Sharp isoperimetric inequalities via the ABP

Given an arbitrary convex cone of Rn, we find a geometric class of homogeneous weights for which balls centered at the origin and intersected with the cone are minimizers of the weighted isoperimetric problem in the convex cone. This leads to isoperimetric inequalities with the optimal constant that were unknown even for a sector of the plane. Our result applies to all nonnegative homogeneous weights in Rn satisfying a concavity condition in the cone. The condition is equivalent to a natural curvature-dimension bound and also to the nonnegativeness of a Bakry-Emery Ricci tensor. Even that our weights are nonradial, still balls are minimizers of the weighted isoperimetric problem. A particular important case is that of monomial weights. Our proof uses the ABP method applied to an appropriate linear Neumann problem. We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. As a particular case of our results, we give new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and PacellaPeer ReviewedPostprint (published version

Local maximum‐entropy approximation schemes: a seamless bridge between finite elements and meshfree methods

We present a one‐parameter family of approximation schemes, which we refer to as local maximum‐entropy approximation schemes, that bridges continuously two important limits: Delaunay triangulation and maximum‐entropy (max‐ent) statistical inference. Local max‐ent approximation schemes represent a compromise—in the sense of Pareto optimality—between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width. Local max‐ent approximation schemes are entirely defined by the node set and the domain of analysis, and the shape functions are positive, interpolate affine functions exactly, and have a weak Kronecker‐delta property at the boundary. Local max‐ent approximation may be regarded as a regularization, or thermalization, of Delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation. Local max‐ent approximation schemes can be taken as a convenient basis for the numerical solution of PDEs in the style of meshfree Galerkin methods. In test cases characterized by smooth solutions we find that the accuracy of local max‐ent approximation schemes is vastly superior to that of finite elements

Winterbottom Construction for Finite Range Ferromagnetic Models: An L_1 Approach

We provide a rigorous microscopic derivation of the thermodynamic description of equilibrium crystal shapes in the presence of a substrate, first studied by Winterbottom. We consider finite range ferromagnetic Ising models with pair interactions in dimensions greater or equal to 3, and model the substrate by a finite-range boundary magnetic field acting on the spins close to the bottom wall of the box

Enhancement of Sandwich Algorithms for Approximating Higher Dimensional Convex Pareto Sets

In many fields, we come across problems where we want to optimize several conflicting objectives simultaneously. To find a good solution for such multi-objective optimization problems, an approximation of the Pareto set is often generated. In this paper, we con- sider the approximation of Pareto sets for problems with three or more convex objectives and with convex constraints. For these problems, sandwich algorithms can be used to de- termine an inner and outer approximation between which the Pareto set is 'sandwiched'. Using these two approximations, we can calculate an upper bound on the approximation error. This upper bound can be used to determine which parts of the approximations must be improved and to provide a quality guarantee to the decision maker. In this paper, we extend higher dimensional sandwich algorithms in three different ways. Firstly, we introduce the new concept of adding dummy points to the inner approx- imation of a Pareto set. By using these dummy points, we can determine accurate inner and outer approximations more e±ciently, i.e., using less time-consuming optimizations. Secondly, we introduce a new method for the calculation of an error measure which is easy to interpret. The combination of easy calculation and easy interpretation makes this measure very suitable for sandwich algorithms. Thirdly, we show how transforming cer- tain objective functions can improve the results of sandwich algorithms and extend their applicability to certain non-convex problems. The calculation of the introduced error measure when using transformations will also be discussed. To show the effect of these enhancements, we make a numerical comparison using four test cases, including a four-dimensional case from the field of intensity-modulated radiation therapy (IMRT). The results of the different cases show that we can indeed achieve an accurate approximation using significantly fewer optimizations by using the enhancements.Convexity;e-efficiency;e-Pareto optimality;Geometric programming;Higher dimensional;Inner and outer approximation;IMRT;Pareto set;Multi-objective optimiza- tion;Sandwich algorithms;Transformations