Generating probability distributions on intervals and spheres with application to finite element method
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 . However, results below can be extended easily to an arbitrary interval by a linear transformation. Given a continuous or integrable function supported in an interval and a point on the -dimensional unit sphere , the zonal function is an integrable function on the sphere, where denotes inner product in the Euclidean space. The convolution on the sphere is defined as where is a
A probabilistic Galerkin scheme
Suppose that we have a domain decomposition of the sphere into disjoint with the properties that where is the size(diameter) of , is the radius of the largest ball inscribed in , and for all neighboring patches and there exist positive uniform constants and such that where we as usual use the arc length as the geodesic distance. We further assume that there exist -diffeomorphisms that map either a cube 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 and truncate the sequences at degree even though at degree forty the figures have been quite good. The first simulation is given to a fast decay sequence in Example 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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