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Unit# Algorithmes numeriques et calcul haute performance - Chaire CADMOS

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Algorithm

In mathematics and computer science, an algorithm (ˈælɡərɪðəm) is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algo

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathema

Low-rank approximation

In mathematics, low-rank approximation is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable)

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Daniel Kressner, Christoph Max Strössner

Global spectral methods offer the potential to compute solutions of partial differential equations numerically to very high accuracy. In this work, we develop a novel global spectral method for linear partial differential equations on cubes by extending the ideas of Chebop2 (Townsend, A. & Olver, S. (2015) The automatic solution of partial differential equations using a global spectral method. J. Comput. Phys., 299, 106-123) to the three-dimensional setting utilizing expansions in tensorized polynomial bases. Solving the discretized partial differential equation involves a linear system that can be recast as a linear tensor equation. Under suitable additional assumptions, the structure of these equations admits an efficient solution via the blocked recursive solver (Chen, M. & Kressner, D. (2020) Recursive blocked algorithms for linear systems with Kronecker product structure. Numer. Algorithms, 84, 1199-1216). In the general case, when these assumptions are not satisfied, this solver is used as a preconditioner to speed up computations.

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Numerical continuation in the context of optimization can be used to mitigate convergence issues due to a poor initial guess. In this work, we extend this idea to Riemannian optimization problems, that is, the minimization of a target function on a Riemannian manifold. For this purpose, a suitable homotopy is constructed between the original problem and a problem that admits an easy solution. We develop and analyze a path-following numerical continuation algorithm on manifolds for solving the resulting parameter-dependent equation. To illustrate our developments, we consider two typical classical applications of Riemannian optimization: the computation of the Karcher mean and low-rank matrix completion. We demonstrate that numerical continuation can yield improvements for challenging instances of both problems.

Alice Cortinovis, Daniel Kressner, Ulf David Persson

This paper is concerned with two improved variants of the Hutch++ algorithm for estimating the trace of a square matrix, implicitly given through matrix-vector products. Hutch++ combines randomized low-rank approximation in a first phase with stochastic trace estimation in a second phase. In turn, Hutch++ only requires O (epsilon(-1)) matrix-vector products to approximate the trace within a relative error\varepsilon with high probability, provided that the matrix is symmetric positive semidefinite. This compares favorably with the O (epsilon(-2)) matrix-vector products needed when using stochastic trace estimation alone. In Hutch++, the number of matrix-vector products is fixed a priori and distributed in a prescribed fashion among the two phases. In this work, we derive an adaptive variant of Hutch++, which outputs an estimate of the trace that is within some prescribed error tolerance with a controllable failure probability, while splitting the matrix-vector products in a near-optimal way among the two phases. For the special case of a symmetric positive semidefinite matrix, we present another variant of Hutch++, called Nystrom++, which utilizes the so-called Nystrom approximation and requires only one pass over the matrix, as compared to two passes with Hutch++. We extend the analysis of Hutch++ to Nystrom++. Numerical experiments demonstrate the effectiveness of our two new algorithms.