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Let F be a compact set of a Banach space X. This paper analyzes the "generalized empirical interpolation method," which, given a function f ∊ F, builds an interpolant Jn[f] in an n dimensional subspace Xn ⊂ X with the knowledge of n outputs (σi(f))ni=1, where σi ∊ X' and X' is the dual space of X. The space Xn is built with a greedy algorithm that is adapted to F in the sense that it is generated by elements of F itself. The algorithm also selects the linear functionals (σi)ni=1 from a dictionary Σ ⊂ X'. In this paper, we study the interpolation error maxf∊F ‖f - Jn[f]‖X by comparing it with the best possible performance on an n dimensional space, i.e., the Kolmogorov n-width of F in X, dn(F,X). For polynomial or exponential decay rates of dn(F,X), we prove that the interpolation error has the same behavior modulo the norm of the interpolation operator. Sharper results are obtained in the case where X is a Hilbert space. [ABSTRACT FROM AUTHOR]

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]