Related papers: Cubature Methods For Stochastic (Partial) Differen…
In this paper we are concerned with restricted additive Schwarz with local impedance transformation conditions for a family of Helmholtz problems in two dimensions. These problems are discretized by the finite element method with conforming…
In the present work, strong approximation errors are analyzed for both the spatial semi-discretization and the spatio-temporal fully discretization of stochastic wave equations (SWEs) with cubic polynomial nonlinearities and additive…
High dimensional integrals can be approximated well by quasi-Monte Carlo methods. However, determining the number of function values needed to obtain the desired accuracy is difficult without some upper bound on an appropriate semi-norm of…
This paper addresses uncertainty quantification (UQ) for problems where scalar (or low-dimensional vector) response quantities are insufficient and, instead, full-field (very high-dimensional) responses are of interest. To do so, an…
In this paper invariant subspace method has been employed for solving linear and non-linear fractional partial differential equations involving Caputo derivative. A variety of illustrative examples are solved to demonstrate the…
We propose, analyze, and implement interpolatory approximations and Filon-type cubature for efficient and accurate evaluation of a class of wideband generalized Fourier integrals on the sphere. The analysis includes derivation of (i)…
In this paper, authors successfully construct a new algorithm for the new higher order scheme of weak approximation of SDEs. The algorithm presented here is based on [1][2]. Although this algorithm shares some features with the algorithm…
In this paper, we employ a Schauder-type estimate method, as developed in \cite{CHN}, to establish critical well-posedness result for the Fractional Fokker-Planck Equation. This equation serves as a fundamental model in kinetic theory and…
We introduce and analyze a numerical approximation of the porous medium equation with fractional potential pressure introduced by Caffarelli and V\'azquez: \[ \partial_t u = \nabla \cdot (u^{m-1}\nabla (-\Delta)^{-\sigma}u) \qquad…
We develop algorithms for multivariate integration and approximation in the weighted half-period cosine space of smooth non-periodic functions. We use specially constructed tent-transformed rank-1 lattice points as cubature nodes for…
A convergence theorem for the continuous weak approximation of the solution of stochastic differential equations by general one step methods is proved, which is an extension of a theorem due to Milstein. As an application, uniform second…
Cubature formulas, asymptotically optimal with respect to accuracy, are derived for calculating multidimensional weakly singular integrals. They are used for developing a universal code for calculating capacitances of conductors of…
We consider a fully discrete scheme for nonlinear stochastic partial differential equations with non-globally Lipschitz coefficients driven by multiplicative noise in a multi-dimensional setting. Our method uses a polynomial based spectral…
The paper is devoted to the efficient computation of high-order cubature formulas for volume potentials obtained within the framework of approximate approximations. We combine this approach with modern methods of structured tensor product…
In this paper, we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations (PDEs) with uncertainties. The new approach is realized in the semi-discrete finite-volume framework and is based…
The coefficients in a second order parabolic linear stochastic partial differential equation (SPDE) are estimated from multiple spatially localised measurements. Assuming that the spatial resolution tends to zero and the number of…
We prove a weak error estimate for the approximation in space and time of a semilinear stochastic Volterra integro-differential equation driven by additive space-time Gaussian noise. We treat this equation in an abstract framework, in which…
This paper establishes a comprehensive well-posedness and regularity theory for time-fractional stochastic partial differential equations on $\mathbb{R}^d$ driven by mixed Wiener--L\'evy noises. The equations feature a Caputo time…
In this paper, we deal with several aspects of the universal Frolov cubature method, that is known to achieve optimal asymptotic convergence rates in a broad range of function spaces. Even though every admissible lattice has this favorable…
In this note we provide conditions for local invariance of finite dimensional submanifolds for solutions to stochastic partial differential equations (SPDEs) in the framework of the variational approach. For this purpose, we provide a…