Related papers: Information Theory and Quadrature Rules
We describe an algorithm for controlling the relative error in the numerical evaluation of a bivariate integral, without prior knowledge of the magnitude of the integral. In the event that the magnitude of the integral is less than unity,…
A randomised trapezoidal quadrature rule is proposed for continuous functions which enjoys less regularity than commonly required. Indeed, we consider functions in some fractional Sobolev space. Various error bounds for this randomised rule…
Some Gauss-type quadrature rules over [0, 1], which involve values and/or the derivative of the integrand at 0 and/or 1, are investigated
For the purpose of uncertainty propagation a new quadrature rule technique is proposed that has positive weights, has high degree, and is constructed using only samples that describe the probability distribution of the uncertain parameters.…
We introduce algorithmic information theory, also known as the theory of Kolmogorov complexity. We explain the main concepts of this quantitative approach to defining `information'. We discuss the extent to which Kolmogorov's and Shannon's…
A framework is presented to compute approximations of an integral $I(f)=\displaystyle \int_a^b f(x) dx$ from a pair of companion rules and its associate rule. We show that an associate rule is a weighted mean of two companion rules. In…
The quadratic decaying property of the information rate function states that given a fixed conditional distribution $p_{\mathsf{Y}|\mathsf{X}}$, the mutual information between the (finite) discrete random variables $\mathsf{X}$ and…
A novel method is proposed to infer Bayesian predictions of computationally expensive models. The method is based on the construction of quadrature rules, which are well-suited for approximating the weighted integrals occurring in Bayesian…
A scoring rule is a loss function measuring the quality of a quoted probability distribution $Q$ for a random variable $X$, in the light of the realized outcome $x$ of $X$; it is proper if the expected score, under any distribution $P$ for…
In this paper, we introduce and analyze arbitrarily high-order quadrature rules for evaluating the two-dimensional singular integrals of the forms \begin{align} I_{i,j} = \int_{\mathbb{R}^2}\phi(x)\frac{x_ix_j}{|x|^{2+\alpha}} \d x, \quad…
After introducing the definitions of positive, negative and companion rules, from a given pair of companion rules we construct a new rule with higher degree of precision The scheme is generalized giving rise to a transformation which we…
In inverse problems, one attempts to infer spatially variable functions from indirect measurements of a system. To practitioners of inverse problems, the concept of "information" is familiar when discussing key questions such as which parts…
Importance sampling (IS) and numerical integration methods are usually employed for approximating moments of complicated target distributions. In its basic procedure, the IS methodology randomly draws samples from a proposal distribution…
In this paper we obtain $L^1$-weighted norms of classical orthogonal polynomials (Hermite, Laguerre and Jacobi polynomials) in terms of the zeros of these orthogonal polynomials; these expressions are usually known as quadrature rules. In…
We propose a new interpretation of measures of information and disorder by connecting these concepts to group theory in a new way. Entropy and group theory are connected here by their common relation to sets of permutations. A combinatorial…
This article presents a novel approach to enhance the accuracy of classical quadrature rules by incorporating correction terms. The proposed method is particularly effective when the position of an isolated discontinuity in the function and…
The implicit boundary integral method (IBIM) provides a framework to construct quadrature rules on regular lattices for integrals over irregular domain boundaries. This work provides a systematic error analysis for IBIMs on uniform…
Sum rules are elegant formulas that relate entropy functionals to coefficients associated with orthogonal polynomials [Sim11]. In a series of paper (see for example [GNR16], [GNR17], [BSZ18a], [BSZ18b]), interesting connections have been…
Symmetry of information establishes a relation between the information that x has about y (denoted I(x : y)) and the information that y has about x (denoted I(y : x)). In classical information theory, the two are exactly equal, but in…
In this work, we present some new integration formulas for any order of accuracy as an application of the B-spline relations obtained in [1]. The resulting rules are defined as a perturbation of the trapezoidal integration method. We prove…