Constructive Quantization and Multilevel Algorithms for Quadrature of SDEs

对随机微分方程组(SDE)求积问题的不同看法导致了不同的算法方法，如基于Fokker-Planck方程的PDE方法，可显式求解的确定性或随机化高维数值积分，以及SDE的蒙特卡罗模拟。在这个项目中，我们使用概率分布近似的概念作为SDE求积的基础，用于构造新的确定性和随机化算法，以及建立最优结果。我们的研究致力于(I)SDE系统的构造性量子化(II)不连续泛函的求积(III)Lévy驱动的SDE的多水平方法(IV)序列空间上的多水平求积和Malliavin正则性(V)OpenCL/CUDAt中的一个数值工具箱(I)关于构造离散分布以逼近目标分布及其在求积问题中的应用。在(Ii)中，分析了不连续泛函求积的Malliavin技巧和高阶数值格式。在(III)中，我们发展了L过程的模拟技术来设计多水平方案，并对连续泛函和不连续泛函进行了分析。高维积分的最新进展推动了(IV)中的研究。我们构建了(Ii)和(Iii)中的算法的并行实现，充分利用了隐藏在当今显卡中的大量并行处理单元。

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.