ISC 5935 Spring 2015 Study plan for Mike Schneier Supervisor: John Burkardt 10 February 2015 -------------------------------------------------------------------------------- Mike Schneier proposes the following special topic activity for his participation in the course. In previous work he has examined the use of reduced basis methods for solving stochastic partial differential equations (SPDE) modeled by a finite element method. This method allows for the accurate solution of an SPDE, while significantly cutting down on this computational cost. This work required both writing numerical software as well as mathematical analysis. As a follow up to this work he has become interested in the application of the reduced basis method to other already established methods in the field of SPDE’s. Specifically he is interested in introducing the reduced basis method into the recently developed multilevel stochastic collation method. In addition he has previously taken a course on the FENICS program, a software package that helps to automate the definition and solution of systems of partial differential equations modeled by the finite element method. This software allows him to examine the implementation of these previously mentioned techniques for more complex problems than he had previously been able to. For his project Mike proposes to implement the previously mentioned reduced basis method into the multilevel stochastic collation method. This will require writing new code as well as in depth mathematical analysis of this new method. Depending on these results and time permitting he will attempt to implement this method into python allowing for the use of FENIC’s so he may also examine more complex problems. At the end of the course, Mike will deliver a written report or working draft of a paper depending on the success of this project. There is a possibility that, during this project, Mike may wish to change the direction of his research; this is acceptable as long as we discuss it beforehand and can agree on a corresponding modification of this existing agreement. -------------------------------------------------------------------------------- Anders Logg, Kent-Andre Mardal, Garth Wells, Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Lecture Notes in Computational Science and Engineering, Springer, 2011, ISBN13: 978-3642230981 Rozza, G., Hunyh, D.B.O, Patera, A.t., Reduced basis approximation and a posterior error estimation for affinetly parameterized elliptic coercive partial differential equations., Arch. Comput. Methods eng.15(3), 229-275(2008) 2014 H.-W. van Wyk Multilevel Sparse Grid Methods for Elliptic Partial Differential Equations with Random Coefficients, Computers & Mathematics with Applications (submitted arxiv:1404.0963).