Constructive Quantization and Multilevel Algorithms for Quadrature of SDEs

对随机微分方程求积问题的不同观点导致了相当不同的算法方法，例如，PDE方法基于福克-普朗克方程，确定性或随机高维数值积分显式可解的SDES，和蒙特卡洛模拟的SDES。在这个项目中，我们采用的概率分布近似的概念作为基础的正交的SDES，无论是，构建新的确定性和随机算法，以及建立最优性的结果。我们的研究致力于（I）系统的构造性量化（II）不连续泛函的求积（III）Lévy驱动的SDEs的多级方法（IV）序列空间上的多级求积和Malliavin正则性（V）OpenCL/CUDAPart中的数值计算（I）关注离散分布的构造以逼近目标分布及其在求积问题中的应用。在（II）中，分析了高阶Malliavin技巧和数值格式对不连续泛函求积的影响。在（III）中，我们开发了Lévy过程的模拟技术，用于设计多级方案，并对连续和不连续泛函进行了分析。最近的进展，在高维集成激励的研究（IV）。我们建立并行实现的算法（II）和（III），充分利用隐藏在今天的图形卡的大规模并行处理单元。

Different views on quadrature problems for stochastic differential equations (SDEs) lead to rather different algorithmic approaches, e.g., PDE methods based on the Fokker-Planck equation, deterministic or randomized high-dimensional numerical integration for explicitly solvable SDEs, and Monte Carlo simulation of SDEs. In this project we employ the concept of approximation of probability distributions as the basis for quadrature of SDEs, both, for constructing new deterministic and randomized algorithms, as well as for establishing optimality results. Our research is devoted to(I) Constructive Quantization of Systems of SDEs(II) Quadrature of Discontinuous Functionals(III) Multilevel Methods for Lévy-driven SDEs(IV) Multilevel Quadrature on the Sequence Space and Malliavin Regularity(V) A Numerical Toolbox in OpenCL/CUDAPart (I) is concerned with the construction of discrete distributions to approximate a target distribution and its application to the quadrature problem. In (II), Malliavin techniques and numerical schemes of higher order are analyzed for quadrature of discontinuous functionals. In (III), we develop simulation techniques for Lévy processes for the design of multilevel schemes and an analysis for continuous and discontinuous functionals is conducted. Recent progress in high-dimensional integration motivates the study in (IV). We build parallel implementations of the algorithms from (II) and (III) that make full use of the massive parallel processing units hidden in today’s graphic cards.