An important problem in applications is the approximation of a function $f$ from a finite set of randomly scattered data $f(x_j)$. A common and powerful approach is to construct a trigonometric least squares approximation based on the set of exponentials $\{e^{2πi kx}\}$. This leads to fast numerical algorithms, but suffers from disturbing boundary effects due to the underlying periodicity assumption on the data, an assumption that is rarely satisfied in practice. To overcome this drawback we impose Neumann boundary conditions on the data. This implies the use of cosine polynomials $\cos (πkx)$ as basis functions. We show that scattered data approximation using cosine polynomials leads to a least squares problem involving certain Toeplitz+Hankel matrices. We derive estimates on the condition number of these matrices. Unlike other Toeplitz+Hankel matrices, the Toeplitz+Hankel matrices arising in our context cannot be diagonalized by the discrete cosine transform, but they still allow a fast matrix-vector multiplication via DCT which gives rise to fast conjugate gradient type algorithms. We show how the results can be generalized to higher dimensions. Finally we demonstrate the performance of the proposed method by applying it to a two-dimensional geophysical scattered data problem.
Spherical Gauss-Laguerre (SGL) basis functions, i. e., normalized functions of the type $L_{n-l-1}^{(l + 1/2)}(r^2) r^l Y_{lm}(\vartheta,φ)$, $|m| \leq l < n \in \mathbb{N}$, $L_{n-l-1}^{(l + 1/2)}$ being a generalized Laguerre polynomial, $Y_{lm}$ a spherical harmonic, constitute an orthonormal polynomial basis of the space $L^2$ on $\mathbb{R}^3$ with radial Gaussian (multivariate Hermite) weight $\exp(-r^2)$. We have recently described fast Fourier transforms for the SGL basis functions based on an exact quadrature formula with certain grid points in $\mathbb{R}^3$. In this paper, we presen…
Choosing models from a hypothesis space is a frequent task in approximation theory and inverse problems. Cross-validation is a classical tool in the learner's repertoire to compare the goodness of fit for different reconstruction models. Much work has been dedicated to computing this quantity in a fast manner but tackling its theoretical properties occurs to be difficult. So far, most optimality results are stated in an asymptotic fashion. In this paper we propose a concentration inequality on the difference of cross-validation score and the risk functional with respect to the squared error. T…
We present a non-nested multilevel algorithm for solving the Poisson equation discretized at scattered points using polyharmonic radial basis function (PHS-RBF) interpolations. We append polynomials to the radial basis functions to achieve exponential convergence of discretization errors. The interpolations are performed over local clouds of points and the Poisson equation is collocated at each of the scattered points, resulting in a sparse set of discrete equations for the unkown variables. To solve this set of equations, we have developed a non-nested multilevel algorithm utilizing multiple …
We prove quantitative norm bounds for a family of operators involving impedance boundary conditions on convex, polygonal domains. A robust numerical construction of Helmholtz scattering solutions in variable media via the Dirichlet-to-Neumann operator involves a decomposition of the domain into a sequence of rectangles of varying scales and constructing impedance-to-impedance boundary operators on each subdomain. Our estimates in particular ensure the invertibility, with quantitative bounds in the frequency, of the merge operators required to reconstruct the original Dirichlet-to-Neumann opera…
We solve first-kind Fredholm boundary integral equations arising from Helmholtz and Laplace problems on bounded, smooth screens in three-dimensions with either Dirichlet or Neumann conditions. The proposed Galerkin-Bubnov method takes as discretization elements pushed-forward weighted azimuthal projections of standard spherical harmonics onto the canonical disk. We show that these bases allow for spectral convergence and provide fully discrete error analysis. Numerical experiments support our claims, with results comparable to Nyström-type and $hp$-methods.
In this paper we propose an approximation method for high-dimensional $1$-periodic functions based on the multivariate ANOVA decomposition. We provide an analysis on the classical ANOVA decomposition on the torus and prove some important properties such as the inheritance of smoothness for Sobolev type spaces and the weighted Wiener algebra. We exploit special kinds of sparsity in the ANOVA decomposition with the aim to approximate a function in a scattered data or black-box approximation scenario. This method allows us to simultaneously achieve an importance ranking on dimensions and dimensio…
We present an efficient method to solve the narrow capture and narrow escape problems for the sphere. The narrow capture problem models the equilibrium behavior of a Brownian particle in the exterior of a sphere whose surface is reflective, except for a collection of small absorbing patches. The narrow escape problem is the dual problem: it models the behavior of a Brownian particle confined to the interior of a sphere whose surface is reflective, except for a collection of small patches through which it can escape. Mathematically, these give rise to mixed Dirichlet/Neumann boundary value prob…
This paper is concerned with the error estimation of the fast multipole method (FMM) for scattering problems in 2-D. The FMM error is caused by truncating Graf's addition theorem in each step of the algorithm, including two expansions and three translations. We first give a novel bound on the truncation error of Graf's addition theorem by the limiting forms of Bessel and Neumann functions, and then estimate the error of the FMM. Explicit error bound and its convergence order are derived. The method proposed in this paper can also be used to the FMM for other problems, such as potential problem…
A solver for the Poisson equation for 1D, 2D and 3D regular grids is presented. The solver applies the convolution theorem in order to efficiently solve the Poisson equation in spectral space over a rectangular computational domain. Conversion to and from the spectral space is achieved through the use of discrete Fourier transforms, allowing for the application of highly optimised O(NlogN) algorithms. The data structure is configured to be modular such that the underlying interface for operations to, from and within the spectral space may be interchanged. For computationally demanding tasks, t…
In the present paper we describe a simple black box algorithm for efficiently and accurately solving scattering problems related to the scattering of time-harmonic waves from radially-symmetric potentials in two dimensions. The method uses FFTs to convert the problem into a set of decoupled second-kind Fredholm integral equations for the Fourier coefficients of the scattered field. Each of these integral equations are solved using scattering matrices, which exploit certain low-rank properties of the integral operators associated with the integral equations. The performance of the algorithm is …
We develop a practical approach to semidefinite programming (SDP) that includes the von Neumann entropy, or an appropriate variant, as a regularization term. In particular we solve the dual of the regularized program, demonstrating how a carefully chosen randomized trace estimator can be used to estimate dual gradients effectively. We also introduce specialized optimization approaches for common SDP, specifically SDP with diagonal constraint and the problem of the determining the spectral projector onto the span of extremal eigenvectors. We validate our approach on such problems with applicati…
We consider the problem of constructing transparent boundary conditions for the time-dependent Schrödinger equation with a compactly supported binding potential and, if desired, a spatially uniform, time-dependent electromagnetic vector potential. Such conditions prevent nonphysical boundary effects from corrupting a numerical solution in a bounded computational domain. We use ideas from potential theory to build exact nonlocal conditions for arbitrary piecewise-smooth domains. These generalize the standard Dirichlet-to-Neumann and Neumann-to-Dirichlet maps known for the equation in one dimens…
INI-VPINN: A Variational Physics-Informed Neural Network with Implicit Neumann and Interface Handling for Multi-Material Domains with Geometric Singularities
Finding a good regularization parameter for Tikhonov regularization problems is a though yet often asked question. One approach is to use leave-one-out cross-validation scores to indicate the goodness of fit. This utilizes only the noisy function values but, on the downside, comes with a high computational cost. In this paper we present a general approach to shift the main computations from the function in question to the node distribution and, making use of FFT and FFT-like algorithms, even reduce this cost tremendously to the cost of the Tikhonov regularization problem itself. We apply this …
