Christoffel Function Research Articles

Data-driven polynomial chaos expansions: A weighted least-square approximation

In this work, we combine the idea of data-driven polynomial chaos expansions with the weighted least-square approach to solve uncertainty quantification (UQ) problems. The idea of data-driven polynomial chaos is to use statistical moments of the input random variables to develop an arbitrary polynomial chaos expansion, and then use such data-driven bases to perform UQ computations. Here we adopt the bases construction procedure by following (Ahlfeld et al. (2016), [1]), where the bases are computed by using matrix operations on the Hankel matrix of moments. Different from previous works, in the postprocessing part, we propose a weighted least-squares approach to solve UQ problems. This approach includes a sampling strategy and a least-squares solver. The main features of our approach are two folds: On one hand, our sampling strategy is independent of the random input. More precisely, we propose to sampling with the equilibrium measure, and this measure is also independent of the data-driven bases. Thus, this procedure can be done in prior (or in a off-line manner). On the other hand, we propose to solve a Christoffel function weighted least-square problem, and this strategy is quasi-linearly stable – the required number of PDE solvers depends linearly (up to a logarithmic factor) on the number of (data-driven) bases. This new approach is thus promising in dealing with a class of problems with epistemic uncertainties. A number of numerical tests are presented to show the effectiveness of our approach.

On Numerical Estimation of Joint Probability Distribution from Lebesgue Integral Quadratures

An important application of Lebesgue integral quadrature arXiv:1807.06007 is developed. Given two random processes, $f(x)$ and $g(x)$, two generalized eigenvalue problems can be formulated and solved. In addition to obtaining two Lebesgue quadratures (for $f$ and $g$) from two eigenproblems, the projections of $f$- and $g$- eigenvectors on each other allow to build a joint distribution estimator, the most general form of which is a density-matrix correlation. Examples of the density-matrix correlation can be a value-correlation $V_{f^{[i]};g^{[j]}}$, similar to a regular correlation concept, and a new one, a probability-correlation $P_{f^{[i]};g^{[j]}}$. If Christoffel function average is used instead of regular average the approach can be extended to an estimation of joint probability of three and more random processes. The theory is implemented numerically; the software is available under the GPLv3 license.

Weighted Approximate Fekete Points: Sampling for Least-Squares Polynomial Approximation

We propose and analyze a weighted greedy scheme for computing deterministic sample configurations in multidimensional space for performing least-squares polynomial approximations on $L^2$ spaces weighted by a probability density function. Our procedure is a particular weighted version of the approximate Fekete points method, with the weight function chosen as the (inverse) Christoffel function. Our procedure has theoretical advantages: when linear systems with optimal condition number exist, the procedure finds them. In the one-dimensional setting with any density function, our greedy procedure almost always generates optimally-conditioned linear systems. Our method also has practical advantages: our procedure is impartial to compactness of the domain of approximation, and uses only pivoted linear algebraic routines. We show through numerous examples that our sampling design outperforms competing randomized and deterministic designs when the domain is both low and high dimensional.

On the Christoffel Function for the Generalized Jacobi Measures on a Quasidisk

We establish the exact (up to the constants) double inequality for the Christoffel function for a measure supported on a Jordan domain bounded by a quasiconformal curve. We show that this quasiconformality of the boundary cannot be omitted.

Upper estimates of Christoffel function on convex domains

New upper bounds on the pointwise behavior of Christoffel function on convex domains in Rd are obtained. These estimates are established by explicitly constructing the corresponding “needle”-like algebraic polynomials having small integral norm on the domain, and are stated in terms of few easy-to-measure geometric characteristics of the location of the point of interest in the domain. Sharpness of the results is shown and examples of applications are given.

A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions

In this paper we propose an algorithm for recovering sparse orthogonal polynomials using stochastic collocation. Our approach is motivated by the desire to use generalized polynomial chaos expansions (PCE) to quantify uncertainty in models subject to uncertain input parameters. The standard sampling approach for recovering sparse polynomials is to use Monte Carlo (MC) sampling of the density of orthogonality. However MC methods result in poor function recovery when the polynomial degree is high. Here we propose a general algorithm that can be applied to any admissible weight function on a bounded domain and a wide class of exponential weight functions defined on unbounded domains. Our proposed algorithm samples with respect to the weighted equilibrium measure of the parametric domain, and subsequently solves a preconditioned $\ell^1$-minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. Numerical examples are also provided that demonstrate that our proposed Christoffel Sparse Approximation algorithm leads to comparable or improved accuracy even when compared with Legendre and Hermite specific algorithms.

A Christoffel function weighted least squares algorithm for collocation approximations

We propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation frame- work. Our method is motivated by generalized Polynomial Chaos approximation in uncertainty quantification where a polynomial approximation is formed from a combination of orthogonal polynomials. A standard Monte Carlo approach would draw samples according to the density of orthogonality. Our proposed algorithm samples with respect to the equilibrium measure of the parametric domain, and subsequently solves a weighted least-squares problem, with weights given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest.

On Nikol’skii Inequalities for Domains in $${\mathbb {R}}^d$$ R d

Nikol'skii inequalities for various sets of functions, domains and weights will be discussed. Much of the work is dedicated to the class of algebraic polynomials of total degree $n$ on a bounded convex domain $D$. That is, we study $\sigma:= \sigma(D,d)$ for which \[ \|P\|_{L_q(D)}\le c n^{\sigma(\frac1p-\frac1q)}\|P\|_{L_p(D)},\quad 0<p\le q\le\infty, \] where $P$ is a polynomial of total degree $n$. We use geometric properties of the boundary of $D$ to determine $\sigma(D,n)$ with the aid of comparison between domains. Computing the asymptotics of the Christoffel function of various domains is crucial in our investigation. The methods will be illustrated by the numerous examples in which the optimal $\sigma(D,n)$ will be computed explicitly.

