Partial differential equations with random coefficients and input data arise in many real world applications. What they often have in common is that the data describing the PDE model are subject to uncertainties. The numerical approximation of statistics of this random solution poses several challenges, in particular when the number of random parameters is large and/or the parameter-to-solution map is complex. Therefore, effective surrogate or reduced models are of great need. We consider a class of time dependent PDEs with random parameters and search for an approximate solution in a separable form, i.e. at each time instant expressed as a linear combination of linearly independent spatial functions multiplied by linearly independent random variables (low rank approximation) in the spirit of a truncated Karhunen-Loeve expansion. Since the optimal deterministic and stochastic modes can significantly change over time, static versions, such as proper orthogonal decomposition or polynomial chaos expansion, may lose their effectiveness. Instead, here we consider a dynamical approach in which those modes are computed on-the-fly as solutions of suitable auxiliary evolution equations. From a geometric point of view, this approach corresponds to constraining the original dynamics to the manifold of fixed rank functions. The original equations are projected onto the tangent space of this manifold along the approximate trajectory. In this poster we recall the construction of the DLR method and give some implementation details. The spatial discretization is carried out by the finite element method and the discretization of the random variables relies on an adaptive choice of sparse grid. We will present some numerical test cases including the heat equation with a random diffusion coefficient and initial condition.
|ScienceOpen disciplines:||Applied mathematics, Applications, Statistics, Data analysis, Mathematics, Mathematical modeling & Computation|
|Keywords:||Uncertainty quantification, Dynamical low rank approximation, Dynamically orthogonal approximation|