The decomposition of optimal transportation problems with convex cost

Given a positive l.s.c. convex function $\mathtt c : \R^d \to \R^d$ and an optimal transference plane $\underline{\pi}$ for the transportation problem \begin{equation*}\int \mathtt c(x'-x) \pi(dxdx'), \end{equation*} we show how the results of \cite{biadan} on the existence of a \emph{Sudakov decomposition} for norm cost $\mathtt c= |\cdot|$ can be extended to this case. More precisely, we prove that there exists a partition of $\R^d$ into a family of disjoint sets \{S^h_\a\}_{h,\a} together with the projection \{O^h_\a\}_{h,\a} on $\R^d$ of proper extremal faces of \epi\, \mathtt c, $h = 0,\dots,d$ and a \in \A^h \subset \R^{d-h}, such that \begin{itemize} \item S^h_\a is relatively open in its affine span, and has affine dimension $h$; \item O^h_\a has affine dimension $h$ and is parallel to S^h_\a; \item \mathcal L^d(\R^d \setminus \cup_{h,\a} S^h_\a) = 0, and the disintegration of $\mathcal L^d$, \mathcal L^d = \sum_h \int \xi^h_\a \eta^h(d\a), w.r.t. S^h_\a has conditional probabilities \xi^h_\a \ll \mathcal H^h \llcorner_{S^h_\a};\item the sets S^h_\a are essentially cyclically connected and cannot be further decomposed. \end{itemize} The last point is used to prove the existence of an optimal transport map. The main idea is to recast the problem in $(t,x) \in [0,\infty] \times \R^d$ with an $1$-homogeneous norm $\bar{\mathtt c}(t,x) := t \mathtt c(- \frac{x}{t})$ and to extend the regularity estimates of \cite{biadan} to this case

Deal2lkit: a Toolkit Library for High Performance Programming in deal.II

We propose a software design for the efficient and flexible handling of the building blocks used in high performance finite element simulations, through the pervasive use of parameters (parsed through parameter files). In the proposed design, all the building blocks of a high performance finite element program are built according to the command and composite design patterns.We present version 1.1.0 of the deal2lkit (deal.II ToolKit) library, which is a collection of modules and classes aimed at providing high level interfaces to several deal.II classes and functions, obeying the command and composite design patterns, and controlled via parameter files. Keywords: Object-orientation, Software design, Finite element methods, C+

deal2lkit: deal2lkit Version 1.0 - doi

A ToolKit library for deal.I

dealii/dealii: deal.II version 9.4.1

All download files are mirrored at https://dealii.43-1.org/downloads/ This is a minor update to 9.4.0 with the following changes: cmake: always export compile_commands.json in deal.II and user projects doxygen: fix various errors in formulas doxygen: fix SymmetricTensor friends cmake: fix PETSc version detection base: fix some VectorizedArrayTypes for non-default vectorization gitignore: ignore clangd files and directories change ConsensusAlgorithm deprecations to early deprecated step-81: Mention example step in the tutorial lists bugfix: use correct tolerance in MappingCartesian chec

dealii/dealii: deal.II version 9.4.0

All download files are mirrored at https://dealii.43-1.org/downloads/ The major changes of this release are: Three new tutorial programs: 81, 82, 85 demonstrating solving a variety of equations and illustrating functionality in deal.II. Advances in simplex- and mixed-mesh support Repartitioning of distributed meshes Advances in matrix-free infrastructure Advances in multigrid infrastructure CutFEM support Experimental integration of the Computational Geometry Algorithms Library (CGAL) Performance improvements in the particle infrastructure Improvements to unstructured communication For more information see The preprint at https://www.dealii.org/deal94-preprint.pdf The list of changes at https://www.dealii.org/developer/doxygen/deal.II/changes_between_9_3_3_and_9_4_0.htm

dealii/dealii: deal.II version 9.4.2

All download files are mirrored at https://dealii.43-1.org/downloads/ This is a minor update to 9.4.1 with the following changes: a compilation issue with step-70 has been resolved CMake: prefer -pthread for posix thread support a type mismatch for suitesparse has been fixed that lead to compilation failures on certain platforms a number of Microsoft Visual Code compatibility fixes concerning extern declaration

dealii: deal.II pre-release version 8.4.0-rc3

The development repository for the deal.II finite element library