相关论文: Dirichlet forms methods, an application to the pro…

In this work, we present an abstract error analysis framework for the approximation of linear partial differential equation (PDE) problems in weak formulation. We consider approximation methods in fully discrete formulation, where the…

In this article, we present a solution to the problem: "Which type of linear operators can be realized by the Dirichlet-to-Neumann operator associated with the operator $-\Delta-a(z)\frac{\partial^{2}}{\partial z^2}$ on an extension…

Sharp $L^\infty$ estimates are obtained for general classes of fully non-linear PDE's on non-K\"ahler manifolds, complementing the theory developed earlier by the authors in joint work with F. Tong for the K\"ahler case. The key idea is…

We investigate error of the Euler scheme in the case when the right-hand side function of the underlying ODE satisfies nonstandard assumptions such as local one-sided Lipschitz condition and local H\"older continuity. Moreover, we assume…

Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method…

We examine numerical rounding errors of some deterministic solvers for systems of ordinary differential equations (ODEs). We show that the accumulation of rounding errors results in a solution that is inherently random and we obtain the…

We propose a simpler derivation of the probability density function of Feller Diffusion using the Fourier Transform and solving the resulting equation via the Method of Characteristics. We also discuss simulation algorithms and confirm key…

Diffusion models are recent state-of-the-art methods for image generation and likelihood estimation. In this work, we generalize continuous-time diffusion models to arbitrary Riemannian manifolds and derive a variational framework for…

This paper focuses on the construction and analysis of explicit numerical methods of high dimensional stochastic nonlinear Schrodinger equations (SNLSEs). We first prove that the classical explicit numerical methods are unstable and suffer…

Employing a limiting case of a conjecture for constructing piecewise separable-variables functions, the elements of the Pseudoanalytic Function Theory are used for numerically approaching solutions of the forward Dirichlet boundary value…

We study a class of stochastic semilinear damped wave equations driven by additive Wiener noise. Owing to the damping term, under appropriate conditions on the nonlinearity, the solution admits a unique invariant distribution. We apply…

The Schauder estimates are among the oldest and most useful tools in the modern theory of elliptic partial differential equations (PDEs). Their influence may be felt in practically all applications of the theory of elliptic boundary-value…

Dirichlet distributions are probability measures on the unit simplex. They are often used as prior distributions in modeling categorical data, such as in topic analysis of text data. Motivated by this application, we consider Monte Carlo…

Approximation in measure is employed to solve an asymptotic Dirichlet problem on arbitrary open sets and to show that many functions, including the Riemann zeta-function, are universal in measure. Connections with the Riemann Hypothesis are…

We present an implicit Split-Step explicit Euler type Method (dubbed SSM) for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of superlinear growth in space, Lipschitz in measure and non-constant…

A new symbolic algorithmic implementation of the general scheme of the exponentially convergent functional-discrete (FD-) method is developed and justified for the Sturm-Liouville problem on a finite interval for the Schr\"odinger equation…

We investigate discretizations of the integrable discrete nonlinear Schr\"odinger dynamical system and related symplectic structures. We develop an effective scheme of invariant reducing the corresponding infinite system of ordinary…

By generalizing the classical Selberg-Chowla formula, we establish the analytic continuation and functional equation for a large class of Epstein zeta functions. This continuation is studied in order to provide new classes of theorems…

General stochastic Euler schemes for ordinary differential equations are studied. We give proofs on the consistency, the rate of convergence and the asymptotic normality of these procedures.

This paper focuses on analyzing the error of the randomized Euler algorithm when only noisy information about the coefficients of the underlying stochastic differential equation (SDE) and the driving Wiener process is available. Two classes…