337 research outputs found
Linear convergence of accelerated conditional gradient algorithms in spaces of measures
A class of generalized conditional gradient algorithms for the solution of
optimization problem in spaces of Radon measures is presented. The method
iteratively inserts additional Dirac-delta functions and optimizes the
corresponding coefficients. Under general assumptions, a sub-linear
rate in the objective functional is obtained, which is sharp
in most cases. To improve efficiency, one can fully resolve the
finite-dimensional subproblems occurring in each iteration of the method. We
provide an analysis for the resulting procedure: under a structural assumption
on the optimal solution, a linear convergence rate is
obtained locally.Comment: 30 pages, 7 figure
Stochastic collocation on unstructured multivariate meshes
Collocation has become a standard tool for approximation of parameterized
systems in the uncertainty quantification (UQ) community. Techniques for
least-squares regularization, compressive sampling recovery, and interpolatory
reconstruction are becoming standard tools used in a variety of applications.
Selection of a collocation mesh is frequently a challenge, but methods that
construct geometrically "unstructured" collocation meshes have shown great
potential due to attractive theoretical properties and direct, simple
generation and implementation. We investigate properties of these meshes,
presenting stability and accuracy results that can be used as guides for
generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure
Compressive Fourier collocation methods for high-dimensional diffusion equations with periodic boundary conditions
High-dimensional Partial Differential Equations (PDEs) are a popular
mathematical modelling tool, with applications ranging from finance to
computational chemistry. However, standard numerical techniques for solving
these PDEs are typically affected by the curse of dimensionality. In this work,
we tackle this challenge while focusing on stationary diffusion equations
defined over a high-dimensional domain with periodic boundary conditions.
Inspired by recent progress in sparse function approximation in high
dimensions, we propose a new method called compressive Fourier collocation.
Combining ideas from compressive sensing and spectral collocation, our method
replaces the use of structured collocation grids with Monte Carlo sampling and
employs sparse recovery techniques, such as orthogonal matching pursuit and
minimization, to approximate the Fourier coefficients of the PDE
solution. We conduct a rigorous theoretical analysis showing that the
approximation error of the proposed method is comparable with the best -term
approximation (with respect to the Fourier basis) to the solution. Using the
recently introduced framework of random sampling in bounded Riesz systems, our
analysis shows that the compressive Fourier collocation method mitigates the
curse of dimensionality with respect to the number of collocation points under
sufficient conditions on the regularity of the diffusion coefficient. We also
present numerical experiments that illustrate the accuracy and stability of the
method for the approximation of sparse and compressible solutions.Comment: 33 pages, 9 figure
Positive solutions of nonlinear elliptic boundary value problems
This dissertation focuses on the study of positive steady states to classes of nonlinear reaction diffusion (elliptic) systems on bounded domains as well as on exterior domains with Dirichlet boundary conditions. In particular, we study such systems in the challenging case when the reaction terms are negative at the origin, referred in the literature as semipositone problems. For the last 30 years, study of elliptic partial differential equations with semipositone structure has flourished not only for the semilinear case but also for quasilinear case. Here we establish several results that directly contribute to and enhance the literature of semipositone problems. In particular, we discuss existence, non-existence and multiplicity results for classes of superlinear as well as sublinear systems. We establish our results via the method of sub-super solutions, degree theory arguments, a priori bounds and energy analysis
Neural PDE Solvers for Irregular Domains
Neural network-based approaches for solving partial differential equations
(PDEs) have recently received special attention. However, the large majority of
neural PDE solvers only apply to rectilinear domains, and do not systematically
address the imposition of Dirichlet/Neumann boundary conditions over irregular
domain boundaries. In this paper, we present a framework to neurally solve
partial differential equations over domains with irregularly shaped
(non-rectilinear) geometric boundaries. Our network takes in the shape of the
domain as an input (represented using an unstructured point cloud, or any other
parametric representation such as Non-Uniform Rational B-Splines) and is able
to generalize to novel (unseen) irregular domains; the key technical ingredient
to realizing this model is a novel approach for identifying the interior and
exterior of the computational grid in a differentiable manner. We also perform
a careful error analysis which reveals theoretical insights into several
sources of error incurred in the model-building process. Finally, we showcase a
wide variety of applications, along with favorable comparisons with ground
truth solutions
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