The multi-level Monte Carlo finite element method for a stochastic Brinkman Problem

We present the formulation and the numerical analysis of the Brinkman problem derived in Allaire (Arch Rational Mech Anal 113(3): 209-259,1990. doi: 10.1007/BF00375065 , Arch Rational Mech Anal 113(3): 261-298, 1990. doi: 10.1007/BF00375066 ) with a lognormal random permeability. Specifically, the permeability is assumed to be a lognormal random field taking values in the symmetric matrices of size $$d\times d$$ , where $$d$$ denotes the spatial dimension of the physical domain $$D$$ . We prove that the solutions admit bounded moments of any finite order with respect to the random input's Gaussian measure. We present a Mixed Finite Element discretization in the physical domain $$D$$ , which is uniformly stable with respect to the realization of the lognormal permeability field. Based on the error analysis of this mixed finite element method (MFEM), we develop a multi-level Monte Carlo (MLMC) discretization of the stochastic Brinkman problem and prove that the MLMC-MFEM allows the estimation of the statistical mean field with the same asymptotical accuracy versus work as the MFEM for a single instance of the stochastic Brinkman problem. The robustness of the MFEM implies in particular that the present analysis also covers the Darcy diffusion limit. Numerical experiments confirm the theoretical result

Finite element methods for flow in porous media

This thesis studies the application of finite element methods to porous flow problems. Particular attention is paid to locally mass conserving methods, which are very well suited for typical multiphase flow applications in porous media. The focus is on the Brinkman model, which is a parameter dependent extension of the classical Darcy model for porous flow taking the viscous phenomena into account. The thesis introduces a mass conserving finite element method for the Brinkman flow, with complete mathematical analysis of the method. In addition, stochastic material parameters are considered for the Brinkman flow, and parameter dependent Robin boundary conditions for the underlying Darcy flow. All of the theoretical results in the thesis are also verified with extensive numerical testing. Furthermore, many implementational aspects are discussed in the thesis, and computational viability of the methods introduced, both in terms of usefulness and computational complexity, is taken into account

Mini-Workshop: Numerical Upscaling for Media with Deterministic and Stochastic Heterogeneity

This minisymposium was third in series of similar events, after two very successful meetings in 2005 and 2009. The aim was to provide a forum for an extensive discussion on the theoretical aspects and on the areas of application and validity of numerical upscaling approaches for heterogeneous problems with deterministic and stochastic coefficients. The intensive discussions during the meeting contributed to a better understanding of upscaling approaches for multiscale problems with stochastic coefficients, and for synergy between scientists coming to this topic from the area of deterministic multiscale problems on one hand, and those coming from the area of SPDE on the other hand. Recent advanced results on upscaling approaches for deterministic multiscale problems were presented, well mixed with strong presentations on SDE and SPDE. The open problems in these areas were discussed, with emphasis on the case of stochastic coefficients brainstorming numerous numerical upscaling approaches. A number of young researchers, very actively working in these areas, were involved in the workshop discussing the links between scales., thus ensuring the continuity between the generations of researchers

Low-Rank Iterative Solvers for Large-Scale Stochastic Galerkin Linear Systems

Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, Dissertation, 2016von Dr. rer. pol. Akwum Agwu OnwuntaLiteraturverzeichnis: Seite 135-14

Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods

Many problems in computational science and engineering are simultaneously characterized by the following challenging issues: uncertainty, nonlinearity, nonstationarity and high dimensionality. Existing numerical techniques for such models would typically require considerable computational and storage resources. This is the case, for instance, for an optimization problem governed by time-dependent Navier-Stokes equations with uncertain inputs. In particular, the stochastic Galerkin finite element method often leads to a prohibitively high dimensional saddle-point system with tensor product structure. In this paper, we approximate the solution by the low-rank Tensor Train decomposition, and present a numerically efficient algorithm to solve the optimality equations directly in the low-rank representation. We show that the solution of the vorticity minimization problem with a distributed control admits a representation with ranks that depend modestly on model and discretization parameters even for high Reynolds numbers. For lower Reynolds numbers this is also the case for a boundary control. This opens the way for a reduced-order modeling of the stochastic optimal flow control with a moderate cost at all stages.Comment: 29 page

Mini-Workshop: Stochastic Differential Equations: Regularity and Numerical Analysis in Finite and Infinite Dimensions

This Mini-Workshop is devoted to regularity and numerical analysis of stochastic ordinary and partial differential equations (SDEs for both). The standard assumption in the literature on SDEs is global Lipschitz continuity of the coefficient functions. However, many SDEs arising from applications fail to have globally Lipschitz continuous coefficients. Recent years have seen a prosper growth of the literature on regularity and numerical approximations for SDEs with non-globally Lipschitz coefficients. Some surprising results have been obtained – e.g., the Euler–Maruyama method diverges for a large class of SDEs with super-linearly growing coefficients, and the limiting equation of a spatial discretization of the stochastic Burgers equation depends on whether the discretization is symmetric or not. Several positive results have been obtained. However the regularity of numerous important SDEs and the closely related question of convergence and convergence rates of numerical approximations remain open. The aim of this workshop is to bring together the main contributers in this direction and to foster significant progress