We consider numerical methods for a “quasi-boundary value” regularization of the backward parabolic problem given by { ut + Au = 0 , 0 < t < T u(T ) = f, where A is positive self-adjoint and unbounded. The regularization, due to Clark and Oppenheimer, perturbs the final value u(T ) by adding αu(0), where α is a small parameter. We show how this leads very naturally to a reformulation of the pro...

In this paper we investigate the nature of the adapted solutions to a class of forward-backward stochastic diierential equations (SDEs for short) in which the forward equation is non-degenerate. We prove that in this case the adapted solution can always be sought in an \ordinary" sense over an arbitrarily prescribed time duration, via a direct \Four Step Scheme". Using this scheme, we further p...

The martingale representation theorem in a Brownian filtration represents any square integrable r.v. ξ as a stochastic integral with respect to the Brownian motion. This is the simplest Backward SDE with nul generator and final data ξ, which can be seen as the non-Markov counterpart of the Cauchy problem in second order parabolic PDEs. Similarly, the notion of Second order BSDEs is the non-Mark...

We discuss a quadratic criterion optimal control problem for stochastic linear system with delay in both state and control variables. This problem will lead to a kind of generalized forward-backward stochastic differential equations FBSDEs with Itô’s stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we p...

Abstract. In this paper, the author propose a numerical method to compute the solution of the Cauchy problem: wt − (w m wx)x = w , the initial condition is a nonnegative function with compact support, m > 0, 1 < p < m + 1. The problem is split in two parts: A hyperbolic term solved by using the Hopf and Lax formula and a parabolic term solved by a backward linearized Euler method in time and a ...

We study a discretization technique for the parabolic fractional obstacle problem in bounded domains. The fractional Laplacian is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic equation posed on a semi-infinite cylinder, which recasts our problem as a quasi-stationary elliptic variational inequality with a dynamic boundary condition. The rapid decay of the solution suggest...

In this paper, we consider the cascadic multigrid method for a parabolic type equation. Backward Euler approximation in time and linear finite element approximation in space are employed. A stability result is established under some conditions on the smoother. Using new and sharper estimates for the smoothers that reflect the precise dependence on the time step and the spatial mesh parameter, t...

Linear quasi-parabolic integro-differential equations and their control appear in many scientific problems and engineering applications such as biology mechanics, nuclear reaction dynamics, heat conduction in materials with memory, and visco-elasticity, etc.. The existence and uniqueness of the solution of the linear quasi-parabolic integro-differential equations have been studied by Wheeler M....

In this paper we studied the finite difference scheme for solution of one dimensional parabolic inverse problem finding function and unknown positive coefficient . The Backward time centered space (BTCS) which is unconditionally stable its convergent proved using application discrete maximum principle. Error estimates to give clear overview methodology several model problems are solved numerica...

In this article we construct and analyze multigrid preconditioners for discretizations of operators of the form Dλ + K K, where Dλ is the multiplication with a relatively smooth function λ > 0 and K is a compact linear operator. These systems arise when applying interior point methods to the minimization problem minu 1 2 (||Ku − f || + β||u||) with box-constraints u 6 u 6 u on the controls. The...