73 research outputs found
A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations
In this paper the numerical solution of non-autonomous semilinear stochastic
evolution equations driven by an additive Wiener noise is investigated. We
introduce a novel fully discrete numerical approximation that combines a
standard Galerkin finite element method with a randomized Runge-Kutta scheme.
Convergence of the method to the mild solution is proven with respect to the
-norm, . We obtain the same temporal order of
convergence as for Milstein-Galerkin finite element methods but without
imposing any differentiability condition on the nonlinearity. The results are
extended to also incorporate a spectral approximation of the driving Wiener
process. An application to a stochastic partial differential equation is
discussed and illustrated through a numerical experiment.Comment: 31 pages, 1 figur
On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients
In this paper we introduce a randomized version of the backward Euler method,
that is applicable to stiff ordinary differential equations and nonlinear
evolution equations with time-irregular coefficients. In the finite-dimensional
case, we consider Carath\'eodory type functions satisfying a one-sided
Lipschitz condition. After investigating the well-posedness and the stability
properties of the randomized scheme, we prove the convergence to the exact
solution with a rate of in the root-mean-square norm assuming only that
the coefficient function is square integrable with respect to the temporal
parameter.
These results are then extended to the numerical solution of
infinite-dimensional evolution equations under monotonicity and Lipschitz
conditions. Here we consider a combination of the randomized backward Euler
scheme with a Galerkin finite element method. We obtain error estimates that
correspond to the regularity of the exact solution. The practicability of the
randomized scheme is also illustrated through several numerical experiments.Comment: 37 pages, 3 figure
The Galerkin Analysis for the Random Periodic Solution of Semilinear Stochastic Evolution Equations
In this paper, we study the numerical method for approximating the random periodic solution of semilinear stochastic evolution equations. The main challenge lies in proving a convergence over an infinite time horizon while simulating infinite-dimensional objects. We first show the existence and uniqueness of the random periodic solution to the equation as the limit of the pull-back flows of the equation, and observe that its mild form is well defined in the intersection of a family of decreasing Hilbert spaces. Then, we propose a Galerkin-type exponential integrator scheme and establish its convergence rate of the strong error to the mild solution, where the order of convergence directly depends on the space (among the family of Hilbert spaces) for the initial point to live. We finally conclude with a best order of convergence that is arbitrarily close to 0.5
Nonlinear Evolution Equations: Analysis and Numerics
The qualitative theory of nonlinear evolution equations is an
important tool for studying the dynamical behavior of systems in
science and technology. A thorough understanding of the complex
behavior of such systems requires detailed analytical and numerical
investigations of the underlying partial differential equations
The Galerkin analysis for the random periodic solution of semilinear stochastic evolution equations
In this paper we study the numerical method for approximating the random periodic solution of semilinear stochastic evolution equations. The main challenge lies in proving a convergence over an infinite time horizon while simulating infinite-dimensional objects. We first show the existence and uniqueness of the random periodic solution to the equation as the limit of the pull-back flows of the equation, and observe that its mild form is well-defined in the intersection of a family of decreasing Hilbert spaces. Then we propose a Galerkin-type exponential integrator scheme and establish its convergence rate of the strong error to the mild solution, where the order of convergence directly depends on the space (among the family of Hilbert spaces) for the initial point to live. We finally conclude with a best order of convergence that is arbitrarily close to 0.5
EQUADIFF 15
Equadiff 15 – Conference on Differential Equations and Their Applications – is an international conference in the world famous series Equadiff running since 70 years ago. This booklet contains conference materials related with the 15th Equadiff conference in the Czech and Slovak series, which was held in Brno in July 2022. It includes also a brief history of the East and West branches of Equadiff, abstracts of the plenary and invited talks, a detailed program of the conference, the list of participants, and portraits of four Czech and Slovak outstanding mathematicians
Stochastic Proximal Gradient Methods for Nonconvex Problems in Hilbert Spaces
For finite-dimensional problems, stochastic approximation methods have long
been used to solve stochastic optimization problems. Their application to
infinite-dimensional problems is less understood, particularly for nonconvex
objectives. This paper presents convergence results for the stochastic proximal
gradient method applied to Hilbert spaces, motivated by optimization problems
with partial differential equation (PDE) constraints with random inputs and
coefficients. We study stochastic algorithms for nonconvex and nonsmooth
problems, where the nonsmooth part is convex and the nonconvex part is the
expectation, which is assumed to have a Lipschitz continuous gradient. The
optimization variable is an element of a Hilbert space. We show almost sure
convergence of strong limit points of the random sequence generated by the
algorithm to stationary points. We demonstrate the stochastic proximal gradient
algorithm on a tracking-type functional with a -penalty term constrained
by a semilinear PDE and box constraints, where input terms and coefficients are
subject to uncertainty. We verify conditions for ensuring convergence of the
algorithm and show a simulation
Snapshot-Based Methods and Algorithms
An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science
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