We propose and analyze sparse deterministic-stochastic tensor Galerkin finite element methods (sparse sGFEMs) for the numerical solution of elliptic partial differential equations (PDEs) with random coefficients in a physical domain D ⊂ R d . In tensor product sGFEMs, the variational solution to the boundary value problem is approximated in tensor product finite element spaces V Γ ⊗ V D , where V Γ and V D denote suitable finite dimensional subspaces of the stochastic and deterministic function spaces, respectively. These approaches lead to sGFEM algorithms of. . of finite dimensional spaces to approximate the law of the random solution. The hierarchies of approximation spaces allow us to define sparse tensor product spaces V Γ ⊗ V D , = 1, 2, . . . , yielding algorithms of O(N Γ log N D + N D log N Γ ) work and memory. We estimate the convergence rate of sGFEM for algebraic decay of the input random field Karhunen-Loève coefficients. We give an algorithm for an input adapted a-priori selection of deterministic and stochastic discretization spaces. The convergence rate in terms of the total number of degrees of freedom of the proposed method is superior to Monte Carlo approximations. Numerical examples illustrate the theoretical results and demonstrate superiority of the sparse tensor product discretization proposed here versus the full tensor product approach.Key words. stochastic partial differential equations, uncertainty quantification, stochastic finite element methods, multilevel approximations, sparse tensor products AMS subject classifications. 35R60, 60H15, 65C20, 65N12, 65N15, 65N30
We design and analyze algorithms for the efficient sensitivity computation of eigenpairs of parametric elliptic self-adjoint eigenvalue problems on high-dimensional parameter spaces. We quantify the analytic dependence of eigenpairs on the parameters. For the efficient approximate evaluation of parameter sensitivities of isolated eigenpairs on the entire parameter space we propose and analyze a sparse tensor spectral collocation method on an anisotropic sparse grid in the parameter domain. The stable numerical implementation of these methods is discussed and their error analysis is given. Applications to parametric elliptic eigenvalue problems with infinitely many parameters arising from elliptic differential operators with random coefficients are presented.
We consider the simultaneous estimation of an optical flow field and an illumination source term in a movie sequence. The particular optical flow equation is obtained by assuming that the image intensity is a conserved quantity up to possible sources and sinks which represent varying illumination. We formulate this problem as an energy minimization problem and propose a space–time simultaneous discretization for the optimality system in saddle-point form. We investigate a preconditioning strategy that renders the discrete system well-conditioned uniformly in the discretization resolution. Numerical experiments complement the theory.
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