizing linearized partial differential equations (PDEs) is a fundamental challenge in numerical analysis and control theory. In this talk, we present advanced strategies for enhancing the stability of linearized PDEs. We begin by reminding the concept of linear instability, illustrated through the example of the Klein-Gordon equation. Next, we introduce an optimal stabilization approach, based on the minimization of the Frobenius norm of the discrete linearized operator and its associated second-order eigenvalue problem. Building on this, we explore automatic stabilization using PyTorch, demonstrating how machine learning techniques can adaptively stabilize PDEs. We showcase the effectiveness of these methods through a case study, of the Klein-Gordon and the Korteweg-de Vries equation. Finally, we discuss broader implications, including stabilization at every time step and connections to approximate controllability. This is a joint work with J.P. Chehab and M. Raydan.

Continuous optimization strategies for inverse structured matrix problems with prescribed spectrum

• Marcos Raydan

Room: Sophie Germain The talk will consider several continuous optimization schemes to solve inverse structured symmetric matrix problems with prescribed spectrum. Some entries in the desired matrix are assigned in advance and cannot be altered, and some others should be nonzero. The rest of the entries are free. The reconstructed matrix must satisfy these requirements and its eigenvalues must be the given ones. This inverse eigenvalue problem, that appears in several applications, is related to the problem of determining the graph with weights on the undirected edges, of the matrix associated with its sparse pattern. Our optimization schemes are based on considering the eigenvector matrix as the only unknown and moving iteratively on the manifold of orthogonal matrices, forcing the additional structural requirements through a change of variables and a convenient differentiable objective function in the space of square matrices. We propose and analyze Riemannian gradient-type methods combined with either a penalization or an augmented Lagrangian strategy. We also present a block alternating technique that takes advantage of a proper separation of variables. We present some numerical results to demonstrate the effectiveness of our proposals. Note: This is a joint work with Jean-Paul Chehab (Amiens, France) and Harry Oviedo (Chile).