Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe boundary layer problem in Bayesian adaptive quadrature
71
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The boundary layer of a finite domain [a, b] covers mesoscopic lateral neighbourhoods, inside [a, b], of the endpoints a and b. The correct diagnostic of the integrand behaviour at a and b, based on its sampling inside the boundary layer, is the first from a set of hierarchically ordered criteria allowing a priori Bayesian inference on efficient mesh generation in automatic adaptive quadrature.
rate research
Read More
Adaptive Bayesian quadrature (ABQ) is a powerful approach to numerical integration that empirically compares favorably with Monte Carlo integration on problems of medium dimensionality (where non-adaptive quadrature is not competitive). Its key ingredient is an acquisition function that changes as a function of previously collected values of the integrand. While this adaptivity appears to be empirically powerful, it complicates analysis. Consequently, there are no theoretical guarantees so far for this class of methods. In this work, for a broad class of adaptive Bayesian quadrature methods, we prove consistency, deriving non-tight but informative convergence rates. To do so we introduce a new concept we call weak adaptivity. Our results identify a large and flexible class of adaptive Bayesian quadrature rules as consistent, within which practitioners can develop empirically efficient methods.
Numerical integration and emulation are fundamental topics across scientific fields. We propose novel adaptive quadrature schemes based on an active learning procedure. We consider an interpolative approach for building a surrogate posterior density, combining it with Monte Carlo sampling methods and other quadrature rules. The nodes of the quadrature are sequentially chosen by maximizing a suitable acquisition function, which takes into account the current approximation of the posterior and the positions of the nodes. This maximization does not require additional evaluations of the true posterior. We introduce two specific schemes based on Gaussian and Nearest Neighbors (NN) bases. For the Gaussian case, we also provide a novel procedure for fitting the bandwidth parameter, in order to build a suitable emulator of a density function. With both techniques, we always obtain a positive estimation of the marginal likelihood (a.k.a., Bayesian evidence). An equivalent importance sampling interpretation is also described, which allows the design of extended schemes. Several theoretical results are provided and discussed. Numerical results show the advantage of the proposed approach, including a challenging inference problem in an astronomic dynamical model, with the goal of revealing the number of planets orbiting a star.
We developed a fast numerical algorithm for solving the three dimensional vectorial Helmholtz equation that arises in electromagnetic scattering problems. The algorithm is based on electric field integral equations and is essentially a boundary element method. Nystroms quadrature rule with a triangular grid is employed to linearize the integral equations, which are then solved by using a right-preconditioned iterative method. We apply the fast multipole technique to accelerate the matrix-vector multiplications in the iterations. We demonstrate the broad applications and accuracy of this method with practical examples including dielectric, plasmonic and metallic objects. We then apply the method to investigate the plasmonic properties of a silver torus and a silver split-ring resonator under the incidence of an electromagnetic plane wave. We show the silver torus can be used as a trapping tool to bind small dielectric or metallic particles.
In this work we present an adaptive boundary element method for computing the electromagnetic response of wave interactions in hyperbolic metamaterials. One unique feature of hyperbolic metamaterial is the strongly directional wave in its propagating cone, which induces sharp transition for the solution of the integral equation across the cone boundary when wave starts to decay or grow exponentially. In order to avoid a global refined mesh over the whole boundary, we employ a two-level a posteriori error estimator and an adaptive mesh refinement procedure to resolve the singularity locally for the solution of the integral equation. Such an adaptive procedure allows for the reduction of the degree of freedom significantly for the integral equation solver while achieving desired accuracy for the solution. In addition, to resolve the fast transition of the fundamental solution and its derivatives accurately across the propagation cone boundary, adaptive numerical quadrature rules are applied to evaluate the integrals for the stiff matrices. Finally, in order to formulate the integral equations over the boundary, we also derive the limits of layer potentials and their derivatives in the hyperbolic media when the target points approach the boundary.
Marginalising over families of Gaussian Process kernels produces flexible model classes with well-calibrated uncertainty estimates. Existing approaches require likelihood evaluations of many kernels, rendering them prohibitively expensive for larger datasets. We propose a Bayesian Quadrature scheme to make this marginalisation more efficient and thereby more practical. Through use of the maximum mean discrepancies between distributions, we define a kernel over kernels that captures invariances between Spectral Mixture (SM) Kernels. Kernel samples are selected by generalising an information-theoretic acquisition function for warped Bayesian Quadrature. We show that our framework achieves more accurate predictions with better calibrated uncertainty than state-of-the-art baselines, especially when given limited (wall-clock) time budgets.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
