In this note we derive the backward (automatic) differentiation (adjoint [automatic] differentiation) for an algorithm containing a conditional expectation operator. As an example we consider the backward algorithm as it is used in Bermudan product valuation, but the method is applicable in full generality. The method relies on three simple properties: 1. a forward or backward (automatic) diffe...

In this paper, we consider the construction of a new class of numerical methods based on the backward differentiation formulas (BDFs) that be equipped by including two off--step points. We represent these methods from general linear methods (GLMs) point of view which provides an easy process to improve their stability properties and implementation in a variable stepsize mode. These superioritie...

The backward differentiation formula (BDF) is a useful family of implicit methods for the numerical integration stiff differential equations. It well noticed that stability and convergence $A$-stable BDF1 BDF2 schemes parabolic equations can be directly established by using standard discrete energy analysis. However, such classical analysis technique seems not applicable to BDF-$\mathbf{k}$ $3\...

In this paper, we propose and analyze a linear second-order numerical method for solving the Allen-Cahn equation with general mobility. The proposed fully-discrete scheme is carefully constructed based on combination of first backward differentiation formulas nonuniform time steps temporal approximation central finite difference spatial discretization. discrete maximum bound principle proved by...