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Exploiting Sparsity for Semi-Algebraic Set Volume Computation
Foundations of Computational Mathematics ( IF 3 ) Pub Date : 2021-03-26 , DOI: 10.1007/s10208-021-09508-w
Matteo Tacchi , Tillmann Weisser , Jean Bernard Lasserre , Didier Henrion

We provide a systematic deterministic numerical scheme to approximate the volume (i.e., the Lebesgue measure) of a basic semi-algebraic set whose description follows a correlative sparsity pattern. As in previous works (without sparsity), the underlying strategy is to consider an infinite-dimensional linear program on measures whose optimal value is the volume of the set. This is a particular instance of a generalized moment problem which in turn can be approximated as closely as desired by solving a hierarchy of semidefinite relaxations of increasing size. The novelty with respect to previous work is that by exploiting the sparsity pattern we can provide a sparse formulation for which the associated semidefinite relaxations are of much smaller size. In addition, we can decompose the sparse relaxations into completely decoupled subproblems of smaller size, and in some cases computations can be done in parallel. To the best of our knowledge, it is the first contribution that exploits correlative sparsity for volume computation of semi-algebraic sets which are possibly high-dimensional and/or non-convex and/or non-connected.



中文翻译:

利用稀疏性进行半代数集体积计算

我们提供了一种系统的确定性数值方案,以近似其描述遵循相关稀疏性模式的基本半代数集的体积(即Lebesgue测度)。与先前的工作(无稀疏性)一样,其基本策略是考虑对最优值是集合量的度量进行无穷维线性规划。这是广义矩问题的一个特殊实例,而广义矩问题又可以通过解决逐渐增大的半确定松弛的层次结构,按需要近似逼近。与以前的工作有关的新颖之处在于,通过利用稀疏模式,我们可以提供稀疏的公式,其相关的半确定松弛的大小要小得多。此外,我们可以将稀疏松弛分解为较小大小的完全解耦的子问题,在某些情况下,可以并行进行计算。据我们所知,这是利用相关稀疏性对半代数集(可能是高维和/或非凸和/或非连通的)进行体积计算的第一项贡献。

更新日期:2021-03-27
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