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Domain decomposition methods have been proved to be an effective strategy to reduce the dimension of parametric partial differential equations (PDEs). However, existing domain decomposition methods for parametric PDEs are usually intrusive, which means domain decomposition based solvers need to be implemented from scratch for each target parametric PDE. To address this issue, we develop a new non‐intrusive domain‐decomposition model reduction method for linear steady‐state PDEs with random‐field coefficients. As a variant of our previous work by Mu and Zhang, the new method only needs access to the final linear system, that is, the global stiffness matrix and the right hand side, of a deterministic PDE solver, in order to build a domain‐decomposition‐based reduced model without intrusive implementation from scratch. The key idea is to remove the interface condition between sub‐domains and rely on the correlation between columns of the linear system to couple the sub‐domains. The non‐intrusive feature enables the applicability of the proposed method to a broader class of uncertainty quantification problems, where many legacy codes/solvers can be fully reused by our method. Two numerical examples including diffusion equations with random diffusivity and convection‐dominated transport with random velocity, are provided to demonstrate the effectiveness and efficiency of our method. [ABSTRACT FROM AUTHOR]

The exit time probability, which gives the likelihood that an initial condition leaves a prescribed region of the phase space of a dynamical system at, or before, a given time, is arguably one of the most natural and important transport problems. Here we present an accurate and efficient numerical method for computing this probability for systems described by non-autonomous (time-dependent) stochastic differential equations (SDEs) or their equivalent Fokker-Planck partial differential equations. The method is based on the direct approximation of the Feynman-Kac formula that establishes a link between the adjoint Fokker-Planck equation and the forward SDE. The Feynman-Kac formula is approximated using the Gauss-Hermite quadrature rules and piecewise cubic Hermite interpolating polynomials, and a GPU accelerated matrix representation is used to compute the entire time evolution of the exit time probability using a single pass of the algorithm. The method is unconditionally stable, exhibits second order convergence in space, first order convergence in time, and it is straightforward to parallelize. Applications are presented to the advection diffusion of a passive tracer in a fluid flow exhibiting chaotic advection, and to the runaway acceleration of electrons in a plasma in the presence of an electric field, collisions, and radiation damping. Benchmarks against analytical solutions as well as comparisons with explicit and implicit finite difference standard methods for the adjoint Fokker-Planck equation are presented. [ABSTRACT FROM AUTHOR]

This work analyzes the overall computational complexity of the stochastic Galerkin finite element method (SGFEM) for approximating the solution of parameterized elliptic partial differential equations with both affine and non-affine random coefficients. To compute the fully discrete solution, such approaches employ a Galerkin projection in both the deterministic and stochastic domains, produced here by a combination of finite elements and a global orthogonal basis, defined on an isotopic total degree index set, respectively. To account for the sparsity of the resulting system, we present a rigorous cost analysis that considers the total number of coupled finite element systems that must be simultaneously solved in the SGFEM. However, to maintain sparsity as the coefficient becomes increasingly nonlinear in the parameterization, it is necessary to also approximate the coefficient by an additional orthogonal expansion. In this case we prove a rigorous complexity estimate for the number of floating point operations (FLOPs) required per matrix–vector multiplication of the coupled system. Based on such complexity estimates we also develop explicit cost bounds in terms of FLOPs to solve the stochastic Galerkin (SG) systems to a prescribed tolerance, which are used to compare with the minimal complexity estimates of a stochastic collocation finite element method (SCFEM), shown in our previous work (Galindo et al., 2015). Finally, computational evidence complements the theoretical estimates and supports our conclusion that, in the case that the coefficient is affine, the coupled SG system can be solved more efficiently than the decoupled SC systems. However, as the coefficient becomes more nonlinear, it becomes prohibitively expensive to obtain an approximation with the SGFEM. [ABSTRACT FROM AUTHOR]

The construction, analysis, and application of reduced-basis methods for uncertainty quantification problems involving nonlocal diffusion problems with random input data is the subject of this work. Because of the lack of sparsity of discretized nonlocal models relative to analogous local partial differential equation models, the need for reduced-order modeling is much more acute in the nonlocal setting. In this effort, we develop reduced-basis approximations for nonlocal diffusion equations with affine random coefficients. Efficiency estimates of the proposed greedy reduced-basis methods are provided. Numerical examples are used to illustrate the effect varying various model parameters have on the efficiency and accuracy of the reduced-basis method relative to sparse-grid interpolation using the full finite element method. It is shown that the proposed reduced-basis approach can indeed provide substantial savings over standard sparse-grid methods. [ABSTRACT FROM AUTHOR]