Information-Based Complexity of Multivariate Problems

多元问题出现在许多不同的应用中；然而，在最坏的情况下，这些问题往往难以解决。研究的一个重要部分将集中在研究多变量问题在平均情况下的设置。特别是，通过从最坏情况切换到平均情况设置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情况复杂性的理解，并将提供新的算法，以小的期望成本和小的期望误差来解决最坏情况下的棘手问题。与一般情况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，并应该揭示比蒙特卡罗方法（常用的）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域上的多元加权积分和逼近。研究的一部分将处理这类问题的复杂性。除了理论结果外，研究还将为多元积分/近似、加权积分/近似和路径积分等问题提供高效的软件。

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.