Direct numerical simulation of turbulence on a Connection Machine CM-5

In this paper we report on our first experiences with direct numerical simulation of turbulent flow on a 16-node Connection Machine CM-5. The CM-5 has been programmed at a global level using data parallel Fortran. A two-dimensional direct simulation, where the pressure is solved using a Conjugate Gradient method without preconditioning, runs at 23% of the peak. Due to higher communication costs, 3D simulations run at 13% of the peak. A diagonalwise re-ordered Incomplete Choleski Conjugate Gradient method cannot compete with a standard CG-method on the CM-5.

Entropic optimal transport is maximum-likelihood deconvolution

We give a statistical interpretation of entropic optimal transport by showing that performing maximum-likelihood estimation for Gaussian deconvolution corresponds to calculating a projection with respect to the entropic optimal transport distance. This structural result gives theoretical support for the wide adoption of these tools in the machine learning community

Local Maximum Stable Sets Greedoids Stemmed from Very Well-Covered Graphs

A maximum stable set in a graph G is a stable set of maximum cardinality. S is called a local maximum stable set of G if S is a maximum stable set of the subgraph induced by the closed neighborhood of S. A greedoid (V,F) is called a local maximum stable set greedoid if there exists a graph G=(V,E) such that its family of local maximum stable sets coinsides with (V,F). It has been shown that the family local maximum stable sets of a forest T forms a greedoid on its vertex set. In this paper we demonstrate that if G is a very well-covered graph, then its family of local maximum stable sets is a greedoid if and only if G has a unique perfect matching.Comment: 12 pages, 12 figure

Nonlinear Parabolic Equations arising in Mathematical Finance

This survey paper is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models of stochastic dynamic portfolio optimization leading to the Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both problems can be represented by solutions to nonlinear parabolic equations. Qualitative analysis will be focused on issues concerning the existence and uniqueness of solutions. In the numerical part we discuss a stable finite-volume and finite difference schemes for solving fully nonlinear parabolic equations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0387

Extremal Lipschitz functions in the deviation inequalities from the mean

We obtain an optimal deviation from the mean upper bound \begin{equation} D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\ x\in\R\label{abstr} \end{equation} where \F is the class of the integrable, Lipschitz functions on probability metric (product) spaces. As corollaries we get exact solutions of \eqref{abstr} for Euclidean unit sphere $S^{n-1}$ with a geodesic distance and a normalized Haar measure, for $\R^n$ equipped with a Gaussian measure and for the multidimensional cube, rectangle, torus or Diamond graph equipped with uniform measure and Hamming distance. We also prove that in general probability metric spaces the $\sup$ in \eqref{abstr} is achieved on a family of distance functions.Comment: 7 page

Mathematical models of avascular cancer

This review will outline a number of illustrative mathematical models describing the growth of avascular tumours. The aim of the review is to provide a relatively comprehensive list of existing models in this area and discuss several representative models in greater detail. In the latter part of the review, some possible future avenues of mathematical modelling of avascular tumour development are outlined together with a list of key questions