A new ParaDiag time-parallel time integration method

Time-parallel time integration has received a lot of attention in the high performance computing community over the past two decades. Indeed, it has been shown that parallel-in-time techniques have the potential to remedy one of the main computational drawbacks of parallel-in-space solvers. In particular, it is well-known that for large-scale evolution problems space parallelization saturates long before all processing cores are effectively used on today's large scale parallel computers. Among the many approaches for time-parallel time integration, ParaDiag schemes have proved themselves to be a very effective approach. In this framework, the time stepping matrix or an approximation thereof is diagonalized by Fourier techniques, so that computations taking place at different time steps can be indeed carried out in parallel. We propose here a new ParaDiag algorithm combining the Sherman-Morrison-Woodbury formula and Krylov techniques. A panel of diverse numerical examples illustrates the potential of our new solver. In particular, we show that it performs very well compared to different ParaDiag algorithms recently proposed in the literature

Resolução da equação da onda utilizando métodos multigrid espaço-tempo

Orientador: Prof. Dr. Marcio Augusto Villela PintoCoorientador: Prof. Dr. Sebastião Romero FrancoTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Métodos Numéricos em Engenharia. Defesa : Curitiba, 10/04/2023Inclui referências: p. 109-117Resumo: Neste trabalho apresenta-se a avaliação de diferentes formas de solução para problemas modelados pela equação da onda, para os casos 1D e 2D. Utiliza-se para discretização espacial, o Método das Diferenças Finitas ponderado por um parâmetro n em diferentes estágios de tempo, para obter-se um esquema de solução implícito. Com isso, propõe-se a utilização de diferentes varreduras no tempo, a fim de gerar métodos robustos e eficientes, desde a clássica Time-Stepping, até outra varredura que envolve simultaneamente o espaço e o tempo, como Waveform Relaxation. Neste trabalho, combina-se o método dos Subdomínios com a estratégia Waveform Relaxation para reduzir as fortes oscilações que ocorrem o início do processo iterativo. Obtém-se excelentes resultados ao aplicar o método Multigrid para esta classe de problemas, já que, melhora-se muito os fatores de convergência calculados a partir das soluções aproximadas do sistema de equações resultante das discretizações. Na verificação das metodologias propostas e suas características, apresentamse simulações de propagação de ondas envolvendo problemas uni e bidimensionais, onde analisa-se os erros de discretização, ordens efetiva e aparente, fator de convergência, ordens de complexidade e tempo computacional.Abstract: In this thesis presents the evaluation of different forms of solution for probelms modeled by the wave equation, for the 1D and 2D cases. The Finite Difference Method is used for the spatial discretization, weighted by a parameter n at different time steps, in order to obtain an implicit solution. With this, it is proposed the use of different sweeps in time, in order to generate robust and efficient methods, from the classical Time-Stepping, to other less usual sweep as Waveform Relaxation. In this work, the Subdomains method is combined with the Waveform Relaxation strategy to reduce the strong oscillations that occur early in the iterative process. Excellent results are obtained when applying the Multigrid method for this class of problems, since the convergence factors calculated from the approximate solutions of the system of equations resulting from the discretizations are greatly improved. In the verification of the proposed methodologies and their respective advantages, simulations of wave propagation involving one- and two-dimensional problems are presented, where the discretization errors, effective and apparent orders, convergence factor, complexity orders and computational time are analyzed

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal