Towards High-order Methods for Stochastic Di erential ...

1 Towards High-order Methods for Stochastic DifferentialEquations with White Noise: A Spectral ApproachbyZhongqiang ZhangA dissertation submitted in partial fulfillment of therequirements for the degree of Doctor of Philosophyin the Division of Applied Mathematics at Brown UniversityPROVIDENCE, RHODE ISLANDMay 2014 Abstract of Towards High-order Methods for Stochastic Differential equations with White Noise:A Spectral Approach by Zhongqiang Zhang, , Brown University, May 2014We develop a recursive multistage Wiener chaos expansion method (WCE) and a recursive multi-stage Stochastic collocation method (SCM) for numerical integration of linear Stochastic advection-diffusion-reaction equations with multiplicative white noise.

2 We show that both Methods are com-parably efficient in computing the first two moments of solutions for long time intervals, comparedto a direct application of WCE and SCM while both Methods are more efficient than the standardMonte Carlo method if high accuracy is Methods belong to Wong-Zakai approximation, where Brownian motion is truncated usingits spectral expansion before any discretization in time and space. For computational convenience,WCE is associated with the Ito formulation of underlying equations and SCM is associated withthe Stratonovich apply SCM using Smolyak s sparse grid construction to obtain the shock location of theone-dimensional piston problem, which is modeled by Stochastic Euler equations with multiplica-tive white noise.

3 We show numerically that SCM is efficient for short time simulations and forsmall magnitudes of noises and quasi-Monte Carlo Methods are efficient for moderate large-timesimulations. We also illustrate the efficiency of SCM through error estimates for a linear further investigate the effect of a spectral approximation of Brownian motion, rather thana piecewise linear approximation, for both spatial and temporal noise. For spatial noise, we con-sider semilinear elliptic equations with additive noise and show that when the solution is smoothenough, the spectral approximation is superior to the piecewise linear approximation while bothapproximations are comparable when the solution is not smooth.

4 For temporal noise, we use thisspectral approach to design numerical schemes for Stochastic delay differential equations under theStratonovich formulation. We show that the spectral approach admits higher-order accuracy onlyfor higher-order equations with coefficients of linear growth, we also consider Stochastic ordinary differ-ential equations with coefficients of polynomial growth. We formulate a basic relationship betweenlocal truncation error and global error of numerical Methods for these equations , and apply thisrelationship for our explicit balanced scheme to obtain the convergence order. Copyright 2014 by Zhongqiang ZhangThis dissertation by Zhongqiang Zhang is accepted in its present formby the Division of Applied Mathematics as satisfying thedissertation requirement for the degree of Doctor of Em Karniadakis, AdvisorDateBoris Rozovskii, Co-advisorRecommended to the Graduate CouncilDateMichael V.

5 Tretyakov, ReaderDateMarcus Sarkis, ReaderApproved by the Graduate CouncilDatePeter M. Weber, Dean of the Graduate SchooliiiAcknowledgmentsI would like to thank my advisor Professor George Em Karniadakis and co-advisor ProfessorBoris L. Rozovskii for their guidance and support during my study at Brown University. Theirpassion and great vision on scientific research in applied mathematics have inspired me to explore myresearch field in depth. Professor George Em Karniadakis always reminds of the essence of appliedmathematics and shares with me his rich knowledge and research experience, which have helpedme appreciate applications beyond mathematics itself.

6 Professor Boris L. Rozovskii introduced hisworks to me and has encouraged me to be more rigorous in my work. My gratitude to them isactually beyond any is my great fortune to collaborate with Professor Michael V. Tretyakov of University ofNottingham at England. Without valuable discussions with him and invaluable advice from him, Iwould not have this dissertation in current form, several chapters of which are papers we publishedtogether in the last few years. He carefully checked every point leaving nothing to chance, whichgreatly helped me improve my presentation in clarity and express my sincere thanks to my committee members and readers, Professor Marcus Sarkisand Professor Michael V.

7 Tretyakov, for their precious time in carefully reading my thesis andserving as committee would like to thank Professor Johnny Guzm an for discussing finite element Methods for ellipticequations with me and Professor Hongjie Dong for his help on energy estimates of some parabolicequations with minimal regularity. I also thank Professor Chau-Hsing Su for his help on thestochastic piston heart is full of gratitude to people at the Crunch group: Heyrim Cho, Minseok Choi, MinggeDeng, Huan Lei, Xuejin Li, Zhen Li, Yuhang Tang, Daniele Venturi, Alix Witthoft, Xiu Yang, YueYu, Summer Zheng and many others for their help and support in life and in research.

8 I would alsolike to thank Xingjie Li at Brown University, Wanrong Cao at Southeast University, Guang Lin ativPacific Northwest National Laboratory, and Xiaoliang Wan at Louisiana State University for manyhelpful thanks are also due to Ms. Madeline Brewster, Ms. Camille O. Dickson, Ms. Lisa Eklund,Ms. Stephanie Han, Ms. Jean Radican and Ms. Laura Leddy for their timely , I have to express my heartfelt thanks to my parents and my sisters for their love andencouragement. I remain indebted to my wife, Fei Yu, for her emotional support and love. I amalso grateful to my father- and mother-in-law for their love and support, especially during my hardtimes.

9 This dissertation is dedicated to my deceased work was supported by OSD/MURI grant FA9550-09-1-0613, NSF/DMS grant DMS-0915077, NSF/DMS grant DMS-1216437 and also by the Collaboratory on Mathematics for Meso-scopic Modeling of Materials (CM4) which is sponsored by Review of numerical Methods for SPDEs .. semi-discretization Methods for parabolic SPDEs .. approximation for parabolic SPDEs .. Methods for parabolic SPDEs .. Burgers and Navier-Stokes equations .. parabolic SPDEs .. and convergence of existing numerical Methods .. Approximation of Brownian motion .. linear approximation .. approximations.

10 Integration Methods in random space .. Carlo method and its variants .. chaos expansion method .. collocation method .. Objectives of this work .. 25viPart I: Temporal White Noise292 Wiener chaos Methods for linear Stochastic advection-diffusion-reaction Introduction .. WCE of the SPDE solution .. Multistage WCE method .. Algorithm for computing moments .. Numerical tests in one dimension .. problems .. of WCE algorithms to the model problem .. results .. of the WCE algorithm and Monte Carlo-type algorithms .. Numerical tests with passive scalar equation.