104 research outputs found
Near-optimal perfectly matched layers for indefinite Helmholtz problems
A new construction of an absorbing boundary condition for indefinite
Helmholtz problems on unbounded domains is presented. This construction is
based on a near-best uniform rational interpolant of the inverse square root
function on the union of a negative and positive real interval, designed with
the help of a classical result by Zolotarev. Using Krein's interpretation of a
Stieltjes continued fraction, this interpolant can be converted into a
three-term finite difference discretization of a perfectly matched layer (PML)
which converges exponentially fast in the number of grid points. The
convergence rate is asymptotically optimal for both propagative and evanescent
wave modes. Several numerical experiments and illustrations are included.Comment: Accepted for publication in SIAM Review. To appear 201
Asymptotic Uniqueness of Best Rational Approximants to Complex Cauchy Transforms in of the Circle
For all n large enough, we show uniqueness of a critical point in best
rational approximation of degree n, in the L^2-sense on the unit circle, to
functions f, where f is a sum of a Cauchy transform of a complex measure \mu
supported on a real interval included in (-1,1), whose Radon-Nikodym derivative
with respect to the arcsine distribution on its support is Dini-continuous,
non-vanishing and with and argument of bounded variation, and of a rational
function with no poles on the support of \mu.Comment: 28 page
Rational RBF-based partition of unity method for efficiently and accurately approximating 3D objects
We consider the problem of reconstructing 3D objects via meshfree
interpolation methods. In this framework, we usually deal with large data sets
and thus we develop an efficient local scheme via the well-known Partition of
Unity (PU) method. The main contribution in this paper consists in constructing
the local interpolants for the implicit interpolation by means of Rational
Radial Basis Functions (RRBFs). Numerical evidence confirms that the proposed
method is particularly performing when 3D objects, or more in general implicit
functions defined by scattered data, need to be approximated
Sparse Modelling and Multi-exponential Analysis
The research fields of harmonic analysis, approximation theory and computer algebra are seemingly different domains and are studied by seemingly separated research communities. However, all of these are connected to each other in many ways. The connection between harmonic analysis and approximation theory is not accidental: several constructions among which wavelets and Fourier series, provide major insights into central problems in approximation theory. And the intimate connection between approximation theory and computer algebra exists even longer: polynomial interpolation is a long-studied and important problem in both symbolic and numeric computing, in the former to counter expression swell and in the latter to construct a simple data model. A common underlying problem statement in many applications is that of determining the number of components, and for each component the value of the frequency, damping factor, amplitude and phase in a multi-exponential model. It occurs, for instance, in magnetic resonance and infrared spectroscopy, vibration analysis, seismic data analysis, electronic odour recognition, keystroke recognition, nuclear science, music signal processing, transient detection, motor fault diagnosis, electrophysiology, drug clearance monitoring and glucose tolerance testing, to name just a few. The general technique of multi-exponential modeling is closely related to what is commonly known as the Padé-Laplace method in approximation theory, and the technique of sparse interpolation in the field of computer algebra. The problem statement is also solved using a stochastic perturbation method in harmonic analysis. The problem of multi-exponential modeling is an inverse problem and therefore may be severely ill-posed, depending on the relative location of the frequencies and phases. Besides the reliability of the estimated parameters, the sparsity of the multi-exponential representation has become important. A representation is called sparse if it is a combination of only a few elements instead of all available generating elements. In sparse interpolation, the aim is to determine all the parameters from only a small amount of data samples, and with a complexity proportional to the number of terms in the representation. Despite the close connections between these fields, there is a clear lack of communication in the scientific literature. The aim of this seminar is to bring researchers together from the three mentioned fields, with scientists from the varied application domains.Output Type: Meeting Repor
Randomized sketching of nonlinear eigenvalue problems
Rational approximation is a powerful tool to obtain accurate surrogates for
nonlinear functions that are easy to evaluate and linearize. The interpolatory
adaptive Antoulas--Anderson (AAA) method is one approach to construct such
approximants numerically. For large-scale vector- and matrix-valued functions,
however, the direct application of the set-valued variant of AAA becomes
inefficient. We propose and analyze a new sketching approach for such functions
called sketchAAA that, with high probability, leads to much better approximants
than previously suggested approaches while retaining efficiency. The sketching
approach works in a black-box fashion where only evaluations of the nonlinear
function at sampling points are needed. Numerical tests with nonlinear
eigenvalue problems illustrate the efficacy of our approach, with speedups
above 200 for sampling large-scale black-box functions without sacrificing on
accuracy.Comment: 15 page
Stable Computations with Flat Radial Basis Functions Using Vector-Valued Rational Approximations
One commonly finds in applications of smooth radial basis functions (RBFs) that scaling the kernels so they are \u27flat\u27 leads to smaller discretization errors. However, the direct numerical approach for computing with flat RBFs (RBF-Direct) is severely ill-conditioned. We present an algorithm for bypassing this ill-conditioning that is based on a new method for rational approximation (RA) of vector-valued analytic functions with the property that all components of the vector share the same singularities. This new algorithm (RBF-RA) is more accurate, robust, and easier to implement than the Contour-Padé method, which is similarly based on vector-valued rational approximation. In contrast to the stable RBF-QR and RBF-GA algorithms, which are based on finding a better conditioned base in the same RBF-space, the new algorithm can be used with any type of smooth radial kernel, and it is also applicable to a wider range of tasks (including calculating Hermite type implicit RBF-FD stencils). We present a series of numerical experiments demonstrating the effectiveness of this new method for computing RBF interpolants in the flat regime. We also demonstrate the flexibility of the method by using it to compute implicit RBF-FD formulas in the flat regime and then using these for solving Poisson\u27s equation in a 3-D spherical shell
Formulas for Continued Fractions. An Automated Guess and Prove Approach
We describe a simple method that produces automatically closed forms for the
coefficients of continued fractions expansions of a large number of special
functions. The function is specified by a non-linear differential equation and
initial conditions. This is used to generate the first few coefficients and
from there a conjectured formula. This formula is then proved automatically
thanks to a linear recurrence satisfied by some remainder terms. Extensive
experiments show that this simple approach and its straightforward
generalization to difference and -difference equations capture a large part
of the formulas in the literature on continued fractions.Comment: Maple worksheet attache
Matrix methods for Pad\'e approximation: numerical calculation of poles, zeros and residues
A representation of the Pad\'e approximation of the -transform of a signal
as a resolvent of a tridiagonal matrix is given. Several formulas for the
poles, zeros and residues of the Pad\'e approximation in terms of the matrix
are proposed. Their numerical stability is tested and compared. Methods for
computing forward and backward errors are presented
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