Optimal Polynomial Approximation Research Articles

An Optimal Linear-combination-of-unitaries-based Quantum Linear System Solver

Solving systems of linear equations is one of the most important primitives in many different areas, including in optimization, simulation, and machine learning. Quantum algorithms for solving linear systems have the potential to provide a quantum advantage for these problems. In this work, we recall the Chebyshev iterative method and the corresponding optimal polynomial approximation of the inverse. We show that the Chebyshev iteration polynomial can be efficiently evaluated both using quantum singular value transformation (QSVT) as well as linear combination of unitaries (LCU). We achieve this by bounding the 1-norm of the coefficients of the polynomial expressed in the Chebyshev basis. This leads to a considerable constant-factor improvement in the runtime of quantum linear system solvers that are based on LCU or QSVT (or, conversely, a several orders of magnitude smaller error with the same runtime/circuit depth).

Polynomial-Based Approach for Efficient Function Computation of Symmetric Matrices Using Restarted Heavy Ball Method

This paper presents a polynomial-based approach for efficiently computing functions ofsymmetric matrices by leveraging the Restarted Heavy Ball (RHB) method. The RHB method is employedto overcome the slow convergence issue commonly encountered when computing functions of symmetricmatrices. The key idea of our approach is to approximate the desired function using a polynomial. Byrepresenting the function as a polynomial, we can leverage the efficient computation of polynomials toaccelerate the overall function computation process. We introduce a systematic methodology forconstructing an optimal polynomial approximation that minimizes the approximation error. To furtherenhance the convergence speed, we incorporate the Restarted Heavy Ball method into our polynomialbased approach. The Restarted Heavy Ball iteration is applied after a certain number of iterations to resetthe computation process and mitigate the slow convergence behavior. The experimental results and analysisvalidate the effectiveness and practicality of our approach, highlighting its potential for various applicationsinvolving function computations of symmetric matrices. Overall, our polynomial-based approach,integrated with the Restarted Heavy Ball method, offers an efficient and accurate solution for computingfunctions of symmetric matrices. The experimental results and analysis validate the effectiveness andpracticality of our approach, highlighting its potential for various applications involving functioncomputations of symmetric matrices.

General Optimal Polynomial Approximants, Stabilization, and Projections of Unity

General Optimal Polynomial Approximants, Stabilization, and Projections of Unity

Density theorems with applications in quantum signal processing

We study the approximation capabilities of two families of univariate polynomials that arise in applications of quantum signal processing. Although approximation only in the domain [0,1] is physically desired, these polynomial families are defined by bound constraints not just in [0,1], but also with additional bound constraints outside [0,1]. One might wonder then if these additional constraints inhibit their approximation properties within [0,1]. The main result of this paper is that this is not the case — the additional constraints do not hinder the ability of these polynomial families to approximate arbitrarily well any continuous function f:[0,1]→[0,1] in the supremum norm, provided f also matches any polynomial in the family at 0 and 1. We additionally study the specific problem of approximating the step function on [0,1] (with the step from 0 to 1 occurring at x=12) using one of these families, and propose two subfamilies of monotone and non-monotone approximations. For the non-monotone case, under some additional assumptions, we provide an iterative heuristic algorithm that finds the optimal polynomial approximation.

Arbitrarily Sparse Polynomial Chaos Expansion for High-Dimensional Parametric Problems: Parametric and Probabilistic Power Flow as an Example

