22 research outputs found
Quadrature Strategies for Constructing Polynomial Approximations
Finding suitable points for multivariate polynomial interpolation and
approximation is a challenging task. Yet, despite this challenge, there has
been tremendous research dedicated to this singular cause. In this paper, we
begin by reviewing classical methods for finding suitable quadrature points for
polynomial approximation in both the univariate and multivariate setting. Then,
we categorize recent advances into those that propose a new sampling approach
and those centered on an optimization strategy. The sampling approaches yield a
favorable discretization of the domain, while the optimization methods pick a
subset of the discretized samples that minimize certain objectives. While not
all strategies follow this two-stage approach, most do. Sampling techniques
covered include subsampling quadratures, Christoffel, induced and Monte Carlo
methods. Optimization methods discussed range from linear programming ideas and
Newton's method to greedy procedures from numerical linear algebra. Our
exposition is aided by examples that implement some of the aforementioned
strategies
Far-Field Compression for Fast Kernel Summation Methods in High Dimensions
We consider fast kernel summations in high dimensions: given a large set of
points in dimensions (with ) and a pair-potential function (the
{\em kernel} function), we compute a weighted sum of all pairwise kernel
interactions for each point in the set. Direct summation is equivalent to a
(dense) matrix-vector multiplication and scales quadratically with the number
of points. Fast kernel summation algorithms reduce this cost to log-linear or
linear complexity.
Treecodes and Fast Multipole Methods (FMMs) deliver tremendous speedups by
constructing approximate representations of interactions of points that are far
from each other. In algebraic terms, these representations correspond to
low-rank approximations of blocks of the overall interaction matrix. Existing
approaches require an excessive number of kernel evaluations with increasing
and number of points in the dataset.
To address this issue, we use a randomized algebraic approach in which we
first sample the rows of a block and then construct its approximate, low-rank
interpolative decomposition. We examine the feasibility of this approach
theoretically and experimentally. We provide a new theoretical result showing a
tighter bound on the reconstruction error from uniformly sampling rows than the
existing state-of-the-art. We demonstrate that our sampling approach is
competitive with existing (but prohibitively expensive) methods from the
literature. We also construct kernel matrices for the Laplacian, Gaussian, and
polynomial kernels -- all commonly used in physics and data analysis. We
explore the numerical properties of blocks of these matrices, and show that
they are amenable to our approach. Depending on the data set, our randomized
algorithm can successfully compute low rank approximations in high dimensions.
We report results for data sets with ambient dimensions from four to 1,000.Comment: 43 pages, 21 figure
Sensitivity analysis of a coupled hydrodynamic-vegetation model using the effectively subsampled quadratures method (ESQM v5.2)
Abstract. Coastal hydrodynamics can be greatly affected by the presence of submerged aquatic vegetation. The effect of vegetation has been incorporated into the Coupled Ocean–Atmosphere–Wave–Sediment Transport (COAWST) modeling system. The vegetation implementation includes the plant-induced three-dimensional drag, in-canopy wave-induced streaming, and the production of turbulent kinetic energy by the presence of vegetation. In this study, we evaluate the sensitivity of the flow and wave dynamics to vegetation parameters using Sobol' indices and a least squares polynomial approach referred to as the Effective Quadratures method. This method reduces the number of simulations needed for evaluating Sobol' indices and provides a robust, practical, and efficient approach for the parameter sensitivity analysis. The evaluation of Sobol' indices shows that kinetic energy, turbulent kinetic energy, and water level changes are affected by plant stem density, height, and, to a lesser degree, diameter. Wave dissipation is mostly dependent on the variation in plant stem density. Performing sensitivity analyses for the vegetation module in COAWST provides guidance to optimize efforts and reduce exploration of parameter space for future observational and modeling work.
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