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Max-plus methods have previously been used to solve deterministic control problems. The methods are based on max-plus (or min-plus) expansions and can yield curse-of-dimensionalityfree numerical methods. In this paper, we explore min-plus methods for continuous-time stochastic control on a finite-time horizon. We first approximate the original value function via timediscretization. By generalizing the min-plus distributive property to continuum spaces, we obtain an algorithm for recursive computation of the time-discretized values, which we refer to as the idempotent distributed dynamic programming principle (IDDPP). Under the IDDPP, the value function at each step can be represented as an infimum of functions in a certain class. This is a min-plus expansion for the value function. For the specific class of problems considered here, we see that the class can be taken as that consisting of the quadratic functions. A means for reducing the numbers of constituent quadratic functions is discussed. [ABSTRACT FROM AUTHOR]

Multidimensional positive definite advection transport algorithm (MPDATA) is an iterative process for approximating the advection equation, which uses a donor cell approximation to compensate for the truncation error of the originally specified donor cell scheme. This step may be repeated an arbitrary number of times, leading to successively more accurate solutions to the advection equation. In this paper, we show how to sum the successive approximations analytically to find a single antidiffusive velocity that represents the effects of an arbitrary number of passes. The analysis is first done in one dimension to illustrate the method and then is repeated in two dimensions. The existence of cross terms in the truncation analysis of the two-dimensional equations introduces an extra complication into the calculation. We discuss the implementation of our new antidiffusive velocities and provide some examples of applications, including a third-order accurate scheme. [ABSTRACT FROM AUTHOR]