Sparse grid

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

Volume Ray Casting on Sparse Grids

Abstract: These days sparse grids are of increasing interest in numerical simulations. Based upon hierarchical tensor product bases, the sparse grid approach is a very efficient one improving the ratio of invested storage and computing time to the achieved accuracy for many problems in the area of numerical solution of partial differential equations. The volume visualization algorithms that are available so far cannot cope with sparse grids. Now we present an approach that directly works on sparse grids. As a second aspect in this paper, we suggest to use sparse grids as a data compression method in order to visualize huge data sets even on workstations with low main memory. Because the size of data sets used in numerical simulations is still growing, this feature makes it possible that workstations can continue to handle these data sets. Besides the standard sparse grid interpolation algorithm and the so called combination approach, we have developed a new sparse grid interpolation method, which harnesses the texture-mapping hardware of Silicon Graphics workstations for accelerating purposes. Therefore, hardware based volume rendering becomes possible on compressed data sets at interactive frame rates.

continuous simulation of hypothetical physics processes with multiple free parameters

Abstract: We present a new approach to simulate Beyond-Standard-Model (BSM) processes which are defined by multiple parameters. In contrast to the traditional grid-scan method where a large number of events are simulated at each point of a sparse grid in the parameter space, this new approach simulates only a few events at each of a selected number of points distributed randomly over the whole parameter space. In subsequent analysis, we rely on the fitting by the Bayesian Neural Network (BNN) technique to obtain accurate estimation of the acceptance distribution. With this new approach, the signal yield can be estimated continuously, while the required number of simulation events is greatly reduced.

Arbitrarily Sparse Polynomial Chaos Expansion for High-Dimensional Parametric Problems: Parametric and Probabilistic Power Flow as an Example

Abstract: Parameters, no matter whether they are uncontrollable uncertainty factors or controllable variables, have a great impact on the states and performances of continuous engineering systems. Acquiring an explicit expression of the complicated implicit function between these parameters and system states, called parametric problem in this article, will facilitate immediate analysis of parameters’ impact on the system, optimal parameter design, and uncertainty quantification. Polynomial chaos expansion (PCE) is a globally optimal polynomial approximation method for parametric problems, but it suffers from the curse of dimensionality, i.e., the number of basis functions and consequent computational burden increases rapidly with the number of parameters (i.e., dimensions). This article proposes a PCE method characterized by the arbitrarily sparse basis and novel generalized Smolyak sparse grid quadrature and, hence, effectively relieves the curse of dimensionality for high-dimensional problems. This basis is not restricted by any existing fixed-form truncation and can merely incorporate a few important basis functions. The novel generalized sparse grid can remarkably reduce the number of required collocation points compared with the prevailing classic sparse grid while achieving high accuracy. The proposed method is universal for parametric problems in engineering systems and is exemplified by parametric and probabilistic power flow problems of electrical power systems. Its effectiveness is validated by computational results on the IEEE 30-bus, 118-bus, and 500-parameter 9241pegase systems.

A Dimension-adaptive Combination Technique for Uncertainty Quantification

Abstract: We present an adaptive algorithm for the computation of quantities of interest involving the solution of a stochastic elliptic PDE where the diffusion coefficient is parametrized by means of a Karhunen-Lo\`eve expansion. The approximation of the equivalent parametric problem requires a restriction of the countably infinite-dimensional parameter space to a finite-dimensional parameter set, a spatial discretization and an approximation in the parametric variables. We consider a sparse grid approach between these approximation directions in order to reduce the computational effort and propose a dimension-adaptive combination technique. In addition, a sparse grid quadrature for the high-dimensional parametric approximation is employed and simultaneously balanced with the spatial and stochastic approximation. Our adaptive algorithm constructs a sparse grid approximation based on the benefit-cost ratio such that the regularity and thus the decay of the Karhunen-Lo\`eve coefficients is not required beforehand. The decay is detected and exploited as the algorithm adjusts to the anisotropy in the parametric variables. We include numerical examples for the Darcy problem with a lognormal permeability field, which illustrate a good performance of the algorithm: For sufficiently smooth random fields, we essentially recover the rate of the spatial discretization as asymptotic convergence rate with respect to the computational cost.

Modelling Gene Regulatory Networks Using Galerkin Techniques Based on State Space Aggregation and Sparse Grids

Abstract: An important driver of the dynamics of gene regulatory networks is noise generated by transcription and translation processes involving genes and their products. As relatively small numbers of copies of each substrate are involved, such systems are best described by stochastic models. With these models, the stochastic master equations, one can follow the time development of the probability distributions for the states defined by the vectors of copy numbers of each substance. Challenges are posed by the large discrete state spaces, and are mainly due to high dimensionality.