Sparse grid

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

Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs

Abstract: By combining a certain approximation property in the spatial domain, and weighted 𝓁2 -summability of the Hermite polynomial expansion coefficients in the parametric domain obtained in Bachmayr et al. [ESAIM: M2AN 51 (2017) 341–363] and Bachmayr et al. [SIAM J. Numer. Anal. 55 (2017) 2151–2186], we investigate linear non-adaptive methods of fully discrete polynomial interpolation approximation as well as fully discrete weighted quadrature methods of integration for parametric and stochastic elliptic PDEs with lognormal inputs. We construct such methods and prove convergence rates of the approximations by them. The linear non-adaptive methods of fully discrete polynomial interpolation approximation are sparse-grid collocation methods which are certain sums taken over finite nested Smolyak-type indices sets of mixed tensor products of dyadic scale successive differences of spatial approximations of particular solvers, and of successive differences of their parametric Lagrange interpolating polynomials. The Smolyak-type sparse interpolation grids in the parametric domain are constructed from the roots of Hermite polynomials or their improved modifications. Moreover, they generate in a natural way fully discrete weighted quadrature formulas for integration of the solution to parametric and stochastic elliptic PDEs and its linear functionals, and the error of the corresponding integration can be estimated via the error in the Bochner space L 1 (ℝ ∞ , V , γ ) norm of the generating methods where γ is the Gaussian probability measure on ℝ∞ and V is the energy space. We also briefly consider similar problems for parametric and stochastic elliptic PDEs with affine inputs, and problems of non-fully discrete polynomial interpolation approximation and integration. In particular, the convergence rates of non-fully discrete polynomial interpolation approximation and integration obtained in this paper significantly improve the known ones.

Option pricing with sparse grids

Abstract: We investigate the suitability of sparse grids for solving high-dimensional option pricing and interest rate models numerically. Starting from the partial differential equation, we try to - at least partially - break the curse of dimensionality through sparse grids which will result from a multi-level splitting of the solution. We make use of an adaptive algorithm in spacetime exploiting the smoothness of the solution. In order to compute sensitivities (the so-calles "greeks"), we avail of interpolets as a smooth basis function leading to faster convergence. Finite differences allow us to adjust the order of consistency. The code providing the results of the paper was designed for fast solving making use of an efficient preconditioner and parallelization. The specific choice of boundary conditions is crucial to obtaining good approximations to the true solution. Different types will be compared here. Our findings suggest the usage of locally full grids in order to approximate the singularity in the initial data. However, this modification does not lead to a deterioration of the speed of convergence which will yield a rate of 4 for the solution. That means the Gamma sensitivity converges as a second derivative at a rate of 2

EXAHD: A massively parallel fault tolerant sparse grid approach for high-dimensional turbulent plasma simulations

Abstract: Plasma fusion is one of the promising candidates for an emission-free energy source and is heavily investigated with high-resolution numerical simulations. Unfortunately, these simulations suffer from the curse of dimensionality due to the five-plus-one-dimensional nature of the equations. Hence, we propose a sparse grid approach based on the sparse grid combination technique which splits the simulation grid into multiple smaller grids of varying resolution. This enables us to increase the maximum resolution as well as the parallel efficiency of the current solvers. At the same time we introduce fault tolerance within the algorithmic design and increase the resilience of the application code. We base our implementation on a manager-worker approach which computes multiple solver runs in parallel by distributing tasks to different process groups. Our results demonstrate good convergence for linear fusion runs and show high parallel efficiency up to 180k cores. In addition, our framework achieves accurate results with low overhead in faulty environments. Moreover, for nonlinear fusion runs, we show the effectiveness of the combination technique and discuss existing shortcomings that are still under investigation.

Sparse grid approximation with Gaussians

Abstract: Motivated by the recent multilevel sparse kernel-based interpolation (MuSIK) algorithm proposed in [Georgoulis, Levesley and Subhan, SIAM J. Sci. Comput., 35(2), pp. A815-A831, 2013], we introduce the new quasi-multilevel sparse interpolation with kernels (Q-MuSIK) via the combination technique. The Q-MuSIK scheme achieves better convergence and run time in comparison with classical quasi-interpolation; namely, the Q-MuSIK algorithm is generally superior to the MuSIK methods in terms of run time in particular in high-dimensional interpolation problems, since there is no need to solve large algebraic systems. We subsequently propose a fast, low complexity, high-dimensional quadrature formula based on Q-MuSIK interpolation of the integrand. We present the results of numerical experimentation for both interpolation and quadrature in R, for d = 2, d = 3 and d = 4. In this work we also consider the convergence rates for multilevel quasiinterpolation of periodic functions using Gaussians on a grid. Initially, we have given the single level quasi-interpolation error by using the shifting properties of Gaussian kernel, and have then found an estimate for the multilevel error using the multilevel algorithm for unit function.