Numerical Approximation Of Integrals Research Articles

Optimal numerical integration and approximation of functions on ℝd equipped with Gaussian measure

Abstract We investigate the numerical approximation of integrals over $\mathbb{R}^{d}$ equipped with the standard Gaussian measure $\gamma $ for integrands belonging to the Gaussian-weighted Sobolev spaces $W^{\alpha }_{p}(\mathbb{R}^{d}, \gamma )$ of mixed smoothness $\alpha \in \mathbb{N}$ for $1 < p < \infty $. We prove the asymptotic order of the convergence of optimal quadratures based on $n$ integration nodes and propose a novel method for constructing asymptotically optimal quadratures. As for related problems, we establish by a similar technique the asymptotic order of the linear, Kolmogorov and sampling $n$-widths in the Gaussian-weighted space $L_{q}(\mathbb{R}^{d}, \gamma )$ of the unit ball of $W^{\alpha }_{p}(\mathbb{R}^{d}, \gamma )$ for $1 \leq q < p < \infty $ and $q=p=2$.

High-order asymptotic expansions of Gaussian quadrature rules with classical and generalized weight functions

Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and explicitly list asymptotic expressions for the points and weights of Gaussian quadrature rules for three general classes of positive weight functions: analytic functions on a bounded interval with algebraic singularities at the endpoints, analytic weight functions on the halfline with exponential decay at infinity and an algebraic singularity at the finite endpoint, and analytic functions on the real line with exponential decay in both directions at infinity. The results include the Gaussian rules of classical orthogonal polynomials (Legendre, Jacobi, Laguerre and Hermite) as special cases. Explicit expressions for these cases are included in the appendix. We present experiments indicating the range of the number of points at which these expressions achieve high precision. We provide an algorithm that can compute arbitrarily many terms in these expansions for the classical cases, and many though not all terms for the generalized cases.

Efficient Time-Stepping for Numerical Integration Using Reinforcement Learning

Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge–Kutta pairs, have been developed which adapt the step size based on local error estimations at each step. While the classical schemes apply very generally and are highly efficient on regular systems, they can behave suboptimally when an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly, it generalizes better to unseen initial data as our approach employs classical numerical schemes as base methods. At the same time it can make use of identified structures of a problem class and, therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior efficiency. Source code is available at https://github.com/lueckem/quadrature-ML.

Target-aware Bayesian inference via generalized thermodynamic integration

In Bayesian inference, we are usually interested in the numerical approximation of integrals that are posterior expectations or marginal likelihoods (a.k.a., Bayesian evidence). In this paper, we focus on the computation of the posterior expectation of a function f(textbf{x}). We consider a target-aware scenario where f(textbf{x}) is known in advance and can be exploited in order to improve the estimation of the posterior expectation. In this scenario, this task can be reduced to perform several independent marginal likelihood estimation tasks. The idea of using a path of tempered posterior distributions has been widely applied in the literature for the computation of marginal likelihoods. Thermodynamic integration, path sampling and annealing importance sampling are well-known examples of algorithms belonging to this family of methods. In this work, we introduce a generalized thermodynamic integration (GTI) scheme which is able to perform a target-aware Bayesian inference, i.e., GTI can approximate the posterior expectation of a given function. Several scenarios of application of GTI are discussed and different numerical simulations are provided.

Numerical Approximation of Integrals in Presence of Nearby Singularities

The quadrature formulas meant for the numerical approximation of integrals of one-dimensional real variables need to be modified for the sake of accuracy when a singularity is present in the proximity of the path of integration. The required corrective factor has been constructed and some existing quadrature rule has been applied with the corrective factor to obtain better accuracy.

A Continuous-Discrete Finite Memory Observer Design for a Class of Nonlinear Systems: Application to Fault Diagnosis

This paper aims to develop a continuous-discrete finite memory observer (CD-FMO) for a class of nonlinear dynamical systems modeled by ordinary differential equations (ODEs) with discrete measurements. The nonlinear systems under consideration are at least locally Lipschitz, which guarantees the existence and uniqueness of solution at each time instant. The proposed nonlinear observer uses a finite number of collected measurements to estimate the system state in the presence of measurement noise. Besides, a one-step prediction algorithm incorporated with an iterative-update scheme is performed to solve the integral problem caused by system nonlinearity, and an analysis of the numerical integration approximation error is given. The properties of estimation performance have been further proved in deterministic case and been analyzed by Monte Carlo simulation in stochastic cases. It is worth noting that the presented method has a finite-time convergence, while most nonlinear observers are usually asymptotically convergent. Another advantage of CD-FMO is that it has no initial value problem. For the application purpose, residuals are generated to implement fault detection cooperated with Cumulative Sum (CUSUM) control charts, while a bank of CD-FMOs is adopted to realize fault isolation for different sensor and actuator faults of the considered nonlinear robotic arm. The robustness and effectiveness of the proposed approach are illustrated via the simulation results.