Parameters, no matter whether they are uncontrollable uncertainty factors or controllable variables, have a great impact on the states and performances of continuous engineering systems. Acquiring an explicit expression of the complicated implicit function between these parameters and system states, called parametric problem in this article, will facilitate immediate analysis of parameters’ impact on the system, optimal parameter design, and uncertainty quantification. Polynomial chaos expansion (PCE) is a globally optimal polynomial approximation method for parametric problems, but it suffers from the curse of dimensionality, i.e., the number of basis functions and consequent computational burden increases rapidly with the number of parameters (i.e., dimensions). This article proposes a PCE method characterized by the arbitrarily sparse basis and novel generalized Smolyak sparse grid quadrature and, hence, effectively relieves the curse of dimensionality for high-dimensional problems. This basis is not restricted by any existing fixed-form truncation and can merely incorporate a few important basis functions. The novel generalized sparse grid can remarkably reduce the number of required collocation points compared with the prevailing classic sparse grid while achieving high accuracy. The proposed method is universal for parametric problems in engineering systems and is exemplified by parametric and probabilistic power flow problems of electrical power systems. Its effectiveness is validated by computational results on the IEEE 30-bus, 118-bus, and 500-parameter 9241pegase systems.

Accurate Compression of Tabulated Chemistry Models with Partition of Unity Networks

ABSTRACT Tabulated chemistry models are widely used to simulate large-scale turbulent fires in applications including energy generation and fire safety. Tabulation via piecewise Cartesian interpolation suffers from the curse-of-dimensionality, leading to a prohibitive exponential growth in parameters and memory usage as more dimensions are considered. Artificial neural networks (ANNs) have attracted attention for constructing surrogates for chemistry models due to their ability to perform high-dimensional approximation. However, due to well-known pathologies regarding the realization of suboptimal local minima during training, in practice they do not converge and provide unreliable accuracy. Partition of unity networks (POUnets) are a recently introduced family of ANNs which preserve notions of convergence while performing high-dimensional approximation, discovering a mesh-free partition of space which may be used to perform optimal polynomial approximation. We assess their performance with respect to accuracy and model complexity in reconstructing unstructured flamelet data representative of nonadiabatic pool fire models. Our results show that POUnets can provide the desirable accuracy of classical spline-based interpolants with the low memory footprint of traditional ANNs while converging faster to significantly lower errors than ANNs. For example, we observe POUnets obtaining target accuracies in two dimensions with 40 to 50 times less memory and roughly double the compression in three dimensions. We also address the practical matter of efficiently training accurate POUnets by studying convergence over key hyperparameters, the impact of partition/basis formulation, and the sensitivity to initialization.

Zeros of optimal polynomial approximants in [formula omitted

The study of inner and cyclic functions in ℓAp spaces requires a better understanding of the zeros of the so-called optimal polynomial approximants. We determine that a point of the complex plane is the zero of an optimal polynomial approximant for some element of ℓAp if and only if it lies outside of a closed disk (centered at the origin) of a particular radius which depends on the value of p. We find the value of this radius for p≠2. In addition, for each positive integer d there is a polynomial fd of degree at most d that minimizes the modulus of the root of its optimal linear polynomial approximant. We develop a method for finding these extremal functions fd and discuss their properties. The method involves the Lagrange multiplier method and a resulting dynamical system.

Equidistribution of zeros of some polynomials related to cyclic functions

In the study of the cyclicity of a function f in reproducing kernel Hilbert spaces an important role is played by sequences of polynomials {p_n}_{nin mathbb {N}} called optimal polynomial approximants (o.p.a.). For many such spaces and when the functions f generating those o.p.a. are polynomials without zeros inside the disk but with some zeros on its boundary, we find that the weakly asympotic distribution of the zeros of 1-p_nf is the uniform measure on the unit circle.

Optimal Polynomial Approximants in Lp

Abstract Over the past several years, optimal polynomial approximants (OPAs) have been studied in many different function spaces. In these settings, numerous papers have been devoted to studying the properties of their zeros. In this paper, we introduce the notion of optimal polynomial approximant in the space Lp , 1 ≤ p ≤ ∞. We begin our treatment by showing existence and uniqueness for 1 < p < ∞. For the extreme cases of p = 1 and p = ∞, we show that uniqueness does not necessarily hold. We continue our development by elaborating on the special case of L 2. Here, we create a test to determine whether or not a given 1st degree OPA is zero-free in ̄𝔻. Afterward, we shed light on an orthogonality condition in Lp . This allows us to study OPAs in Lp with the additional tools from the L 2 setting. Throughout this paper, we focus many of our discussions on the zeros of OPAs. In particular, we show that if 1 < p < ∞, f ∈ Hp , and f(0) ≠ 0, then there exists a disk, centered at the origin, in which all the associated OPAs are zero-free. Toward the end of this paper, we use the orthogonality condition to compute the coefficients of some OPAs in Lp . To inspire further research in the general theory, we pose several open questions throughout our discussions.

