arXiv subjects

Miranda Boutilier

Publications and source records attributed to Miranda Boutilier.

4 recordsLinked to original sources

arXiv2024

Nonlinear Multilevel Solution Strategies for Diffusive Wave Flood Models in Perforated Domains

This article investigates the numerical solution of the Diffusive Wave equation posed on domains containing a large number of polygonal perforations, motivated by urban flood modeling. Such geometries induce strong multiscale effects driven by geometric complexity, which significantly challenge the robustness of standard nonlinear and linear solvers. The work builds on a multiscale coarse space previously introduced by the authors for linear Poisson problems on perforated domains. This low-dimensional space, constructed on a coarse polygonal partition and spanned by locally discrete harmonic (Trefftz-type) basis functions, is shown to remain effective for the linearized Diffusive Wave problems arising within Newton iterations. This enables the construction of robust two-level preconditioners for the resulting sequence of linear systems. Beyond linearization, the main focus of this work is on the effective solution of the fully nonlinear problem. We assess and combine several Schwarz-based nonlinear preconditioning strategies, including a two-level RASPEN method and a two-step nonlinear method, using the same multiscale coarse space to ensure scalability. While the individual components are drawn from the existing literature, their combination provides a robust and practical solution strategy for a challenging nonlinear problem posed on highly perforated domains. A systematic comparison of the methods and a discussion of algorithmic complexity are presented. The proposed approaches are validated through numerical experiments, including a realistic test case based on topographical data from the city of Nice.

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arXiv2024

Learning local Dirichlet-to-Neumann maps of nonlinear elliptic PDEs with rough coefficients

Partial differential equations (PDEs) involving high contrast and oscillating coefficients are common in scientific and industrial applications. Numerical approximation of these PDEs is a challenging task that can be addressed, for example, by multi-scale finite element analysis. For linear problems, multi-scale finite element method (MsFEM) is well established and some viable extensions to non-linear PDEs are known. However, some features of the method seem to be intrinsically based on linearity-based. In particular, traditional MsFEM rely on the reuse of computations. For example, the stiffness matrix can be calculated just once, while being used for several right-hand sides, or as part of a multi-level iterative algorithm. Roughly speaking, the offline phase of the method amounts to pre-assembling the local linear Dirichlet-to-Neumann (DtN) operators. We present some preliminary results concerning the combination of MsFEM with machine learning tools. The extension of MsFEM to nonlinear problems is achieved by means of learning local nonlinear DtN maps. The resulting learning-based multi-scale method is tested on a set of model nonlinear PDEs involving the $p-$Laplacian and degenerate nonlinear diffusion.

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arXiv2024

Robust Methods for Multiscale Coarse Approximations of Diffusion Models in Perforated Domains

For the Poisson equation posed in a domain containing a large number of polygonal perforations, we propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain. Similarly to other multiscale numerical methods, this coarse space is spanned by locally discrete harmonic basis functions. Along the subdomain boundaries, the basis functions are piecewise polynomial. The main contribution of this article is an error estimate regarding the H1-projection over the coarse space which depends only on the regularity of the solution over the edges of the coarse partitioning. For a specific edge refinement procedure, the error analysis establishes superconvergence of the method even if the true solution has a low general regularity. Combined with domain decomposition (DD) methods, the coarse space leads to an efficient two-level iterative linear solver which reaches the fine-scale finite element error in few iterations. It also bodes well as a preconditioner for Krylov methods and provides scalability with respect to the number of subdomains. Numerical experiments showcase the increased precision of the coarse approximation as well as the efficiency and scalability of the coarse space as a component of a DD algorithm.

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arXiv2023

A Trefftz-like coarse space for the two-level Schwarz method on perforated domains

We consider a new coarse space for the ASM and RAS preconditioners to solve elliptic partial differential equations on perforated domains, where the numerous polygonal perforations represent structures such as walls and buildings in urban data. With the eventual goal of modelling urban floods by means of the nonlinear Diffusive Wave equation, this contribution focuses on the solution of linear problems on perforated domains. Our coarse space uses a polygonal subdomain partitioning and is spanned by Trefftz-like basis functions that are piecewise linear on the boundary of a subdomain and harmonic inside it. It is based on nodal degrees of freedom that account for the intersection between the perforations and the subdomain boundaries. As a reference, we compare this coarse space to the well-studied Nicolaides coarse space with the same subdomain partitioning. It is known that the Nicolaides space is unable to prevent stagnation in convergence when the subdomains are not connected; we work around this issue by separating each subdomain by disconnected component. Scalability and robustness are tested for multiple data sets based on realistic urban topography. Numerical results show that the new coarse space is very robust and accelerates the number of Krylov iterations when compared to Nicolaides, independent of the complexity of the data.

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