142 research outputs found
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Mini-Workshop: Numerical Analysis for Non-Smooth PDE-Constrained Optimal Control Problems
This mini-workshop brought together leading experts working on various aspects of numerical analysis for optimal control problems with nonsmoothness. Fifteen extended abstracts summarize the presentations at this mini-workshop
The conforming virtual element method for polyharmonic and elastodynamics problems: a review
In this paper, we review recent results on the conforming virtual element
approximation of polyharmonic and elastodynamics problems. The structure and
the content of this review is motivated by three paradigmatic examples of
applications: classical and anisotropic Cahn-Hilliard equation and phase field
models for brittle fracture, that are briefly discussed in the first part of
the paper. We present and discuss the mathematical details of the conforming
virtual element approximation of linear polyharmonic problems, the classical
Cahn-Hilliard equation and linear elastodynamics problems.Comment: 30 pages, 7 figures. arXiv admin note: text overlap with
arXiv:1912.0712
Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities
We prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs)
in for weighted analytic function classes in certain polytopal
domains , in space dimension . Functions in these classes are
locally analytic on open subdomains , but may exhibit isolated
point singularities in the interior of or corner and edge
singularities at the boundary . The exponential expression
rate bounds proved here imply uniform exponential expressivity by ReLU NNs of
solution families for several elliptic boundary and eigenvalue problems with
analytic data. The exponential approximation rates are shown to hold in space
dimension on Lipschitz polygons with straight sides, and in space
dimension on Fichera-type polyhedral domains with plane faces. The
constructive proofs indicate in particular that NN depth and size increase
poly-logarithmically with respect to the target NN approximation accuracy
in . The results cover in particular solution sets
of linear, second order elliptic PDEs with analytic data and certain nonlinear
elliptic eigenvalue problems with analytic nonlinearities and singular,
weighted analytic potentials as arise in electron structure models. In the
latter case, the functions correspond to electron densities that exhibit
isolated point singularities at the positions of the nuclei. Our findings
provide in particular mathematical foundation of recently reported, successful
uses of deep neural networks in variational electron structure algorithms.Comment: Found Comput Math (2022
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New Discretization Methods for the Numerical Approximation of PDEs
The construction and mathematical analysis of numerical methods for PDEs is a fundamental area of modern applied mathematics. Among the various techniques that have been proposed in the past, some – in particular, finite element methods, – have been exceptionally successful in a range of applications. There are however a number of important challenges that remain, including the optimal adaptive finite element approximation of solutions to transport-dominated diffusion problems, the efficient numerical approximation of parametrized families of PDEs, and the efficient numerical approximation of high-dimensional partial differential equations (that arise from stochastic analysis and statistical physics, for example, in the form of a backward Kolmogorov equation, which, unlike its formal adjoint, the forward Kolmogorov equation, is not in divergence form, and therefore not directly amenable to finite element approximation, even when the spatial dimension is low). In recent years several original and conceptionally new ideas have emerged in order to tackle these open problems.
The goal of this workshop was to discuss and compare a number of novel approaches, to study their potential and applicability, and to formulate the strategic goals and directions of research in this field for the next five years
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