A new approach to energy-based sparse finite-element spaces

We show that the logarithmic factor in the standard error estimate for sparse finite element (FE) spaces in arbitrary dimension d is removable in the energy (H1) norm. Via a penalized sparse grid condition, we then propose and analyse a new version of the energy-based sparse FE spaces introduced first in Bungartz (1992, Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung. Dissertation. Munich, Germany: TU München) and known to satisfy an optimal approximation property in the energy nor

Sparse finite element approximation of high-dimensional transport-dominated diffusion problems

Partial differential equations with nonnegative characteristic form arise in numerous mathematical models in science. In problems of this kind, the exponential growth of computational complexity as a function of the dimension d of the problem domain, the so-called ``curse of dimension'', is exacerbated by the fact that the problem may be transport-dominated. We develop the numerical analysis of stabilized sparse tensor-product finite element methods for such high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations, using piecewise polynomials of degree p > 0. Our convergence analysis is based on new high-dimensional approximation results in sparse tensor-product spaces. By tracking the dependence of the various constants on the dimension $d$ and the polynomial degree p, we show in the case of elliptic transport-dominated diffusion problems that for p > 0 the error constant exhibits exponential decay as d tends to infinity. In the general case when the characteristic form of the partial differential equation is non-negative, under a mild condition relating p to d, the error constant is shown to grow no faster than quadratically in d. In any case, the sparse stabilized finite element method exhibits an optimal rate of convergence with respect to the mesh-size, up to a factor that is polylogarithmic in the mesh-size.\ud \ud Dedicated to Henryk Wozniakowski, on the occasion of his 60th birthday

Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients

A scalar, elliptic boundary-value problem in divergence form with stochastic diffusion coefficient a(x, ω) in a bounded domain D ⊂ ℝd is reformulated as a deterministic, infinite-dimensional, parametric problem by separation of deterministic (x ∈ D) and stochastic (ω ∈ Ω) variables in a(x, ω) via Karhúnen-Loève or Legendre expansions of the diffusion coefficient. Deterministic, approximate solvers are obtained by projection of this problem into a product probability space of finite dimension M and sparse discretizations of the resulting M-dimensional parametric problem. Both Galerkin and collocation approximations are considered. Under regularity assumptions on the fluctuation of a(x, ω) in the deterministic variable x, the convergence rate of the deterministic solution algorithm is analysed in terms of the number N of deterministic problems to be solved as both the chaos dimension M and the multiresolution level of the sparse discretization resp. the polynomial degree of the chaos expansion increase simultaneousl

Sparse finite elements for elliptic problems with stochastic loading

Summary.: We formulate elliptic boundary value problems with stochastic loading in a bounded domain D⊂ℝ d . We show well-posedness of the problem in stochastic Sobolev spaces and we derive a deterministic elliptic PDE in D×D for the spatial correlation of the random solution. We show well-posedness and regularity results for this PDE in a scale of weighted Sobolev spaces with mixed highest order derivatives. Discretization with sparse tensor products of any hierarchic finite element (FE) spaces in D yields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data. Multilevel preconditioning in D×D allows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equatio

Sparse finite elements for elliptic problems with stochastic loading

ISSN:0029-599XISSN:0945-324

Requirements and Use Cases ; Report I on the sub-project Smart Content Enrichment

In this technical report, we present the results of the first milestone phase of the Corporate Smart Content sub-project "Smart Content Enrichment". We present analyses of the state of the art in the fields concerning the three working packages defined in the sub-project, which are aspect-oriented ontology development, complex entity recognition, and semantic event pattern mining. We compare the research approaches related to our three research subjects and outline briefly our future work plan