1,622 research outputs found
Hybrid PDE solver for data-driven problems and modern branching
The numerical solution of large-scale PDEs, such as those occurring in
data-driven applications, unavoidably require powerful parallel computers and
tailored parallel algorithms to make the best possible use of them. In fact,
considerations about the parallelization and scalability of realistic problems
are often critical enough to warrant acknowledgement in the modelling phase.
The purpose of this paper is to spread awareness of the Probabilistic Domain
Decomposition (PDD) method, a fresh approach to the parallelization of PDEs
with excellent scalability properties. The idea exploits the stochastic
representation of the PDE and its approximation via Monte Carlo in combination
with deterministic high-performance PDE solvers. We describe the ingredients of
PDD and its applicability in the scope of data science. In particular, we
highlight recent advances in stochastic representations for nonlinear PDEs
using branching diffusions, which have significantly broadened the scope of
PDD.
We envision this work as a dictionary giving large-scale PDE practitioners
references on the very latest algorithms and techniques of a non-standard, yet
highly parallelizable, methodology at the interface of deterministic and
probabilistic numerical methods. We close this work with an invitation to the
fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European
Journal of Applied Mathematics (EJAM
The Vanishing Moment Method for Fully Nonlinear Second Order Partial Differential Equations: Formulation, Theory, and Numerical Analysis
The vanishing moment method was introduced by the authors in [37] as a
reliable methodology for computing viscosity solutions of fully nonlinear
second order partial differential equations (PDEs), in particular, using
Galerkin-type numerical methods such as finite element methods, spectral
methods, and discontinuous Galerkin methods, a task which has not been
practicable in the past. The crux of the vanishing moment method is the simple
idea of approximating a fully nonlinear second order PDE by a family
(parametrized by a small parameter \vepsi) of quasilinear higher order (in
particular, fourth order) PDEs. The primary objectives of this book are to
present a detailed convergent analysis for the method in the radial symmetric
case and to carry out a comprehensive finite element numerical analysis for the
vanishing moment equations (i.e., the regularized fourth order PDEs). Abstract
methodological and convergence analysis frameworks of conforming finite element
methods and mixed finite element methods are first developed for fully
nonlinear second order PDEs in general settings. The abstract frameworks are
then applied to three prototypical nonlinear equations, namely, the
Monge-Amp\`ere equation, the equation of prescribed Gauss curvature, and the
infinity-Laplacian equation. Numerical experiments are also presented for each
problem to validate the theoretical error estimate results and to gauge the
efficiency of the proposed numerical methods and the vanishing moment
methodology.Comment: 141 pages, 16 figure
A Neumann interface optimal control problem with elliptic PDE constraints and its discretization and numerical analysis
We study an optimal control problem governed by elliptic PDEs with interface,
which the control acts on the interface. Due to the jump of the coefficient
across the interface and the control acting on the interface, the regularity of
solution of the control problem is limited on the whole domain, but smoother on
subdomains. The control function with pointwise inequality constraints is
served as the flux jump condition which we called Neumann interface control. We
use a simple uniform mesh that is independent of the interface. The standard
linear finite element method can not achieve optimal convergence when the
uniform mesh is used. Therefore the state and adjoint state equations are
discretized by piecewise linear immersed finite element method (IFEM). While
the accuracy of the piecewise constant approximation of the optimal control on
the interface is improved by a postprocessing step which possesses
superconvergence properties; as well as the variational discretization concept
for the optimal control is used to improve the error estimates. Optimal error
estimates for the control, suboptimal error estimates for state and adjoint
state are derived. Numerical examples with and without constraints are provided
to illustrate the effectiveness of the proposed scheme and correctness of the
theoretical analysis.Comment: 31pages, 12 figures, 4 table
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