Constructive Quantization and Multilevel Algorithms for Quadrature of SDEs

SDE 求积的构造量化和多级算法

• 批准号: 79644002
• 负责人: Professor Dr. Steffen Dereich
• 金额: --
• 依托单位: Lehrstuhl für Mathematische Stochastik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2008
• 资助国家: 德国
• 起止时间: 2007-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/79644002?language=en
• 关键词: Constructive; Quantization; Multilevel; Algorithms; Quadrature

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.

对随机微分方程（SDEs）的正交问题的不同看法导致了不同的算法方法，例如基于Fokker-Planck方程的PDE方法，明确可解SDEs的确定性或随机高维数值积分，以及SDEs的蒙特卡罗模拟。在这个项目中，我们采用概率分布近似的概念作为SDEs正交的基础，用于构建新的确定性和随机算法，以及建立最优性结果。我们的研究主要集中在(I) SDEs系统的构造量化（II）间断泛函的正交（III） lsamv驱动SDEs的多级方法（IV）序列空间和Malliavin正则上的多级正交(V) OpenCL/CUDAPart中的一个数值工具箱(I)关注离散分布的构造以近似目标分布及其在正交问题中的应用。(2)分析了不连续泛函求积分的Malliavin技术和高阶数值格式。在（III）中，我们开发了用于多级方案设计的lsamvy过程的模拟技术，并对连续和不连续泛函进行了分析。高维整合的最新进展激发了（IV）的研究。我们构建了（II）和（III）算法的并行实现，充分利用了隐藏在当今图形卡中的大规模并行处理单元。