随机数值解研讨会

(Workshop forStochastic Numerics)

主办单位：华中科技大学数学与统计学院

(Sponsor: School of Mathematics and Statistics, HUST)

时间(Time)：2016.4.15-18

地点：华中科技大学数学与统计学院科技楼南楼 702

(Address: No. 702, Keji Building, HUST)

Speakers

1.     洪佳林(Jialin Hong)，(科学院数学与系统科学研究院)

2.     Tony Shardlow, (University of Bath)

3.     邹永魁(Yongkui Zou)(吉林大学)

4.     甘四清(Siqing Gan) (中南大学)

5.     李铁军(Tiejun Li) (北京大学)

6.     赵卫东(Weidong Zhao) (山东大学)

7.     周涛(Tao Zhou) (科学院数学与系统科学研究院)

8.     黄乘明(Chengming Huang)(华中科技大学)

9.     刘偉(Wei Liu)  (上海师范大学)

10.  曹婉容(Wanrong Cao)  (东南大学)

11.  郭谦(Qian Guo) (上海师范大学)

12.  宋明辉(Minghui Song)  (哈尔滨工业大学)

13.  牛原玲(Yuanling Niu) (中南大学)

14.  王小捷(Xiaojie Wang) (中南大学)

15.  陈琳(Lin Chen)          (江西财经大学)

16.  唐晓(Xiao Tang) (湘潭大学)

Organizers

Peter Kloeden(华中科技大学)

吴付科(Fuke Wu)(华中科技大学)

王小捷(Xiaojie Wang)      (中南大学)

Schedule

Numerical analysis of SDE by paths

Image registration with uncertain landmarks

报告（一）

时间：2016年4月16日（星期六）8:30–9:15

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：Tony Shardlow， University of Bath，教授

报告题目： Numerical analysis of SDE by paths

报告摘要：The theory of rough paths has provided a way of examining the sample paths of stochastic differential equations (SDEs) directly, rather than in the mean-square sense inherent in the Ito theory. In this talk, we take some of the ideas from rough path theory and develop a pathwise error analysis for time-stepping methods for SDEs. A simple criterion on the local truncation error is presented that yields pathwise error bounds. We use the theory to develop some adaptive time-stepping methods.

报告（二）

时间：2016年4月16日（星期六）9:15-10:00

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：李铁军，北京大学，教授

报告题目：Energy Landscape and the two-scale large deviations for biological stochastic dynamics

报告摘要：The construction of energy landscape for bio-dynamics is attracting more and more attention recent years. In this talk, I will introduce the strategy to construct the landscape from the connection to rare events, which relies on the large deviation theory for Gillespie-type jump dynamics. In the application to a typical genetic switching model, the two-scale large deviation theory is developed to take into account the fast switching of DNA states. The comparison with other proposals are also discussed. We demonstrate different diffusive limits arise when considering different regimes for genetic translation and switching processes. This is a joint work with Fangting Li, Xianggang Li, Cheng Lv, and Peijie Zhou.

报告（三）

时间：2016年4月16日（星期六）10:30-11:15

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：赵卫东，山东大学，教授

报告题目：Accurate numerical scheme for FBSDEs with applications

报告摘要：In this talk, we will introduce our accurate numerical method for solving forward backward stochastic differential equations (FBSDEs). Some applications of the scheme are discussed, such as in solving second-order FBSDEs, fully nonlinear parabolic-type partial differential equations, and stochastic optimal control.

报告（四）

时间：2016年4月16日（星期六）11:15-11:45

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：周涛，科学院数学与系统科学研究院，教授

报告题目：A generalized sampling and preconditioning scheme for sparse approximation of polynomial chaos expansions

报告摘要：we propose an algorithm for recovering sparse orthogonal polynomials using stochastic collocation. Our approach is motivated by the desire to use generalized polynomial chaos expansions to quantify uncertainty in models subject to uncertain input parameters. The standard sampling approach for recovering sparse polynomials is to use Monte Carlo (MC) sampling of the density of orthogonality.  While we propose a general algorithm that can be applied to any admissible weight function on a bounded domain and a wide class of exponential weight functions defined on unbounded domains. Our proposed algorithm samples with respect to the weighted equilibrium measure of the parametric domain, and subsequently solves a preconditioned l^1-minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest.

