Information-Based Complexity of Multivariate Problems

多变量问题出现在许多不同的应用中；然而，在最糟糕的情况下，它们往往是难以解决的。这项研究的重要部分将集中在平均案例背景下的多变量问题的研究上。特别是，通过从最坏情况切换到平均情况来降低复杂性的研究将是中心焦点之一。这将丰富对重要多变量问题的平均案例复杂性的理解，并将提供新的算法来解决最坏情况下的期望代价和期望误差较小的棘手问题。与平均案例分析相关的是路径积分。这项研究将为他的重要问题提供复杂性界限，并将揭示一些路径积分比蒙特卡罗方法(通常使用的)更有效的新算法。研究路径积分的方法之一是通过分析无界区域上的多元加权积分和逼近。研究的一部分将涉及此类问题的复杂性。除了理论结果，这项研究还将产生有效的软件来解决多变量积分/逼近、加权积分/逼近和路径积分等问题。

Multivariate problems appear in many diverse applications; yet they are often intractable in the worst case setting. A significant part of the research will concentrate on the study of multivariate problems in the average case setting. In particular, the study of the reduction in complexity by switching from the worst case to the average case setting will be one of the central foci. This will enrich understanding of the average case complexity of important multivariate problems and will provide new algorithms that solve worst case intractable problems with small expected cost and small expected error. Related to the average case analysis are path integrals. The research will provide complexity bounds for his important problem and should reveal new algorithms that are more efficient than Monte Carlo methods (that are commonly used) for some path integrals. One of the approaches in studying path integrals is through the analysis of multivariate weighted integration and approximation over unbounded domains. A part of the research will deal with the complexity of such problems. In addition to theoretical results, the research will result in efficient software for such problems as multivariate integration/approximation, weighted integration/approximation, and path integrals.