Sparse grid

About: Sparse grid is a research topic. Over the lifetime, 1013 publications have been published within this topic receiving 20664 citations.

Non-intrusive double-greedy parametric model reduction by interpolation of frequency-domain rational surrogates

Abstract: We propose a model order reduction approach for non-intrusive surrogate modeling of parametric dynamical systems. The reduced model over the whole parameter space is built by combining surrogates in frequency only, built at few selected values of the parameters. This, in particular, requires matching the respective poles by solving an optimization problem. If the frequency surrogates are constructed by a suitable rational interpolation strategy, frequency and parameters can both be sampled in an adaptive fashion. This, in general, yields frequency surrogates with different numbers of poles, a situation addressed by our proposed algorithm. Moreover, we explain how our method can be applied even in high-dimensional settings, by employing locally-refined sparse grids in parameter space to weaken the curse of dimensionality. Numerical examples are used to showcase the effectiveness of the method, and to highlight some of its limitations in dealing with unbalanced pole matching, as well as with a large number of parameters.

Sparse tensor phase space Galerkin approximation for radiative transport.

Abstract: We develop, analyze, and test a sparse tensor product phase space Galerkin discretization framework for the stationary monochromatic radiative transfer problem with scattering. The mathematical model describes the transport of radiation on a phase space of the Cartesian product of a typically three-dimensional physical domain and two-dimensional angular domain. Known solution methods such as the discrete ordinates method and a spherical harmonics method are derived from the presented Galerkin framework. We construct sparse versions of these well-established methods from the framework and prove that these sparse tensor discretizations break the “curse of dimensionality”: essentially (up to logarithmic factors in the total number of degrees of freedom) the solution complexity increases only as in a problem posed in the physical domain alone, while asymptotic convergence orders in terms of the discretization parameters remain essentially equal to those of a full tensor phase space Galerkin discretization. Algorithmically we compute the sparse tensor approximations by the combination technique. In numerical experiments on 2+1 and 3+2 dimensional phase spaces we demonstrate that the advantages of sparse tensorization can be leveraged in applications.

A proof of the consistency of the finite difference technique on sparse grids

Abstract: In this paper, we give a proof of the consistency of the finite difference technique on regular sparse grids [7, 18]. We introduce an extrapolation-type discretization of differential operators on sparse grids based on the idea of the combination technique and we show the consistency of this discretization. The equivalence of the new method with that of [7, 18] is established.

Turbulence suppression by energetic particles: a sensitivity-driven dimension-adaptive sparse grid framework for discharge optimization

Abstract: A newly developed sensitivity-driven approach is employed to study the role of energetic particles in suppressing turbulence-inducing micro-instabilities for a set of realistic JET-like cases with NBI deuterium and ICRH $^3$He fast ions. First, the efficiency of the sensitivity-driven approach is showcased for scans in a $21$-dimensional parameter space, for which only $250$ simulations are necessary. The same scan performed with traditional Cartesian grids with only two points in each of the $21$ dimensions would require $2^{21} = 2,097,152$ simulations. Then, a $14$-dimensional parameter subspace is considered, using the sensitivity-driven approach to find an approximation of the parameter-to-growth rate map averaged over nine bi-normal wave-numbers, indicating pathways towards turbulence suppression. The respective turbulent fluxes, obtained via nonlinear simulations for the optimized set of parameters, are reduced by more than two order of magnitude compared to the reference results.

An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems

Abstract: In this paper we present an asymptotically compatible meshfree method for solving nonlocal equations with random coefficients, describing diffusion in heterogeneous media. In particular, the random diffusivity coefficient is described by a finite-dimensional random variable or a truncated combination of random variables with the Karhunen-Loeve decomposition, then a probabilistic collocation method (PCM) with sparse grids is employed to sample the stochastic process. On each sample, the deterministic nonlocal diffusion problem is discretized with an optimization-based meshfree quadrature rule. We present rigorous analysis for the proposed scheme and demonstrate convergence for a number of benchmark problems, showing that it sustains the asymptotic compatibility spatially and achieves an algebraic or sub-exponential convergence rate in the random coefficients space as the number of collocation points grows. Finally, to validate the applicability of this approach we consider a randomly heterogeneous nonlocal problem with a given spatial correlation structure, demonstrating that the proposed PCM approach achieves substantial speed-up compared to conventional Monte Carlo simulations.