Generating probability distributions on intervals and spheres with application to finite element method

doi:10.1016/j.camwa.2020.10.017

Computers & Mathematics with Applications

Volume 95, 1 August 2021, Pages 282-295

https://doi.org/10.1016/j.camwa.2020.10.017 Get rights and content

Abstract

This work aims to build a bridge between probability methods and finite element methods. It starts with considering probability distributions supported in an interval $[a, b]$ , which incorporate the traditional probability distributions defined on the whole real space as limit cases on the one hand, lead to a type of spherical probability models with wide potential applications on the other hand. This type of probability has scaling and symmetry feature, and sufficient conditions under which a density function can be generated through discrete polynomial spectrum are given in this work followed by concrete examples. The density function $ρ$ obtained in this way has the advantage of being positive definite. Computer based numerical simulation shows that the theoretically verified criteria for probability distribution are almost optimal with respect to our testing examples. After the establishment of an approximation theorem in $L^{1}$ space, we propose a probabilistic Galerkin scheme that can be either continuous or discontinuous, which is potentially useful to asymptotically solve some PDEs on the sphere locally and globally.

Section snippets

Introduction to main contribution

Imagine that an earthquake happens somewhere in Asia. To detect the hypocenter and the waves, scientists do not search in the Atlantic ocean. In other words, the likelihood of damage that a tsunami could bring outside certain region is almost zero. This is one of the main reasons we consider in this paper probability distributions defined on an interval, or a bounded domain, instead of the whole real line. Conversely, a probability distribution defined on the whole real line can be well

A criterion for probability distribution on intervals

For convenience, the interval of our consideration is $[- 1, 1]$ . However, results below can be extended easily to an arbitrary interval by a linear transformation. Given a continuous or integrable function $f$ supported in an interval $[- 1, 1]$ and a point $x$ on the $(d - 1)$ -dimensional unit sphere $S^{d - 1}$ , the zonal function $f (x \cdot y)$ is an integrable function on the sphere, where $x \cdot y$ denotes inner product in the Euclidean space. The convolution on the sphere is defined as $f * φ (x) = \int_{S^{d - 1}} f (x \cdot y) φ (y) d y,$ where $φ$ is a

A probabilistic Galerkin scheme

Suppose that we have a domain decomposition of the sphere into disjoint ${U_{j}}_{j \in J}$ with the properties that $⋃_{j} {\bar{U}}_{j} = S^{d - 1}$ $\frac{h_{j}}{q_{j}} \leq C_{1}$ where $h_{j}$ is the size(diameter) of $U_{j}$ , $q_{j}$ is the radius of the largest ball inscribed in $U_{j}$ , and for all neighboring patches $U_{i}$ and $U_{j}$ there exist positive uniform constants $C_{1}$ and $C_{2}$ such that $\frac{1}{C_{2}} h_{j} \leq h_{i} \leq C_{2} h_{j}$ where we as usual use the arc length as the geodesic distance. We further assume that there exist $C^{1}$ -diffeomorphisms $F_{i}$ that map either a cube ${(- 1, 1)}^{d - 1}$ or an open

Numerical simulation

Let us compare three sets of numerical simulation to check the results of our theorem. While the right-hand side pictures are probability distributions generated through different interval characteristic sequences, the left-side ones are their density functions, where we choose $λ = \frac{1}{2}$ and truncate the sequences at degree $N = 80$ even though at degree forty the figures have been quite good. The first simulation is given to a fast decay sequence in Example 2 $g (n) = exp (- n^{2}),$ so that we obtain a positive

Conclusion and further remarks

We have successfully established certain criteria for generating a type of spherical probabilities with symmetry property using the interval characteristic sequences and applied them to probabilistic Galerkin schemes. However, many questions remain unanswered. For instance curious readers would wonder what are spherical probability of type II or type III and what are their special advantages; how is the error estimation and primal formulation with respect to more general models; how to

Acknowledgments

This work is dedicated to celebrating the 250th birthday of Alexander von Humboldt on occasion of the conference “Minimum residual and least squares finite element methods”. The first author would like to thank Emilie and Fleurianne for the invitation. He was partially supported by China Scholarship Council under the Grant No. 201206320164 and Stipendium des Präsident von Technische Universität Berlin, Germany. The second author is funded by National Natural Science Foundation of China under

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Cited by (2)

Generating probability distributions on intervals and spheres: Convex decomposition
2024, Computers and Mathematics with Applications
This work studies the convexity of a class of positive definite probability measures with density functions that are generated through the interval characteristic sequences. We discuss when such measures supported in an interval are convex and demonstrate how convexity is preserved under multiplication by certain multipliers and under the intertwining product. A key message to be delivered from this work is: any integrable function f on the interval with a polynomial expansion that has relatively fast absolute convergence, can be decomposed into a non-unique pair of positive convex interval probabilities $(f_{+}, f_{-}) \in E_{λ}^{'} \times E_{λ}^{'}$ up to normalization, in the sense that $f = f_{+} - f_{-}$ , where $E_{λ}^{'}$ is a class of convex functions. Consequently not only the study of an interval distribution is reduced to the study of the properties of class $E_{λ}^{'}$ , but also the discontinuous probabilistic Galerkin schemes can be simplified. As concrete examples the decomposition of polynomials and truncated normal distribution are computed and graphically illustrated. Such interval measures lead to localized spherical probability measures, and as a convex expression of probability measure is a Jensen type inequality for spherical random variables. Based on the spherical information entropy models, one dependent on the domain partition and the other not, we propose three minimization formulations for the probabilistic finite element method, including one that mixes the least-squares and entropy models.
Recent Advances in Least-Squares and Discontinuous Petrov–Galerkin Finite Element Methods
2021, Computers and Mathematics with Applications

View full text

Generating probability distributions on intervals and spheres with application to finite element method

Abstract

Section snippets

Introduction to main contribution

A criterion for probability distribution on intervals

A probabilistic Galerkin scheme

Numerical simulation

Conclusion and further remarks

Acknowledgments

Comput. Math. Appl.

A necessary and sufficient condition for strictly positive definite functions on spheres

Proc. Amer. Math. Soc.

Positive Jacobi polynomial sums, II

Amer. J. Math.