报告（五）

时间：2016年4月16日（星期六）11:45-12:15

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：郭谦，上海师范大学，副教授

报告题目：Stability analysis of explicit Runge-Kutta Maruyama methods for SDDEs

报告摘要：The stability properties of the numerical solutions generated by the explicit Runge-Kutta Maruyama methods for SDDEs are investigated, and a sufficient condition for stability is obtained and applied to the stochastic Runge-Kutta-Chebyshev methods for SDDEs.

报告（六）

时间： 2016年4月16日（星期六）14:00-14:45

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：邹永魁，吉林大学，教授

报告题目：Numerical analysis of a forth-order evolutionary equation

报告摘要：无

报告（七）

时间： 2016年4月16日（星期六）14:45-15:30

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：甘四清，中南大学，教授

报告题目： Stability of numerical methods for stochastic delay differential equations

报告摘要：This talk is concerned with stability of numerical methods for stochastic delay differential equations (SDDEs).  Firstly, we construct split-step backward Euler (SSBE) method for linear SDDEs. The conditions under which the SSBE method is mean-square stable (MS-stable) and general mean-square stable (GMS-stable) are obtained. Secondly, mean-square stability of θ-Maruyama methods are studied for nonlinear stochastic delay differential equations with variable delay. Under global Lipschitz conditions, it is proved that exponential mean-square stability of SDDEs implies that of the methods for sufficiently small step size h>0. Further, the exponential mean-square stability properties of SDDEs and those of numerical methods are investigated under some non-global Lipschitz conditions on the drift term. It is shown in this setting that the θ-Maruyama method with θ=1 can preserve the exponential mean-square stability for any step size. Additionally, the θ-Maruyama method with 1/2≤θ≤1 is asymptotically mean-square stable for any step size, provided that the underlying system with constant lag is exponentially mean-square stable. Finally, we derive sufficient conditions for the stability, contractivity and asymptotic contractivity in mean square of the solutions for nonlinear SDDEs. The results provide a unified theoretical treatment for SDDEs with constant delay and variable delay (including bounded and unbounded variable delays). Then the stability, contractivity and asymptotic contractivity in mean square are investigated for the backward Euler method. It is shown that the backward Euler method preserves the properties of the underlying SDDEs.

报告（八）

时间： 2016年4月16日（星期六）16:00-16:30

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：宋明辉，哈尔滨工业大学副教授

报告题目：待定

报告摘要：待定

报告（九）

时间： 2016年4月16日（星期六）16:30-17:00

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：陈琳，江西财经大学讲师

报告题目： Stability of numerical solutions for highly-nonlinear SDEs

报告摘要： Numerical methods as a practical and effective method has been developing rapidly,  since almost all of the stochastic differential equations are unable to get analytical solutions. And breaking through the traditional linear growth condition, which make the research results can cover many highly-nonlinear stochastic differential equations, is a valuable research direction. We intend to study trivial stability of stochastic differential equations and its numerical solutions under one-side linear growth，monotone-type or polynomial-type growth conditions, which may defy the traditional linear growth condition.

报告（十）

时间： 2016年4月17日（星期日）8:30-9:15

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：洪佳林，科学院数学与系统科学研究院，教授

报告题目：待定

报告摘要：待定

报告（十一）

时间： 2016年4月17日（星期日）9:15-10:00

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：Tony Shardlow，University of Bath，教授

报告题目：Image registration with uncertain landmarks

报告摘要：Image registration is the process of deforming one image, by a change in co-ordinate system, so it more closely resembles a second image and is an important task in medical imaging and shape analysis. We show how stochastic differential equations (SDEs) can be used to define a prior distribution on a space of image registrations. This leads to some difficult computational challenges  involving bridge diffusions. We describe recent work in this area and some open problems.

