Abstract-Many problems in applied mathematics, physics, and engineering require the solution of the heat equation in unbounded domains. Integral equation methods are particularly appropriate in this setting for several reasons: they are unconditionally stable, they are insensitive to the complexity of the geometry, and they do not require the artificial truncation of the computational domain as do finite difference and finite element techniques. Methods of this type, however, have not become widespread due to the high cost of evaluating heat potentials. When m points are used in the discretization of the initial data, M points are used in the discretization of the boundary, and N time steps are computed, an amount of work of the order O(N 2 M 2 + NMm) has traditionally been required. In this paper, we present an algorithm which requires an amount of work of the order O(NM log M + m log m) and which is based on the evolution of the continuous spectrum of the solution. The method generalizes an earlier technique developed by Greengard and Strain (1990, Comm. Pure Appl. Math. 43, 949) for evaluating layer potentials in bounded domains.
Many problems in Machine Learning can be modeled as submodular optimization problems. Recent work has focused on stochastic or adaptive versions of these problems. We consider the Scenario Submodular Cover problem, which is a counterpart to the Stochastic Submodular Cover problem studied by Golovin and Krause (2011). In Scenario Submodular Cover, the goal is to produce a cover with minimum expected cost, where the expectation is with respect to an empirical joint distribution, given as input by a weighted sample of realizations. In contrast, in Stochastic Submodular Cover, the variables of the input distribution are assumed to be independent, and the distribution of each variable is given as input. Building on algorithms developed by Cicalese et al. (2014) and Golovin and Krause (2011) for related problems, we give two approximation algorithms for Scenario Submodular Cover over discrete distributions. The first achieves an approximation factor of O(log Qm), where m is the size of the sample and Q is the goal utility. The second, simpler algorithm achieves an approximation bound of O(log QW ), where Q is the goal utility and W is the sum of the integer weights. (Both bounds assume an integer-valued utility function.) Our results yield approximation bounds for other problems involving non-independent distributions that are explicitly specified by their support.
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We consider the problem of stochastic monotone submodular function maximization, subject to constraints. We give results on adaptivity gaps, and on the gap between the optimal offline and online solutions. We present a procedure that transforms a decision tree (adaptive algorithm) into a non-adaptive chain. We prove that this chain achieves at least τ times the utility of the decision tree, over a product distribution and binary state space, where τ = min i,j Pr[x i = j]. This proves an adaptivity gap of 1 τ (which is 2 in the case of a uniform distribution) for the problem of stochastic monotone submodular maximization subject to state-independent constraints. For a cardinality constraint, we prove that a simple adaptive greedy algorithm achieves an approximation factor of (1 − 1 e τ ) with respect to the optimal offline solution; previously, it has been proven that the algorithm achieves an approximation factor of (1 − 1 e ) with respect to the optimal adaptive online solution. Finally, we show that there exists a non-adaptive solution for the stochastic max coverage problem that is within a factor (1 − 1 e ) of the optimal adaptive solution and within a factor of τ(1 − 1 e ) of the optimal offline solution.
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