# 401-4606-00L Numerical Analysis of Stochastic Partial Differential Equations

Semester | Spring Semester 2016 |

Lecturers | A. Jentzen |

Periodicity | yearly recurring course |

Language of instruction | English |

Abstract | In this course solutions of semilinear stochastic partial differential equations (SPDEs) of the evolutionary type and some of their numerical approximation methods are investigated. Semilinear SPDEs are a key ingredient in a number of models from economics and the natural sciences. |

Objective | The aim of this course is to teach the students a decent knowledge on solutions of semilinear stochastic partial differential equations (SPDEs), on some numerical approximation methods for such equations and on the functional analytic and probabilistic concepts used to formulate and study such equations. |

Content | The course includes content (i) on the (functional) analytic concepts used to study semilinear stochastic partial differential equations (SPDEs) (e.g., nuclear operators, Hilbert-Schmidt operators, diagonal linear operators on Hilbert spaces, interpolation spaces associated to a diagonal linear operator, semigroups of bounded linear operators, Gronwall-type inequalities), (ii) on the probabilistic concepts used to study SPDEs (e.g., Hilbert space valued random variables, Hilbert space valued stochastic processes, infinite dimensional Wiener processes, stochastic integration with respect to infinite dimensional Wiener processes, infinite dimensional jump processes), (iii) on solutions of SPDEs (e.g., existence, uniqueness and regularity properties of mild solutions of SPDEs, applications involving SPDEs), and (iv) on numerical approximations of SPDEs (e.g., spatial and temporal discretizations, strong convergence, weak convergence). Semilinear SPDEs are a key ingredient in a number of models from economics and the natural sciences. They appear, for example, in models from neurobiology for the approximative description of the propagation of electrical impulses along nerve cells, in models from financial engineering for the approximative pricing of financial derivatives, in models from fluid mechanics for the approximative description of velocity fields in fully developed turbulent flows, in models from quantum field theory for describing the temporal dynamics associated to Euclidean quantum field theories, and in models from chemistry for the approximative description of the temporal evolution of the concentration of an undesired chemical contaminant in the groundwater system. |

Lecture notes | Lecture notes will be available as a PDF file. |

Literature | 1. Stochastic Equations in Infinite Dimensions G. Da Prato and J. Zabczyk Cambridge Univ. Press (1992) 2. Taylor Approximations for Stochastic Partial Differential Equations A. Jentzen and P.E. Kloeden Siam (2011) 3. Numerical Solution of Stochastic Differential Equations P.E. Kloeden and E. Platen Springer Verlag (1992) 4. A Concise Course on Stochastic Partial Differential Equations C. Prévôt and M. Röckner Springer Verlag (2007) 5. Galerkin Finite Element Methods for Parabolic Problems V. Thomée Springer Verlag (2006) |

Prerequisites / Notice | Mandatory prerequisites: Functional analysis, probability theory; Recommended prerequisites: stochastic processes; |