A mixed scheme of product integration rules in (−1,1)

The paper deals with the numerical approximation of integrals of the typeI(f,y):=∫−11f(x)k(x,y)dx,y∈S⊂R where f is a smooth function and the kernel k(x,y) involves some kinds of “pathologies” (for instance, weak singularities, high oscillations and/or endpoint algebraic singularities). We introduce and study a product integration rule obtained by interpolating f by an extended Lagrange polynomial based on Jacobi zeros. We prove that the rule is stable and convergent with the order of the best polynomial approximation of f in suitable function spaces. Moreover, we derive a general recurrence relation for the new modified moments appearing in the coefficients of the rule, just using the knowledge of the usual modified moments. The new quadrature sequence, suitable combined with the ordinary product rule, allows to obtain a “mixed” quadrature scheme, significantly reducing the number of involved samples of f. Numerical examples are provided in order to support the theoretical results and to show the efficiency of the procedure.

Fast Numerical Integration on Polytopic Meshes with Applications to Discontinuous Galerkin Finite Element Methods

In this paper we present efficient quadrature rules for the numerical approximation of integrals of polynomial functions over general polygonal/polyhedral elements that do not require an explicit construction of a sub-tessellation into triangular/tetrahedral elements. The method is based on successive application of Stokes’ theorem; thereby, the underlying integral may be evaluated using only the values of the integrand and its derivatives at the vertices of the polytopic domain, and hence leads to an exact cubature rule whose quadrature points are the vertices of the polytope. We demonstrate the capabilities of the proposed approach by efficiently computing the stiffness and mass matrices arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations.

A non-model-based approach to bandwidth selection for kernel estimators of spatial intensity functions

We propose a new bandwidth selection method for kernel estimators of spatial point process intensity functions. The method is based on an optimality criterion motivated by the Campbell formula applied to the reciprocal intensity function. The new method is fully nonparametric, does not require knowl- edge of higher-order moments, and is not restricted to a specific class of point process. Our approach is computationally straightforward and does not require numerical approximation of integrals.

On the Optimal Order of Integration in Hermite Spaces with Finite Smoothness

We study the numerical approximation of integrals over ${R}^s$ with respect to the standard Gaussian measure for integrands which lie in certain Hermite spaces of functions. The decay rate of the associated sequence is specified by a single integer parameter which determines the smoothness classes and the inner product can be expressed via $L_2$ norms of the derivatives of the function. We map higher order digital nets from the unit cube to a suitable subcube of ${R}^s$ via a linear transformation and show that such rules achieve, apart from powers of $\log N$, the optimal rate of convergence of the integration error.

The N-Point Definite Integral Approximation Formula (N-POINT DIAF)

Various authors have discovered formulae for numerical integration approximation. However these formulae always result to some amount of error which may differ in size depending on the formula. It’s therefore important that a formula with highest precision has been discovered and should be implemented for use in numerical integration approximations problems, especially for the definite integrals which cannot be evaluated by applying the analytical techniques. The present paper therefore explores the derivation of the N-point Definite Integral Approximation Formula (N-point DIAF) which amounts to the discovery of the 2-Point DIAF. This formula will assist in almost accurate evaluation of all definite integrals numerically. The proof of the formula is given, a specific test problem is then solved using the discovered 2-Point DIAF to obtain the solution numerically, which has the highest precision compared to other numerical methods of integration. Further the error terms are obtained and compared with the existing methods. Finally, the effectiveness of the proposed formula is illustrated by means of a numerical example.

Stabilized finite element methods for a blood flow model of arteriosclerosis

In this article, a blood flow model of arteriosclerosis, which is governed by the incompressible Navier–Stokes equations with nonlinear slip boundary conditions, is constructed and analyzed. By means of suitable numerical integration approximation for the nonlinear boundary term in this model, a discrete variational inequality for the model based on stabilized finite elements is proposed. Optimal order error estimates are obtained. Finally, numerical examples are shown to demonstrate the validity of the theoretical analysis and the efficiency of the presented methods. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2063–2079, 2015

Numerical quadrature for high-dimensional singular integrals over parallelotopes

We introduce and analyze a family of algorithms for an efficient numerical approximation of integrals of the form I=∫C(1)∫C(2)F(x,y,y−x)dydx where C(1), C(2) are d-dimensional parallelotopes (i.e. affine images of d-hypercubes) and F has a singularity at y−x=0. Such integrals appear in Galerkin discretization of integral operators in Rd. We construct a family of quadrature rules QN with N function evaluations for a class of integrands F which may have algebraic singularities at y−x=0 and are Gevrey-δ regular for y−x≠0. The main tool is an explicit regularizing coordinate transformation, simultaneously simplifying the singular support and the domain of integration. For the full tensor product variant of the suggested quadrature family we prove that QN achieves the exponential convergence rate O(exp(−rNγ)) with the exponent γ=1/(2dδ+1). In the special case of a singularity of the form ‖y−x‖α with real α we prove that the improved convergence rate of γ=1/(2dδ) is achieved if a certain modified one-dimensional Gauss–Jacobi quadrature rule is used in the singular direction. We give numerical results for various types of the quadrature rules, in particular based on tensor product rules, standard (Smolyak), optimized and adaptive sparse grid quadratures and Sobol’ sequences.