An Orthogonality Property of the Legendre Polynomials

We give a remarkable additional othogonality property of the classical Legendre polynomials on the real interval \([-1,1]\): polynomials up to degree n from this family are mutually orthogonal under the arcsine measure weighted by the normalized degree-n Christoffel function.

Multiple – Instance Learning: Christoffel Function Approach to Distribution Regression Problem

A two – step Christoffel function based solution is proposed to distribution regression problem. On the first step, to model distribution of observations inside a bag, build Christoffel function for each bag of observations. Then, on the second step, build outcome variable Christoffel function, but use the bag’s Christoffel function value at given point as the weight for the bag’s outcome. The approach allows the result to be obtained in closed form and then to be evaluated numerically. While most of existing approaches minimize some kind an error between outcome and prediction, the proposed approach is conceptually different, because it uses Christoffel function for knowledge representation, what is conceptually equivalent working with probabilities only. To receive possible outcomes and their probabilities Gauss quadrature for second–step measure can be built, then the nodes give possible outcomes and normalized weights – outcome probabilities. A library providing numerically stable polynomial basis for these calculations is available, what make the proposed approach practical.

An estimate for derivative of the de la Vallée Poussin mean

In this paper, we discuss derivative of the de la Vallee Poussin mean for exponential weights on real line. When we lead an inequality, an estimate for the Christoffel function plays an important role.

Christoffel functions for doubling measures on quasismooth curves and arcs

Matching two-sided estimates are given for Christoffel functions associated with a doubling measure ν over a quasismooth curve or arc. The size of the the n-th Christoffel function at a point z is given by the ν-measure of the largest disk about z which lies within the 1/n-level line of the Green’s function. The main theorem contains as special case all previously known weak asymptotics for Christoffel functions, and it also gives their size in explicit form about smooth corners. Applications are given for estimating the size of orthonormal polynomials and for Nikolskii-type inequalities.

On generalized Christoffel functions

The generalized Christoffel function λ p,q,n (dμ;x) (0<p<∞, 0≦q<∞) with respect to a measure dμ on R is defined by $$\lambda_{p,q,n}(d\mu;x)=\inf_{Q\in\mathbf{P}_{n-1},\ Q(x)=1}\int_{\mathbf{R}} \big|Q(t)\big|^p {|t-x|}^q\, d\mu(t).$$ The novelty of our definition is that it contains the factor |t−x| q , which is of particular interest. Its properties are discussed and estimates are given. In particular, upper and lower bounds for generalized Christoffel functions with respect to generalized Jacobi weights are also provided.

A trigonometric orthogonality with respect to a nonnegative Borel measure

In this paper we consider trigonometric polynomials of semi-integer degree orthogonal with respect to a linear functional, defined by a nonnegative Borel measure. By using a suitable vector form we consider the corresponding Fourier sums and reproducing kernels for trigonometric polynomials of semi- integer degree. Also, we consider the Christoffel function, and prove that it satisfies extremal property analogous with the algebraic case.

Christoffel functions on curves and domains

Asymptotics for Christoffel functions are established for measures supported on unions of smooth Jordan curves and for area-like measures on unions of smooth Jordan domains. For example, in the former casenntimes thenn-th Christoffel function tends to the Radon-Nikodym derivative of the measure with respect to the equilibrium distribution of the support of the measure.

Inequalities with exponential weights

Let R = ( - ∞ , ∞ ) and let Q ∈ C 2 : R → R + = [ 0 , ∞ ) be an even function. Then in this paper we consider the infinite–finite range inequality, an estimate for the Christoffel function, and the Markov–Bernstein inequality with the exponential weights w ρ ( x ) = | x | ρ e - Q ( x ) , x ∈ R .

The Christoffel function for the Hermite weight is bell-shaped

We show that the Christoffel function λ n associated with the Hermite weight function w H ( x)=exp(− x 2) is bell-shaped. As a consequence, we describe completely how the weights in a Gauss-type quadrature formula associated with w H ( x) are arranged in magnitude.

Asymptotics for Christoffel functions for general measures on the real line

We consider asymptotics of Christoffel functions for measures ν with compact support on the real line. It is shown that under some natural conditionsn times thenth Christoffel function has a limit asn→∞ almost everywhere on the support, and the limit is the Radon-Nikodym derivative of ν with respect to the equilibrium measure of the support of ν. The case in which the support is an interval was settled previously by A. Mate, P. Nevai and the author. The present paper solves the general problem.

Asymptotic Behaviour of Orthogonal Polynomials on the Unit Circle with Asymptotically Periodic Reflection Coefficients: II. Weak Asymptotics

In this paper we study orthogonal polynomials with asymptotically periodic reflection coefficients. It's known that the support of the orthogonality measure of such polynomials consists of several arcs. We are mainly interested in the asymptotic behaviour on the support and derive weak convergence results for the orthogonal polynomials and also for the Christoffel function.

The Christoffel Function for Orthogonal Polynomials on a Circular Arc

For the special type of weight functions on circular arc we study the asymptotic behavior of the Christoffel kernel off the arc and of the Christoffel function inside the arc. We prove Totik's conjecture for the Christoffel function corresponding to such weight functions.