We introduce and study parallel space-time domain decomposition methods for solving deterministic and stochastic parabolic equations. Traditional parallel algorithms solve parabolic problems time step by time step. The parallelism is restricted to each time step, and the algorithms are purely sequential in time. In this paper, we develop some overlapping Schwarz methods whose subdomains cover b...

The twistor transform of a parabolic geometry has two steps: lift up to a geometry of higher dimension, and then descend to a geometry of lower dimension. The first step is a functor, but the second requires some compatibility conditions. Local necessary conditions were uncovered by Andreas Čap [12]. We uncover necessary and sufficient global conditions for complex analytic geometries: rational...

Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming Finite Element, Mixed Finite Element and Finite Volume methods. We show here that this framework can be applied to a family of degenerate non-linear parabolic equations (which contain in particular the Ri...

    In this paper, a variational iteration method (VIM), which is a well-known method for solving nonlinear equations, has been employed to solve an inverse parabolic partial differential equation. Inverse problems in partial differential equations can be used to model many real problems in engineering and other physical sciences. The VIM is to construct correction functional using general Lagr...

The nonlocal boundary value problems for differential operator equations of second order with dependent coefficients are studied. The principal parts of the differential operators generated by these problems are non-selfadjoint. Several conditions for the maximal regularity and the Fredholmness in Banach-valued Lp-spaces of these problems are given. By using these results, the maximal regularit...

We prove comparison results between viscosity sub and supersolutions of degenerate elliptic and parabolic equations associated to, possibly non-linear, Neumann boundary conditions. These results are obtained under more general assumptions on the equation (in particular the dependence in the gradient of the solution) and they allow applications to quasilinear, possibly singular, elliptic or para...

In this paper we explain the notion of stochastic backward differential equations and its relationship with classical (backward) parabolic differential equations of second order. The paper contains a mixture of stochastic processes like Markov processes and martingale theory and semi-linear partial differential equations of parabolic type. Some emphasis is put on the fact that the whole theory ...

We consider the perturbation of parabolic operators of the form ∂t + P (x,D) by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order. We show that, when the dimension is smaller than the order of the elliptic pseu...

It is shown how one can get upper bounds for ju ? vj when u and v are the (viscosity) solutions of ut ? (Dxu))xu = 0 and vt ? (Dxv))xv = 0; respectively, in (0;1) with Dirichlet boundary conditions. Similar results are obtained for some other parabolic equations as well, including certain equations in divergence form. 1. Introduction In this paper we study the problem of how to estimate the dii...