A Cascaded Structure for Generalized Graph Filters

One of the main challenges of graph filters is the stability of their design. While classical graph filters allow for a stable design using optimal polynomial approximation theory, generalized graph filters tend to suffer from the ill-conditioning of the involved system matrix. This issue, accentuated for increasing graph filter orders, naturally leads to very large (small) filter coefficients or error saturation, casting a shadow on the benefits of these richer graph filter structures. In addition to this, data-driven design/learning of graph filters with large filter orders, even in the case of classical graph filters, suffers from the eigenvalue spread of the input data covariance matrix and mode coupling, leading to convergence-related issues as the ones observed when identifying time-domain filters with large orders. To alleviate these conditioning and convergence problems, and to reduce the overall design complexity, in this work, we propose a cascaded implementation of generalized graph filters and an efficient algorithm for designing the graph filter coefficients in both model- and data-driven settings. Further, we establish the connections of this implementation with so-called graph convolutional neural networks and demonstrate the performance of the proposed structure in different network applications. By the proposed approach, further error reduction and better design stability are achieved.

A survey of optimal polynomial approximants, applications to digital filter design, and related open problems

In the last few years, the notion of optimal polynomial approximant has appeared in the mathematics literature in connection with the Hilbert spaces of analytic functions of one or more variables. In the 1970s, researchers in engineering and applied mathematics introduced least-squares inverses in the context of digital filters in signal processing. It turns out that in the Hardy space \(H^{2},\) these objects are identical. This paper is a survey of known results about optimal polynomial approximants. In particular, we will examine their connections with orthogonal polynomials and reproducing kernels in weighted spaces and digital filter design. We will also describe what is known about the zeros of optimal polynomial approximants, their rates of decay, and convergence results. Throughout the paper, we state many open questions that may be of interest.

Optimal approximants and orthogonal polynomials in several variables

AbstractWe discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants and orthogonal polynomials in weighted spaces. Weakly inner functions, whose optimal approximants are all constant, provide extreme cases where nontrivial orthogonal polynomials cannot be recovered from the optimal approximants. Concrete examples are presented to illustrate the general theory and are used to disprove certain natural conjectures regarding zeros of optimal approximants in several variables.

Polynomial approach to cyclicity for weighted $$\ell ^p_A$$

In previous works, an approach to the study of cyclic functions in reproducing kernel Hilbert spaces has been presented, based on the study of so called optimal polynomial approximants. In the present article, we extend such approach to the (non-Hilbert) case of spaces of analytic functions whose Taylor coefficients are in $$\ell ^p(\omega )$$ , for some weight $$\omega $$ . When $$\omega =\{(k+1)^\alpha \}_{k\in \mathbb {N}}$$ , for a fixed $$\alpha \in \mathbb {R}$$ , we derive a characterization of the cyclicity of polynomial functions and, when $$1<p<\infty $$ , we obtain sharp rates of convergence of the optimal norms.

Boundary Behavior of Optimal Polynomial Approximants

In this paper, we provide an efficient method for computing the Taylor coefficients of $$1-p_n f$$ , where $$p_n$$ denotes the optimal polynomial approximant of degree n to 1/f in a Hilbert space $$H^2_\omega $$ of analytic functions over the unit disc $$\mathbb {D}$$ , and f is a polynomial of degree d with d simple zeros. As a consequence, we show that in many of the spaces $$H^2_\omega $$ , the sequence $$\{1-p_nf\}_{n\in \mathbb {N}}$$ is uniformly bounded on the closed unit disc and, if f has no zeros inside $$\mathbb {D}$$ , the sequence $$\{1-p_nf \}$$ converges uniformly to 0 on compact subsets of the complement of the zeros of f in $$\overline{\mathbb {D}}, $$ and we obtain precise estimates on the rate of convergence on compacta. We also obtain the precise constant in the rate of decay of the norm of $$1 - p_n f$$ in the previously unknown case of a function with a single zero of multiplicity greater than 1, when the weights are given by $$\omega _k = (k+1)^{\alpha }$$ for $$\alpha \le 1$$ .

