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Publication# Numerical Mathematics

Abstract

Numerical mathematics is the branch of mathematics that proposes, develops, analyzes and applies methods from scientific computing to several fields including analysis, linear algebra, geometry, approximation theory, functional equations, optimization and differential equations. Other disciplines, such as physics, the natural and biological sciences, engineering, and economics and the financial sciences frequently give rise to problems that need scientific computing for their solutions. As such, numerical mathematics is the crossroad of several disciplines of great relevance in modern applied sciences, and can become a crucial tool for their qualitative and quantitative analysis. One of the purposes of this book is to provide the mathematical foundations of numerical methods, to analyze their basic theoretical properties (stability, accuracy, computational complexity) and demonstrate their performances on examples and counterexamples which outline their pros and cons. This is done using the MATLAB software environment which is user-friendly and widely adopted. Within any specific class of problems, the most appropriate scientific computing algorithms are reviewed, their theoretical analyses are carried out and the expected results are verified on a MATLAB computer implementation. Every chapter is supplied with examples, exercises and applications of the discussed theory to the solution of real-life problems. This book is addressed to senior undergraduate and graduate students with particular focus on degree courses in Engineering, Mathematics, Physics and Computer Sciences. The attention which is paid to the applications and the related development of software makes it valuable also for researchers and users of scientific computing in a large variety of professional fields. In this second edition, the readability of pictures, tables and program headings have been improved. Several changes in the chapters on iterative methods and on polynomial approximation have also been added.

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Simone Brugiapaglia, Fabio Nobile

We present a theoretical analysis of the CORSING (COmpRessed SolvING) method for the numerical approximation of partial differential equations based on compressed sensing. In particular, we show that the best s-term approximation of the weak solution of a PDE with respect to an orthonormal system of N trial functions, can be recovered via a Petrov-Galerkin approach using m < N orthonormal test functions. This recovery is guaranteed if the local a-coherence associated with the bilinear form and the selected trial and test bases fulfills suitable decay properties. The fundamental tool of this analysis is the restricted infsup property, i.e., a combination of the classical inf-sup condition and the well-known restricted isometry property of compressed sensing.

This textbook is an introduction to Scientific Computing, in which several numerical methods for the computer solution of certain classes of mathematical problems are illustrated. The authors show how to compute the zeros or the integrals of continuous functions, solve linear systems, approximate functions by polynomials and construct accurate approximations for the solution of ordinary and partial differential equations. To make the presentation concrete and appealing, the programming environments Matlab and Octave, which is freely distributed, are adopted as faithful companions. The book contains the solutions to several problems posed in exercises and examples, often originating from specific applications. A specific section is devoted to subjects which were not addressed in the book and contains the bibliographical references for a more comprehensive treatment of the material. The second edition features many new problems and examples, as well as more numerical methods for linear and nonlinear systems and ordinary and partial differential equations. This book is presently being translated or has appeared in the following languages: Italian, German, French, Chinese and Spanish. Reviews for "Scientific Computing with MATLAB" - 1st edition: " ... Scientific Computing with MATLAB is written in a clear and concise style, figures, tables and formula boxes complement the explanations... The whole book is an invitation, if not a request, of the authors to the reader to play with MATLAB, apply its powerful menagerie of functions to solve the given (or own) problems - in brief, supervised learning by doing .... is a stimulating introductory textbook about numerical methods that successfully combines mathematical theory with programming experience..." Anselm A.C. Horn, Journal of Molecular Modeling 2004 "... An excellent addition to academic libraries and university bookstores, this book will be useful for self-study and as a complement to other MATLAB-based books. Highly recommended. Upper-division undergraduates through professionals." S.T. Karris, Choice 2003.

Simone Deparis, Stefano Pagani, Alfio Quarteroni, Riccardo Tenderini

In this work, we present a PDE-aware deep learning model for the numerical solution to the inverse problem of electrocardiography. The model both leverages data availability and exploits the knowledge of a physically based mathematical model, expressed by means of partial differential equations (PDEs), to carry out the task at hand. The goal is to estimate the epicardial potential field from measurements of the electric potential at a discrete set of points on the body surface. The employment of deep learning techniques in this context is made difficult by the low amount of clinical data at disposal, as measuring cardiac potentials requires invasive procedures. Suitably exploiting the underlying physically based mathematical model allowed circumventing the data availability issue and led to the development of fast-training and low-complexity models. Physical awareness has been pursued by means of two elements: the projection of the epicardial potential onto a space-time reduced subspace, spanned by the numerical solutions of the governing PDEs, and the inclusion of a tensorial reduced basis solver of the forward problem in the network architecture. Numerical tests have been conducted only on synthetic data, obtained via a full order model approximation of the problem at hand, and two variants of the model have been addressed. Both proved to be accurate, up to an average $\ell^1$-norm relative error on epicardial activation maps of 3.5%, and both could be trained in \approx$$15 min. Nevertheless, some improvements, mostly concerning data generation, are necessary in order to bridge the gap with clinical applications.

2022