The role of interface boundary conditions and sampling strategies for Schwarz-based coupling of projection-based reduced order models

• URL: http://arxiv.org/abs/2410.04668v1
• Date: Mon, 7 Oct 2024 00:44:22 GMT
• Title: The role of interface boundary conditions and sampling strategies for Schwarz-based coupling of projection-based reduced order models
• Authors: Christopher R. Wentland, Francesco Rizzi, Joshua Barnett, Irina Tezaur,
• Abstract summary: We present a framework for the coupling of subdomain-local projection-based reduced order models (PROMs) using the Schwarz alternating method. We show that it is possible to obtain a stable and accurate coupled model utilizing Dirichlet-Dirichlet (rather than Robin-Robin or alternating Dirichlet-Neumann) transmission BCs on the subdomain boundaries. Our numerical results suggest that the proposed methodology has the potential to improve PROM accuracy.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper presents and evaluates a framework for the coupling of subdomain-local projection-based reduced order models (PROMs) using the Schwarz alternating method following a domain decomposition (DD) of the spatial domain on which a given problem of interest is posed. In this approach, the solution on the full domain is obtained via an iterative process in which a sequence of subdomain-local problems are solved, with information propagating between subdomains through transmission boundary conditions (BCs). We explore several new directions involving the Schwarz alternating method aimed at maximizing the method's efficiency and flexibility, and demonstrate it on three challenging two-dimensional nonlinear hyperbolic problems: the shallow water equations, Burgers' equation, and the compressible Euler equations. We demonstrate that, for a cell-centered finite volume discretization and a non-overlapping DD, it is possible to obtain a stable and accurate coupled model utilizing Dirichlet-Dirichlet (rather than Robin-Robin or alternating Dirichlet-Neumann) transmission BCs on the subdomain boundaries. We additionally explore the impact of boundary sampling when utilizing the Schwarz alternating method to couple subdomain-local hyper-reduced PROMs. Our numerical results suggest that the proposed methodology has the potential to improve PROM accuracy by enabling the spatial localization of these models via domain decomposition, and achieve up to two orders of magnitude speedup over equivalent coupled full order model solutions and moderate speedups over analogous monolithic solutions.

Related papers

• Non-overlapping, Schwarz-type Domain Decomposition Method for Physics and Equality Constrained Artificial Neural Networks [0.24578723416255746]
  We present a non-overlapping, Schwarz-type domain decomposition method with a generalized interface condition. Our approach employs physics and equality-constrained artificial neural networks (PECANN) within each subdomain. A distinct advantage our domain decomposition method is its ability to learn solutions to both Poisson's and Helmholtz equations.
  arXiv  Detail & Related papers  (2024-09-20T16:48:55Z)
• Domain Decomposition-based coupling of Operator Inference reduced order models via the Schwarz alternating method [0.4473915603131591]
  We present an approach for coupling together subdomain-local reduced order models (ROMs) with each other and with subdomain-local full order models (FOMs) We demonstrate that the method is capable of coupling together arbitrary combinations of OpInf ROMs and FOMs, and that speed-ups over a monolithic FOM are possible when performing OpInf ROM coupling.
  arXiv  Detail & Related papers  (2024-09-02T19:20:26Z)
• Total Uncertainty Quantification in Inverse PDE Solutions Obtained with Reduced-Order Deep Learning Surrogate Models [50.90868087591973]
  We propose an approximate Bayesian method for quantifying the total uncertainty in inverse PDE solutions obtained with machine learning surrogate models. We test the proposed framework by comparing it with the iterative ensemble smoother and deep ensembling methods for a non-linear diffusion equation.
  arXiv  Detail & Related papers  (2024-08-20T19:06:02Z)
• A Generalized Schwarz-type Non-overlapping Domain Decomposition Method using Physics-constrained Neural Networks [0.9137554315375919]
  We present a meshless Schwarz-type non-overlapping domain decomposition based on artificial neural networks. Our method is applicable to both the Laplace's and Helmholtz equations.
  arXiv  Detail & Related papers  (2023-07-23T21:18:04Z)
• Decomposed Diffusion Sampler for Accelerating Large-Scale Inverse Problems [64.29491112653905]
  We propose a novel and efficient diffusion sampling strategy that synergistically combines the diffusion sampling and Krylov subspace methods. Specifically, we prove that if tangent space at a denoised sample by Tweedie's formula forms a Krylov subspace, then the CG with the denoised data ensures the data consistency update to remain in the tangent space. Our proposed method achieves more than 80 times faster inference time than the previous state-of-the-art method.
  arXiv  Detail & Related papers  (2023-03-10T07:42:49Z)
• Score-based Diffusion Models in Function Space [140.792362459734]
  Diffusion models have recently emerged as a powerful framework for generative modeling. We introduce a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space. We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• A domain-decomposed VAE method for Bayesian inverse problems [0.0]
  This paper proposes a domain-decomposed variational auto-encoder Markov chain Monte Carlo (DD-VAE-MCMC) method to tackle these challenges simultaneously. The proposed method first constructs local deterministic generative models based on local historical data, which provide efficient local prior representations.
  arXiv  Detail & Related papers  (2023-01-09T07:35:43Z)
• Towards a machine learning pipeline in reduced order modelling for inverse problems: neural networks for boundary parametrization, dimensionality reduction and solution manifold approximation [0.0]
  Inverse problems, especially in a partial differential equation context, require a huge computational load. We apply a numerical pipeline that involves artificial neural networks to parametrize the boundary conditions of the problem in hand. It derives a general framework capable to provide an ad-hoc parametrization of the inlet boundary and quickly converges to the optimal solution.
  arXiv  Detail & Related papers  (2022-10-26T14:53:07Z)
• An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equations [68.8204255655161]
  Filtering equations play a central role in many real-life applications, including numerical weather prediction, finance and engineering. One of the classical approaches to approximate the solution of the filtering equations is to use a PDE inspired method, called the splitting-up method. We combine this method with a neural network representation to produce an approximation of the unnormalised conditional distribution of the signal process.
  arXiv  Detail & Related papers  (2022-01-10T11:01:36Z)
• Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems [91.3755431537592]
  We consider a control system of the form $dot x = sum_i=1lF_i(x)u_i$, with linear dependence in the controls. We use the corresponding flow to approximate the action of a diffeomorphism on a compact ensemble of points.
  arXiv  Detail & Related papers  (2021-10-24T08:57:46Z)
• Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable Approach for Continuous Markov Random Fields [53.31927549039624]
  We show that a piecewise discretization preserves better contrast from existing discretization problems. We apply this theory to the problem of matching two images.
  arXiv  Detail & Related papers  (2021-07-13T12:31:06Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.