Stratified Monte Carlo Quadrature for Continuous Random Fields

We consider the problem of numerical approximation of integrals of random fields over a unit hypercube. We use a stratified Monte Carlo quadrature and measure the approximation performance by the mean squared error. The quadrature is defined by a finite number of stratified randomly chosen observations with the partition generated by a rectangular grid (or design). We study the class of locally stationary random fields whose local behaviour is like a fractional Brownian field in the mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large number of the observations. For the Hölder class of random functions, we provide an upper bound for the approximation error. Additionally, for a certain class of isotropic random functions with an isolated singularity at the origin, we construct a sequence of designs eliminating the effect of the singularity point.

Numerical evaluation of integrals of analytic functions

A general formula of degree of precision at least seven for the numerical approximation of integrals of analytic functions along straight line segments has been constructed. Some existing rules as well as some new rules have been obtained as particular cases of the general formula.

Two general methods for the numerical approximation of multidimensional Cauchy principal value integrals

The numerical approximation of integrals containing strongly singular integrals, in particular Cauchy principal value integrals, is a major issue connected, e.g., to the boundary integral approach for many types of partial differential equations. Whereas the one-dimensional problem has been addressed very intensively in recent years, much less attention has been paid to multidimensional problems. In the present paper, we investigate two possible approaches to this problem, corresponding to generalizations of two approaches known in the 1-D case. In principle, both methods can be applied to integration domains of arbitrary shape, although we find that certain combinations of algorithms and domains are more useful than others. In particular, we discuss error estimates and show that the methods are highly competitive. Moreover, in contrast to most of the previously discussed methods, the approaches are very efficient when integrals have to be calculated for various locations of the singularity.

Time-domain analysis methodology for large-scale RLC circuits and its applications

With soaring work frequency and decreasing feature sizes, VLSI circuits with RLC parasitic components are more like analog circuits and should be carefully analyzed in physical design. However, the number of extracted RLC components is typically too large to be analyzed efficiently by using present analog circuit simulators like SPICE. In order to speedup the simulations without error penalty, this paper proposes a novel methodology to compress the time-descritized circuits resulted from numerical integration approximation at every time step. The main contribution of the methodology is the efficient structure-level compression of DC circuits containing many current sources, which is an important complement to present circuit analysis theory. The methodology consists of the following parts: 1) An approach is proposed to delete all intermediate nodes of RL branches. 2) An efficient approach is proposed to compress and back-solve parallel and serial branches so that it is error-free and of linear complexity to analyze circuits of tree topology. 3) The Y to π transformation method is used to error-free reduce and back-solve the intermediate nodes of ladder circuits with the linear complexity. Thus, the whole simulation method is very accurate and of linear complexity to analyze circuits of chain topology. Based on the methodology, we propose several novel algorithms for efficiently solving RLC-model transient power/ground (P/G) networks. Among them, EQU-ADI algorithm of linear-complexity is proposed to solve RLC P/G networks with mesh-tree or mesh-chain topologies. Experimental results show that the proposed method is at least two orders of magnitude faster than SPICE while it can scale linearly in both time-and memory-complexity to solve very large P/G networks.

Marcinkiewicz–Zygmund inequalities and the numerical approximation of singular integrals for exponential weights: methods, results and open problems, some new, some old

In this short article, we explore some methods, results and open problems dealing with weighted Marcinkiewicz–Zygmund inequalities as well as the numerical approximation of integrals for exponential weights on the real line and on finite intervals of the line. The problems posed are based primarily on ongoing work of the author and his collaborators and were presented at the November 2001 Oberwolfach workshop: ‘Numerical integration and Complexity’.

Numerical quadrature of analytic functions

A four‐point quadrature rule with degree of p recision five for the numerical approximation of integrals of analytic functions of a complex variable has been constructed. An asymptotic error estimate and an error bound of the rule constructed have been derived.

Methods for Approximating Integrals in Statistics with Special Emphasis on Bayesian Integration Problems

This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain methods. Each method is discussed, giving an outline of the basic supporting theory and particular features of the technique. Conclusions are drawn concerning the relative merits of the methods based on the discussion and their application to three examples. The following broad recommendations are made. Asymptotic methods should only be considered in contexts where the integrand has a dominant peak with approximate ellipsoidal symmetry. Importance sampling (and preferably adaptive importance sampling) based on a multivariate Student should be used instead of asymptotics methods in such a context. Multiple quadrature and, in particular, subregion adaptive integration are the algorithms of choice for lower-dimensional integrals. Due to the difficulties in assessing convergence to stationarity and the error in estimates, Markov chain methods are recommended only when there is no adequate alternative. In certain very high dimensional problems, however, Markov chain methods are the only hope. The importance of the parameterization of the integral is noted for the success of all the methods, and several useful reparameterizations are presented.