Sparse grid

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

Iterated Crank-Nicolson Procedure With Enhanced Absorption for Nonuniform Domains

Abstract: In multi-dimension problems, huge sparse matrices must be calculated according to Crank-Nicolson (CN) procedure which results in degeneration of efficiency and accuracy. Based on the iterated CN procedure, domain decomposition method, an alternative scheme is proposed for the termination of unbounded uniform finite-difference time-domain domains. Meanwhile, absorbing boundary condition is proposed in the iterated CN procedure which is incorporated with the higher order concept. The proposed scheme employs the explicit scheme during the calculation rather than the implicit one. Thus, the calculation of matrices can be avoided. It shows the advantages especially in efficiency and accuracy compared with implicit schemes. Such conclusion can be further demonstrated through the numerical example. From results, it can be concluded that the proposed scheme shows efficiency improvement and accurate maintenance. Compared with other procedures, it also holds its considerable effectiveness in nonuniform domains. For comparison, it can maintain considerable performance compared with the others.

Numerical Comparison of Leja and Clenshaw-Curtis Dimension-Adaptive Collocation for Stochastic Parametric Electromagnetic Field Problems

Abstract: We consider the problem of approximating the output of a parametric electromagnetic field model in the presence of a large number of uncertain input parameters. Given a sufficiently smooth output with respect to the input parameters, such problems are often tackled with interpolation-based approaches, such as the stochastic collocation method on tensor-product or isotropic sparse grids. Due to the so-called curse of dimensionality, those approaches result in increased or even forbidding computational costs. In order to reduce the growth in complexity with the number of dimensions, we employ a dimension-adaptive, hierarchical interpolation scheme, based on nested univariate interpolation nodes. Clenshaw-Curtis and Leja nodes satisfy the nestedness property and have been found to provide accurate interpolations when the parameters follow uniform distributions. The dimension-adaptive algorithm constructs the approximation based on the observation that not all parameters or interactions among them are equally important regarding their impact on the model's output. Our goal is to exploit this anisotropy in order to construct accurate polynomial surrogate models at a reduced computational cost compared to isotropic sparse grids. We apply the stochastic collocation method to two electromagnetic field models with medium- to high-dimensional input uncertainty. The performances of isotropic and adaptively constructed, anisotropic sparse grids based on both Clenshaw-Curtis and Leja interpolation nodes are examined. All considered approaches are compared with one another regarding the surrogate models' approximation accuracies using a cross-validation error metric.

Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs)

Abstract: We give a convergence proof for the approximation by sparse collocation of Hilbert-space-valued functions depending on countably many Gaussian random variables. Such functions appear as solutions of elliptic PDEs with lognormal diffusion coefficients. We outline a general $L^2$-convergence theory based on previous work by Bachmayr et al. [ESAIM Math. Model. Numer. Anal., 51 (2017), pp. 341--363] and Chen [ESAIM Math. Model. Numer. Anal., in press, 2018, https://doi.org/10.1051/m2an/2018012] and establish an algebraic convergence rate for sufficiently smooth functions assuming a mild growth bound for the univariate hierarchical surpluses of the interpolation scheme applied to Hermite polynomials. We specifically verify for Gauss--Hermite nodes that this assumption holds and also show algebraic convergence with respect to the resulting number of sparse grid points for this case. Numerical experiments illustrate the dimension-independent convergence rate.

Adaptive Stochastic Collocation Method for Parameterized Statistical Timing Analysis with Quadratic Delay Model

Abstract: In this paper, we propose an Adaptive Stochastic Collocation Method for block-based Statistical Static Timing Analysis (SSTA). A novel adaptive method is proposed to perform SSTA with delays of gates and interconnects modeled by quadratic polynomials based on Homogeneous Chaos expansion. In order to approximate the key atomic operator MAX in the full random space during timing analysis, the proposed method adaptively chooses the optimal algorithm from a set of stochastic collocation methods by considering different input conditions. Compared with the existing stochastic collocation methods, including the one using dimension reduction technique and the one using Sparse Grid technique, the proposed method has 10x improvements in the accuracy while using the same order of computation time. The proposed algorithm also shows great improvement in accuracy compared with a moment matching method. Compared with the 10,000 Monte Carlo simulations on ISCAS85 benchmark circuits, the results of the proposed method show less than 1% error in the mean and variance, and nearly 100x speeds up.

Multivariate Periodic Interpolating Wavelets

Abstract: Nested spaces of multivariate periodic functions are investigated. The scaling functions of these spaces are chosen as fundamental polynomials of Lagrange interpolation on a sparse grid. The interpolatory properties are crucial for the approach based on Boolean sums.