Conformally Symmetric Triangular Lattices and Discrete ϑ-Conformal Maps

Two immersed triangulations in the plane with the same combinatorics are considered as preimage and image of a discrete immersion $F$. We compare the cross-ratios $Q$ and $q$ of corresponding pairs of adjacent triangles in the two triangulations. If for every pair the arguments of these cross-ratios (i.e., intersection angles of circumcircles) agree, $F$ is a discrete conformal map based on circle patterns. Similarly, if for every pair the absolute values of the corresponding cross-ratios $Q$ and $q$ (i.e., length cross-ratios) agree, the two triangulations are discretely conformally equivalent. We introduce a new notion, discrete $\vartheta $-conformal maps, which interpolates between these two known notions of discrete conformality for planar triangulations. We prove that there exists an associated variational principle. In particular, discrete $\vartheta $-conformal maps are unique maximizers of a locally defined concave functional ${\mathcal{F}}_{\vartheta}$in suitable variables. Furthermore, we study conformally symmetric triangular lattices that contain examples of discrete $\vartheta $-conformal maps.

AN OPTIMIZATION-BASED DOMAIN DECOMPOSITION METHOD FOR NONLINEAR WALL LAWS IN COUPLED SYSTEMS

We study an optimization-based domain decomposition method for a nonlinear wall law in a coupled system. The problem is restated as a saddle-point problem by introducing as a new variable the displacement jump across the interface. Then the minimization step of the saddle-point problem corresponds to the equilibrium equations stated in each subdomain with Lagrange multiplier as interface force. The maximization step corresponds to maximizing a (nonlinear) strictly concave functional. This could have a lot of applications in geophysical flows such as coupling ocean and atmosphere, free surface and groundwater flows.

Finite-horizon dynamic optimisation when the terminal reward is a concave functional of the distribution of the final state

We consider a problem similar in many respects to a finite horizon Markov decision process, except that the reward to the individual is a strictly concave functional of the distribution of the state of the individual at final time T. Reward structures such as these are of interest to biologists studying the fitness of different strategies in a fluctuating environment. The problem fails to satisfy the usual optimality equation and cannot be solved directly by dynamic programming. We establish equations characterising the optimal final distribution and an optimal policy π*. We show that in general π* will be a Markov randomised policy (or equivalently a mixture of Markov deterministic policies) and we develop an iterative, policy improvement based algorithm which converges to π*. We also consider an infinite population version of the problem, and show that the population cannot do better using a coordinated policy than by each individual independently following the individual optimal policy π*.

Finite-horizon dynamic optimisation when the terminal reward is a concave functional of the distribution of the final state

We consider a problem similar in many respects to a finite horizon Markov decision process, except that the reward to the individual is a strictly concave functional of the distribution of the state of the individual at final time T. Reward structures such as these are of interest to biologists studying the fitness of different strategies in a fluctuating environment. The problem fails to satisfy the usual optimality equation and cannot be solved directly by dynamic programming. We establish equations characterising the optimal final distribution and an optimal policy π*. We show that in general π* will be a Markov randomised policy (or equivalently a mixture of Markov deterministic policies) and we develop an iterative, policy improvement based algorithm which converges to π*. We also consider an infinite population version of the problem, and show that the population cannot do better using a coordinated policy than by each individual independently following the individual optimal policy π*.