报告（十二）

时间： 2016年4月17日（星期日）10:30-11:15

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：黄乘明，华中科技大学，教授

报告题目：Delay-dependent stability analysis of stochastic theta methods for stochastic delay differential equations

报告摘要：This talk is concerned with the numerical solution of stochastic delay differential equations. The focus is on the delay-dependent stability of numerical methods for a linear scalar test equation with real coefficients. By using the so-called root locus technique, the full asymptotic stability region in mean square of stochastic theta methods is obtained, which is characterized by a sufficient and necessary condition in terms of the drift and diffusion coefficients as well as time stepsize and method parameter theta. Then, this condition is compared with the analytical stability condition. It is proved that the Backward Euler method completely preserves the asymptotic mean square stability of the underlying system and the Euler-Maruyama method preserves the instability of the system. Some numerical examples are presented to confirm the theoretical results.

报告（十三）

时间： 2016年4月17日（星期日）11:15-11:45

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：刘偉，上海师范大学，副教授

报告题目：Moment bounds and noise excitations for stochastic heat equations

报告摘要：In this talk, the recent results on moment bounds and noise excitations of a class of stochastic heat equations are reported. We start with the definition of noise excitations, followed by the discussion on the equations driven by white noise. The fractional stochastic heat equations driven by colored noise are studied then. The main techniques and ideas are briefed

报告（十四）

时间： 2016年4月17日（星期日）11:45-12:15

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：曹婉容，东南大学，副教授

报告题目：Numerical methods for stochastic delay differential equations via the Wong-Zakai approximation

报告摘要：We use the Wong–Zakai approximation as an intermediate step to derive numerical schemes for stochastic delay differential equations. By approximating the Brownian motion with its truncated spectral expansion and then using different discretizations in time, we present three schemes: a predictor-corrector scheme, a midpoint scheme, and a Milstein-like scheme. We prove that the predictor-corrector scheme converges with order half in the mean-square sense while the Milstein-like scheme converges with order one. We further consider the mean-square stability of the predictor-corrector scheme and the midpoint scheme. Numerical tests confirm the theoretical prediction and demonstrate that the midpoint scheme is of half-order convergence. Numerical results also show that the predictor-corrector and midpoint schemes can be of first-order convergence under commutative noises when there is no delay in the diffusion coefficients.

报告（十五）

时间： 2016年4月17日（星期日）16：00-16:30

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：牛原玲，中南大学，副教授

报告题目：Modelling biochemical reaction systems by stochastic differential equations with reflection

报告摘要：In this work, we gave a new framework for modelling and simulating biochemical reaction systems by stochastic differential equations with reflection not in a heuristic way but in a mathematical way. The model is computationally efficient compared with the discrete-state Markov chain approach, and it ensures that both analytic and numerical solutions remain in a biologically plausible region. Specifically, our model mathematically ensures that species numbers lie in the domain $D$, which is a physical constraint for biochemical reactions, in contrast to the previous models. The domain $D$ is actually obtained according to the structure of the corresponding chemical Langevin equations, i.e., the boundary is inherent in the biochemical reaction system. A variant of projection method was employed to solve the reflected stochastic differential equation model, and it includes three simple steps, i.e., Euler-Maruyama method was applied to the equations first, and then check whether or not the point lies within the domain $D$, and if not perform an orthogonal projection. It is found that the projection onto the closure $\bar{D}$ is the solution to a convex quadratic programming problem. Thus, existing methods for the convex quadratic programming problem can be employed for the orthogonal projection map. Numerical tests on several important problems in biological systems confirmed the efficiency and accuracy of this approach.

报告（十六）

时间： 2016年4月17日（星期日）16:30-17：00

地点：华中科技大学数学与统计学院科技楼南楼 702

报告人：唐晓，湘潭大学，博士生

报告题目：Efficient stochastic Runge-Kutta methods for stochastic differential equations with small noise

报告摘要：This talk concerns the stochastic Runge-Kutta (SRK) methods for stochastic differential equations (SDEs) with small noise. The mean-square convergence properties of the SRK methods for the Itô and the Stratonovich SDEs with small noise are studied, respectively. Various new efficient explicit SRK methods for SDEs with small multiplicative, commutative, additive and colored noise are proposed. The numerical results are reported to illustrate the theoretical results.