Discrete Least-squares Approximation Research Articles

A least-squares Fourier frame method for nonlocal diffusion models on arbitrary domains

We introduce a least-squares Fourier frame method for solving nonlocal diffusion models with Dirichlet volume constraint on arbitrary domains. The mathematical structure of a frame rather than a basis allows using a discrete least-squares approximation on irregular domains and imposing non-periodic boundary conditions. The method has inherited the one-dimensional integral expression of Fourier symbols of the nonlocal diffusion operator from Fourier spectral methods for any d spatial dimensions. High precision of its solution can be achieved via a direct solver such as pivoted QR decomposition even though the corresponding system is extremely ill-conditioned, due to the redundancy in the frame. The extension of AZ algorithm improves the complexity of solving the rectangular linear system to O(Nlog2⁡N) for 1d problems and O(N2log2⁡N) for 2d problems, compared with O(N3) of the direct solvers, where N is the number of degrees of freedom. We present ample numerical experiments to show the flexibility, fast convergence and asymptotical compatibility of the proposed method.

Chromatic dispersion equalization FIR filter design based on discrete least-squares approximation.

Chromatic dispersion (CD) equalization is one of the core tasks of the digital signal processing (DSP) chain in modern optical coherent receivers. A conventional impulse-invariant method for designing the CD equalization filter is revisited, improved by proper weighting, and reinterpreted as a Fourier series. To improve upon a direct evaluation of the passband least-squares (LS) approximation, we propose to design a CD equalization finite impulse response (FIR) filter based on a discrete LS approximation. The proposed method avoids numerical evaluation of nontrivial functions and relies only on Fourier transform. Its flexibility is corroborated by a filter design demonstration of joint matched filtering and CD equalization.

Function integration, reconstruction and approximation using rank-$1$ lattices

We consider rank-1 lattices for integration and reconstruction of functions with series expansion supported on a finite index set. We explore the connection between the periodic Fourier space and the non-periodic cosine space and Chebyshev space, via tent transform and then cosine transform, to transfer known results from the periodic setting into new insights for the non-periodic settings. Fast discrete cosine transform can be applied for the reconstruction phase. To reduce the size of the auxiliary index set in the associated component-by-component (CBC) construction for the lattice generating vectors, we work with a bi-orthonormal set of basis functions, leading to three methods for function reconstruction in the non-periodic settings. We provide new theory and efficient algorithmic strategies for the CBC construction. We also interpret our results in the context of general function approximation and discrete least-squares approximation.

Prediction of Temperature Distribution for Previous Cement Concrete Pavement with Asphalt Overlay

In this study, the temperature distribution of a pavement was predicted by developing an analytic algorithm. The Laplace and inverse Laplace transform techniques and Gaussian quadratic formula were applied to a pavement system of an asphalt overlay placed over an existing concrete pavement. The temperature distribution of the previous cement concrete pavement with an asphalt overlay can be estimated with the proposed analytical method regardless of the depth and time. To conduct the method, the layer thicknesses, material thermal properties and climatic factors (including air temperature, wind velocity and solar radiation) were firstly input. Then, a discrete least-squares approximation of the interpolatory trigonometric polynomials was used to fit some specific measured climatic factors considered in the surface boundary condition, i.e., the measured solar radiation intensity and air temperature. The pavement surface convection coefficient can be approximately calculated by the wind speed. The temperature solutions are validated with the measured pavement temperature data of two different periods of a whole year (summer and winter). The obtained results demonstrate the feasibility of the developed analytical approach to predict the temperature distribution of the existing cement concrete pavement with an asphalt overlay in different weather conditions with acceptable accuracy.

Computation of induced orthogonal polynomial distributions

We provide a robust and general algorithm for computing distribution functions associated to induced orthogonal polynomial measures. We leverage several tools for orthogonal polynomials to provide a spectrally-accurate method for a broad class of measures, encompassing those associated to classical orthogonal polynomial families, which is stable for polynomial degrees up to at least 1000. Paired with other standard tools such as a numerical root-finding algorithm and inverse transform sampling, this provides a methodology for generating random samples from an induced orthogonal polynomial measure. Generating samples from this measure is one ingredient in optimal numerical methods for certain types of multivariate polynomial approximation. For example, sampling from induced distributions for weighted discrete least-squares approximation has recently been shown to yield convergence guarantees with a minimal number of samples. We also provide publicly-available code that implements the algorithms in this paper for sampling from induced distributions.

