2,109 research outputs found
A rapid and efficient isogeometric design space exploration framework with application to structural mechanics
In this paper, we present an isogeometric analysis framework for design space exploration. While the methodology is presented in the setting of structural mechanics, it is applicable to any system of parametric partial differential equations. The design space exploration framework elucidates design parameter sensitivities used to inform initial and early-stage design. Moreover, this framework enables the visualization of a full system response, including the displacement and stress fields throughout the domain, by providing an approximation to the system solution vector. This is accomplished through a collocation-like approach where various geometries throughout the design space under consideration are sampled. The sampling scheme follows a quadrature rule while the physical solutions to these sampled geometries are obtained through an isogeometric method. A surrogate model to the design space solution manifold is constructed through either an interpolating polynomial or pseudospectral expansion. Examples of this framework are presented with applications to the Scordelis–Lo roof, a Flat L-Bracket, and an NREL 5 MW wind turbine blade
Trigonometric Quadrature Fourier Features for Scalable Gaussian Process Regression
Fourier feature approximations have been successfully applied in the
literature for scalable Gaussian Process (GP) regression. In particular,
Quadrature Fourier Features (QFF) derived from Gaussian quadrature rules have
gained popularity in recent years due to their improved approximation accuracy
and better calibrated uncertainty estimates compared to Random Fourier Feature
(RFF) methods. However, a key limitation of QFF is that its performance can
suffer from well-known pathologies related to highly oscillatory quadrature,
resulting in mediocre approximation with limited features. We address this
critical issue via a new Trigonometric Quadrature Fourier Feature (TQFF)
method, which uses a novel non-Gaussian quadrature rule specifically tailored
for the desired Fourier transform. We derive an exact quadrature rule for TQFF,
along with kernel approximation error bounds for the resulting feature map. We
then demonstrate the improved performance of our method over RFF and Gaussian
QFF in a suite of numerical experiments and applications, and show the TQFF
enjoys accurate GP approximations over a broad range of length-scales using
fewer features
The Surface Laplacian Technique in EEG: Theory and Methods
This paper reviews the method of surface Laplacian differentiation to study
EEG. We focus on topics that are helpful for a clear understanding of the
underlying concepts and its efficient implementation, which is especially
important for EEG researchers unfamiliar with the technique. The popular
methods of finite difference and splines are reviewed in detail. The former has
the advantage of simplicity and low computational cost, but its estimates are
prone to a variety of errors due to discretization. The latter eliminates all
issues related to discretization and incorporates a regularization mechanism to
reduce spatial noise, but at the cost of increasing mathematical and
computational complexity. These and several others issues deserving further
development are highlighted, some of which we address to the extent possible.
Here we develop a set of discrete approximations for Laplacian estimates at
peripheral electrodes and a possible solution to the problem of multiple-frame
regularization. We also provide the mathematical details of finite difference
approximations that are missing in the literature, and discuss the problem of
computational performance, which is particularly important in the context of
EEG splines where data sets can be very large. Along this line, the matrix
representation of the surface Laplacian operator is carefully discussed and
some figures are given illustrating the advantages of this approach. In the
final remarks, we briefly sketch a possible way to incorporate finite-size
electrodes into Laplacian estimates that could guide further developments.Comment: 43 pages, 8 figure
Interpolating the Trace of the Inverse of Matrix
We develop heuristic interpolation methods for the function , where the
matrices and are symmetric and positive definite and
is a real variable. This function is featured in many applications in
statistics, machine learning, and computational physics. The presented
interpolation functions are based on the modification of a sharp upper bound
that we derive for this function, which is a new trace inequality for matrices.
We demonstrate the accuracy and performance of the proposed method with
numerical examples, namely, the marginal maximum likelihood estimation for
linear Gaussian process regression and the estimation of the regularization
parameter of ridge regression with the generalized cross-validation method
- …