Related papers: Estimating Discretization Error with Preset Orders…
Local grid refinement aims to optimise the relationship between accuracy of the results and number of grid nodes. In the context of the finite volume method no single local refinement criterion has been globally established as optimum for…
This paper builds on the algebraic theory in the companion paper [Algebraic Error Analysis for Mixed-Precision Multigrid Solvers] to obtain discretization-error-accurate solutions for linear elliptic partial differential equations (PDEs) by…
Wavelet-based grid adaptation methods use multiresolution analysis for error estimation, offering a mathematically rigorous approach to adaptive grid refinement when solving Partial Differential Equations (PDEs). However, applying these…
DEEPTHINK methods improve reasoning by generating, refining, and aggregating populations of candidate solutions, which enables strong performance on complex mathematical and scientific tasks. However, existing frameworks often lack reliable…
The issue of single-grid discretization error estimator, operating in the postprocessor mode, is addressed in the paper. An ensemble of numerical solutions, obtained using solvers of different accuracy, is shown to provide an upper estimate…
A genetic algorithm procedure is demonstrated that refines the selection of interpolation points of the discrete empirical interpolation method (DEIM) when used for constructing reduced order models for time dependent and/or parametrized…
An adaptive mesh refinement and error estimation method for numerically solving optimal control problems is developed using Legendre-Gauss-Radau direct collocation. In regions of the solution where the desired accuracy tolerance has not…
We introduce a novel Bayesian phase estimation technique based on adaptive grid refinement method. This method automatically chooses the number particles needed for accurate phase estimation using grid refinement and cell merging strategies…
When using the finite element method (FEM) in inverse problems, its discretization error can produce parameter estimates that are inaccurate and overconfident. The Bayesian finite element method (BFEM) provides a probabilistic model for the…
Efficient exploitation of exascale architectures requires rethinking of the numerical algorithms used in many large-scale applications. These architectures favor algorithms that expose ultra fine-grain parallelism and maximize the ratio of…
This paper discusses the error and cost aspects of ill-posed integral equations when given discrete noisy point evaluations on a fine grid. Standard solution methods usually employ discretization schemes that are directly induced by the…
Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error. Often, such systems arise from the discretization of operator…
This work introduces a new method to efficiently solve optimization problems constrained by partial differential equations (PDEs) with uncertain coefficients. The method leverages two sources of inexactness that trade accuracy for speed:…
The state-of-the art proof of a global inf-sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite, second-order linear elliptic PDEs. This paper, therefore, first analyses a nonconforming finite…
We present a hybrid a-priori/a-posteriori goal oriented error estimator for a combination of dynamic iteration-based solution of ordinary differential equations discretized by finite elements. Our novel error estimator combines estimates…
This paper studies the numerical approximation of parametric time-dependent partial differential equations (PDEs) by proper orthogonal decomposition reduced order models (POD-ROMs). Although many papers in the literature consider reduced…
Models incorporating uncertain inputs, such as random forces or material parameters, have been of increasing interest in PDE-constrained optimization. In this paper, we focus on the efficient numerical minimization of a convex and smooth…
Deep neural networks have been shown to be highly miscalibrated. often they tend to be overconfident in their predictions. It poses a significant challenge for safety-critical systems to utilise deep neural networks (DNNs), reliably. Many…
In this paper we present an algorithm for adaptive sparse grid approximations of quantities of interest computed from discretized partial differential equations. We use adjoint-based a posteriori error estimates of the physical…
Iterative refinement -- start with a random guess, then iteratively improve the guess -- is a useful paradigm for representation learning because it offers a way to break symmetries among equally plausible explanations for the data. This…