Discrete Least-Squares Approximations over Optimized Downward Closed Polynomial Spaces in Arbitrary Dimension

We analyze the accuracy of the discrete least-squares approximation of a function \(u\) in multivariate polynomial spaces \(\mathbb {P}_\varLambda :=\mathrm{span} \{y\mapsto y^\nu : \nu \in \varLambda \}\) with \(\varLambda \subset \mathbb {N}_0^d\) over the domain \(\varGamma :=[-1,1]^d\), based on the sampling of this function at points \(y^1,\ldots ,y^m \in \varGamma \). The samples are independently drawn according to a given probability density \(\rho \) belonging to the class of multivariate beta densities, which includes the uniform and Chebyshev densities as particular cases. Motivated by recent results on high-dimensional parametric and stochastic PDEs, we restrict our attention to polynomial spaces associated with downward closed sets \(\varLambda \) of prescribed cardinality n, and we optimize the choice of the space for the given sample. This implies, in particular, that the selected polynomial space depends on the sample. We are interested in comparing the error of this least-squares approximation measured in \(L^2(\varGamma ,d\rho )\) with the best achievable polynomial approximation error when using downward closed sets of cardinality n. We establish conditions between the dimension n and the size m of the sample, under which these two errors are proved to be comparable. Our main finding is that the dimension d enters only moderately in the resulting trade-off between m and n, in terms of a logarithmic factor \(\ln (d)\), and is even absent when the optimization is restricted to a relevant subclass of downward closed sets, named anchored sets. In principle, this allows one to use these methods in arbitrarily high or even infinite dimension. Our analysis builds upon (Chkifa et al. in ESAIM Math Model Numer Anal 49(3):815–837, 2015), which considered fixed and nonoptimized downward closed multi-index sets. Potential applications of the proposed results are found in the development and analysis of efficient numerical methods for computing the solution to high-dimensional parametric or stochastic PDEs, but are not limited to this area.

Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points

We study the accuracy of the discrete least-squares approximation on a finite-dimensional space of a real-valued target function from noisy pointwise evaluations at independent random points distributed according to a given sampling probability measure. The convergence estimates are given in mean-square sense with respect to the sampling measure. The noise may be correlated with the location of the evaluation and may have nonzero mean (offset). We consider both cases of bounded or square-integrable noise/offset. We prove conditions between the number of sampling points and the dimension of the underlying approximation space that ensure a stable and accurate approximation. Particular focus is on deriving estimates in probability within a given confidence level. We analyze how the best approximation error and the noise terms affect the convergence rate and the overall confidence level achieved by the convergence estimate. The proofs of our convergence estimates in probability use arguments from the theory of large deviations to bound the noise term. Finally we address the particular case of multivariate polynomial approximation spaces with any density in the beta family, including uniform and Chebyshev.

Analysis of discrete least squares on multivariate polynomial spaces with evaluations at low-discrepancy point sets

We analyze the stability and accuracy of discrete least squares on multivariate polynomial spaces to approximate a given function depending on a multivariate random variable uniformly distributed on a hypercube. The polynomial approximation is calculated starting from pointwise noise-free evaluations of the target function at low-discrepancy point sets. We prove that the discrete least-squares approximation, in a multivariate anisotropic tensor product polynomial space and with evaluations at low-discrepancy point sets, is stable and accurate under the condition that the number of evaluations is proportional to the square of the dimension of the polynomial space, up to logarithmic factors. This result is analogous to those obtained in Cohen et al. (2013), Migliorati et al. (2014), Migliorati (2013) and Chkifa et al. (in press) for discrete least squares with random point sets, however it holds with certainty instead of just with high probability. The result is further generalized to arbitrary polynomial spaces associated with downward closed multi-index sets, but with a more demanding (and probably nonoptimal) proportionality between the number of evaluation points and the dimension of the polynomial space.

A new stable basis for radial basis function interpolation

It is well known that radial basis function interpolants suffer from bad conditioning if the basis of translates is used. In the recent work by Pazouki and Schaback (2011), [5], the authors gave a quite general way to build stable and orthonormal bases for the native space NΦ(Ω) associated to a kernel Φ on a domain Ω⊂Rs. The method is simply based on the factorization of the corresponding kernel matrix. Starting from that setting, we describe a particular basis which turns out to be orthonormal in NΦ(Ω) and in ℓ2,w(X), where X is a set of data sites of the domain Ω. The basis arises from a weighted singular value decomposition of the kernel matrix. This basis is also related to a discretization of the compact operator TΦ:NΦ(Ω)→NΦ(Ω), TΦ[f](x)=∫ΩΦ(x,y)f(y)dy∀x∈Ω, and provides a connection with the continuous basis that arises from an eigendecomposition of TΦ. Finally, using the eigenvalues of this operator, we provide convergence estimates and stability bounds for interpolation and discrete least-squares approximation.

One-Dimensional Rigid Pavement Temperature Prediction Using Laplace Transformation

AbstractThis paper presents an analytical solution for the prediction of the time-dependent one-dimensional temperature profile in an N-layered rigid pavement system using Laplace transformation. Temperature at any depth in a rigid pavement system can be estimated by using the proposed solution with input data such as pavement layer thicknesses, material thermal properties, and the measured air temperature and solar radiation intensities. The associated initial boundary value problem is governed by the classic heat conduction equation with the air temperature and solar radiation considered in the surface boundary condition. Interpolatory trigonometric polynomials, based on the method of discrete least-squares approximation, are used to fit the measured air temperature and solar radiation intensities during a user-defined time period. The relationship between the constants of integration for the ith layer and the first layer are deduced, which simplifies the solution process. Model validation with the meas...

