The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in...

In this paper we propose a new algorithm for the well known elliptic obstacle problem and for parabolic variational inequalities like oneand two-phase Stefan problem and of obstacle type. Our approach enters the category of xed domain methods and solves just linear elliptic or parabolic equations and their discretization at each iteration. We prove stability and convergence properties. The appr...

In this paper we investigate results on finite-time blow-up of solutions to a nonlocal quasi-linear parabolic problem. We extend some known results to higher dimensions. AMS subject classification: 35K55, 35K65, 35B35.

This paper is concerned with the state-constrained optimal control of the twodimensional thermistor problem, a quasi-linear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Existence, uniqueness and continuity for the state system are derived by employing maximal elliptic and parabo...

The nonlocal boundary value problem for the parabolic differential equation v'(t) + A(t)v(t) = f(t) (0 ≤ t ≤ T), v(0) = v(λ) + φ, 0 < λ ≤ T in an arbitrary Banach space E with the dependent linear positive operator A(t) is investigated. The well-posedness of this problem is established in Banach spaces C 0 (β,γ) (E α-β ) of all E α-β -valued continuous functions φ(t) on [0, T] satisfying a Höld...

Recently, Higham and Wald en, Karlson, and Sun have provided formulas for computing the best backward perturbation bounds for the linear least squares problem. In this paper we provide several backward perturbation bounds that are easier to compute and optimal up to a factor less than 2. We also show that any least squares algorithm that is stable in the sense of Stewart is necessarily a backwa...

In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasi-linear parabolic partial (PDEs), which is related to FBSDEs from Pardoux-Peng theory. The rely on a process by minimizing path-wise difference between two discrete processes, are defined time discretization DNN r...