Simultaneous zero-free approximation and universal optimal polynomial approximants

Let E be a closed subset of the unit circle of measure zero. Recently, Beise and Müller showed the existence of a function in the Hardy space H2 for which the partial sums of its Taylor series approximate any continuous function on E. In this paper, we establish an analogue of this result in a non-linear setting where we consider optimal polynomial approximants of reciprocals of functions in H2 instead of Taylor polynomials. The proof uses a new result on simultaneous zero-free approximation of independent interest. Our results extend to the Dirichlet space D and are expected for more general Dirichlet-type spaces.

Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows

This paper is concerned with the numerical simulation of time-dependent variably-saturated Darcian flow problems described by the Richards equation. We present the adaptive higher-order space-time discontinuous Galerkin (hp-STDG) method which optimizes accuracy and efficiency by balancing the errors that arise from the space and time discretizations and from the resulting nonlinear algebraic system. Convergence problems related to the transition between unsaturated flow and saturated flow are eliminated by regularizing the constitutive formulas. We also present an hp-anisotropic mesh adaptation technique capable of generating unstructured triangular elements with optimal sizes, shapes, and polynomial approximation degrees. Several numerical experiments are presented to demonstrate the accuracy, efficiency, and robustness of the numerical method presented here.

Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems

We study the structure of the zeros of optimal polynomial approximants to reciprocals of functions in Hilbert spaces of analytic functions in the unit disk. In many instances, we find the minimum possible modulus of occurring zeros via a nonlinear extremal problem associated with norms of Jacobi matrices. We examine global properties of these zeros and prove Jentzsch-type theorems describing where they accumulate. As a consequence, we obtain detailed information regarding zeros of reproducing kernels in weighted spaces of analytic functions.

Global optimal polynomial approximation for parametric problems in power systems

The influence of parameters on system states for parametric problems in power systems is to be evaluated. These parameters could be renewable generation outputs, load factor, etc. Polynomial approximation has been applied to express the nonlinear relationship between system states and parameters, governed by the nonlinear and implicit equations. Usually, sampling-based methods are applied, e.g., data fitting methods and sensitivity methods, etc. However, the accuracy and stability of these methods are not guaranteed. This paper proposes an innovative method based on Galerkin method, providing global optimal approximation. Compared to traditional methods, this method enjoys high accuracy and stability. IEEE 9-bus system is used to illustrate its effectiveness, and two additional studies including a 1648-bus system are performed to show its applications to power system analysis.

Remarks on Inner Functions and Optimal Approximants

AbstractWe discuss the concept of inner function in reproducing kernelHilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to 1/f, wherefis a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modiûed to produce inner functions.

High Power LED Parametric Modelling Using a Double Polynomial Approximation

This article describes calculations for an accurate mathematical model of high power LED modules using double optimal polynomial approximation. The model is based on unique tests of various LED types in a thermal chamber, providing a comprehensive list of parametric temperature profiles. This model was then implemented into MATLAB®&amp;Simulink® and Micro-Cap programs as a Spice compatible electronic circuit model, utilising the newly created algorithm. To define an optimal degree of approximation polynomials, Euclidean norm of residues was used. The new algorithm described in this article was verified using real-life data tested at the author’s work site, where the corresponding research takes place. To maximise the test’s efficiency, an automated data collection system was created. This article describes one particular tested LED module whose characteristic was modelled in both the absolute and the normalised form for easy comparison.