34,551 research outputs found

    Efficient Localization of Discontinuities in Complex Computational Simulations

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    Surrogate models for computational simulations are input-output approximations that allow computationally intensive analyses, such as uncertainty propagation and inference, to be performed efficiently. When a simulation output does not depend smoothly on its inputs, the error and convergence rate of many approximation methods deteriorate substantially. This paper details a method for efficiently localizing discontinuities in the input parameter domain, so that the model output can be approximated as a piecewise smooth function. The approach comprises an initialization phase, which uses polynomial annihilation to assign function values to different regions and thus seed an automated labeling procedure, followed by a refinement phase that adaptively updates a kernel support vector machine representation of the separating surface via active learning. The overall approach avoids structured grids and exploits any available simplicity in the geometry of the separating surface, thus reducing the number of model evaluations required to localize the discontinuity. The method is illustrated on examples of up to eleven dimensions, including algebraic models and ODE/PDE systems, and demonstrates improved scaling and efficiency over other discontinuity localization approaches

    Space-Time Tradeoffs for Distributed Verification

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    Verifying that a network configuration satisfies a given boolean predicate is a fundamental problem in distributed computing. Many variations of this problem have been studied, for example, in the context of proof labeling schemes (PLS), locally checkable proofs (LCP), and non-deterministic local decision (NLD). In all of these contexts, verification time is assumed to be constant. Korman, Kutten and Masuzawa [PODC 2011] presented a proof-labeling scheme for MST, with poly-logarithmic verification time, and logarithmic memory at each vertex. In this paper we introduce the notion of a tt-PLS, which allows the verification procedure to run for super-constant time. Our work analyzes the tradeoffs of tt-PLS between time, label size, message length, and computation space. We construct a universal tt-PLS and prove that it uses the same amount of total communication as a known one-round universal PLS, and tt factor smaller labels. In addition, we provide a general technique to prove lower bounds for space-time tradeoffs of tt-PLS. We use this technique to show an optimal tradeoff for testing that a network is acyclic (cycle free). Our optimal tt-PLS for acyclicity uses label size and computation space O((logn)/t)O((\log n)/t). We further describe a recursive O(logn)O(\log^* n) space verifier for acyclicity which does not assume previous knowledge of the run-time tt.Comment: Pre-proceedings version of paper presented at the 24th International Colloquium on Structural Information and Communication Complexity (SIROCCO 2017

    A general framework for coloring problems: old results, new results, and open problems

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    In this survey paper we present a general framework for coloring problems that was introduced in a joint paper which the author presented at WG2003. We show how a number of different types of coloring problems, most of which have been motivated from frequency assignment, fit into this framework. We give a survey of the existing results, mainly based on and strongly biased by joint work of the author with several different groups of coauthors, include some new results, and discuss several open problems for each of the variants
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