Continuous and discrete least-squares approximation by radial basis functions on spheres

In this paper we discuss Sobolev bounds on functions that vanish at scattered points on the n-sphere S n in R n + 1 . The Sobolev spaces involved may have fractional as well as integer order. We then apply these results to obtain estimates for continuous and discrete least-squares surface fits via radial basis functions (RBFs). We also address a stabilization or regularization technique known as spline smoothing.

On numerical methods for discrete least-squares approximation by trigonometric polynomials

Fast, efficient and reliable algorithms for discrete least-squares approximation of a real-valued function given at arbitrary distinct nodes in [ 0 , 2 π ) [0,2\pi ) by trigonometric polynomials are presented. The algorithms are based on schemes for the solution of inverse unitary eigenproblems and require only O ( m n ) (mn) arithmetic operations as compared to O ( m n 2 ) (mn^2) operations needed for algorithms that ignore the structure of the problem. An algorithm which solves this problem with real-valued data and real-valued solution using only real arithmetic is given. Numerical examples are presented that show that the proposed algorithms produce consistently accurate results that are often better than those obtained by general QR decomposition methods for the least-squares problem.

Polynomial Interpolation and Hyperinterpolation over General Regions

This paper studies a generalization of polynomial interpolation: given a continuous function over a rather general manifold, hyperinterpolation is a linear approximation that makes use of values of f on a well chosen finite set. The approximation is a discrete least-squares approximation constructed with the aid of a high-order quadrature rule: the role of the quadrature rule is to approximate the Fourier coefficients of f with respect to an orthonormal basis of the space of polynomials of degree ≤ n. The principal result is a generalization of the result of Erdös and Turan for classical interpolation at the zeros of orthogonal polynomials: for a rule of suitably high order (namely 2 n or greater), the L 2 error of the approximation is shown to be within a constant factor of the error of best uniform approximation by polynomials of degree ≤ n. The L 2 error therefore converges to zero as the degree of the approximating polynomial approaches ∞. An example discussed in detail is the approximation of continuous functions on the sphere in R s by spherical polynomials. In this case the number of quadrature points must exceed the number of degrees of freedom if n > 2 and s ≥ 3. In such a situation the classical interpolation property cannot hold, whereas satisfactory hyperinterpolation approximations do exist.

Finite sequences of orthogonal polynomials connected by a Jacobi matrix

If p 0,…, p n is an orthogonal sequence, with p j a monic polynomial of exact degree j, all j, then the sequence q 0,…, q n of monic polynomials defined by the requirement that p j = c j q n − j−1 p n−1 at the zeros of p n (with c j suitable constants) is also orthogonal (with respect to a possibly differently weighted scalar product). These two sequences can be shown to be the upper left, respectively lower right, principal minors of the matrix t – J, with J a suitable tridiagonal matrix. The sequence q occurs in the characterization of the discrete least-squares approximation to f from span( p 0,…, p j ) in terms of the ( n− j−1)-order divided differences of f p n−1 at the nodes, i.e., the zeros of p n . A complete characterization of the pair ( U,V) of zero sets of p j−1 and q n − j is given, and an application is made to the problem of recovering a Jacobi matrix from such data.

Estimation of the error in the reduced basis method solution of nonlinear equations

The reduced basis method is a projection technique for approximating the solution curve of a finite system of nonlinear algebraic equations by the solution curve of a related system that is typically of much lower dimension. In this paper, the reduced basis error is shown to be dominated by an approximation error. This, in turn, leads to error estimates for projection onto specific subspaces; for example, subspaces related to Taylor, Lagrange and discrete least-squares approximation.

Introduction to Approximation Theory

Introduction: 1 Examples and prospectus 2 Metric spaces 3 Normed linear spaces 4 Inner-product spaces 5 Convexity 6 Existence and unicity of best approximations 7 Convex functions The Tchebycheff Solution of Inconsistent Linear Equations: 1 Introduction 2 Systems of equations with one unknown 3 Characterization of the solution 4 The special case 5 Polya's algorithm 6 The ascent algorithm 7 The descent algorithm 8 Convex programming Tchebycheff Approximation by Polynomials and Other Linear Families: 1 Introduction 2 Interpolation 3 The Weierstrass theorem 4 General linear families 5 the unicity problem 6 Discretization errors: General theory 7 Discretization: Algebraic polynomials. The inequalities of Markoff and Bernstein 8 Algorithms Least-squares Approximation and Related Topics: 1 Introduction 2 Orthogonal systems of polynomials 3 Convergence of orthogonal expansions 4 Approximation by series of Tchebycheff polynomials 5 Discrete least-squares approximation 6 The Jackson theorems Rational Approximation: 1 Introduction 2 Existence of best rational approximations 3 The characterization of best approximations 4 Unicity Continuity of best-approximation operators 5 Algorithms 6 Pade Approximation and its generalizations 7 Continued fractions Some Additional Topics: 1 The Stone approximation theorem 2 The Muntz theorem 3 The converses of the Jackson theorems 4 Polygonal approximation and bases in $C[a, b]$ 5 The Kharshiladze-Lozinski theorems 6 Approximation in the mean Notes References Index.