Greedy forwarding with geographical locations in a wireless sensor network may fail at a local minimum. In this paper we propose to use conformal mapping to compute a new embedding of the sensor nodes in the plane such that greedy forwarding with the virtual coordinates guarantees delivery. In particular, we extract a planar triangulation of the sensor network with non-triangular faces as holes, by either using the nodes' location or using a landmark-based scheme without node location. The conformal map is computed with Ricci flow such that all the non-triangular faces are mapped to perfect circles. Thus greedy forwarding will never get stuck at an intermediate node. The computation of the conformal map and the virtual coordinates is performed at a preprocessing phase and can be implemented by local gossip-style computation. The method applies to both unit disk graph models and quasi-unit disk graph

Consider the single-facility Euclidean -centrum location problem in R n. This problem is a generalization of the classical Euclidean 1-median problem and 1-center problem. In this paper, we develop two efficient algorithms that are particularly suitable for problems where n is large by using unconstrained optimization techniques. The first algorithm is based on the neural networks smooth approximation for the plus function and reduces the problem to an unconstrained smooth convex minimization problem. The second algorithm is based on the FischerBurmeister merit function for the second-order cone complementarity problem and transforms the KKT system of the second-order cone programming reformulation for the problem into an unconstrained smooth minimization problem. Our computational experiments indicate that both methods are extremely efficient for large problems and the first algorithm is able to

Sparse modeling/approximation of images plays an important role in image restoration. Instead of using a fixed system to sparsely model any input image, a more promising approach is using a system that is adaptive to the input image. A non-convex variational model is proposed in [1] for constructing a tight frame that is optimized for the input image, and an alternating scheme is used to solve the resulting non-convex optimization problem. Although it showed good empirical performance in image denoising, the proposed alternating iteration lacks convergence analysis. This paper aims at providing the convergence analysis of the method proposed in [1]. We first established the sub-sequence convergence property of the iteration scheme proposed in [1], i.e., there exists at least one convergent sub-sequence and any convergent sub-sequence converges to a stationary point of the minimization problem. Moreover, we showed that the original method can be modified to have sequence convergence, i.e., the modified algorithm generates a sequence that converges to a stationary point of the minimization problem.

The goal of smooth function reconstruction on a 2D or 3D manifold is to obtain a smooth function on surfaces or higher dimensional manifolds. It is a common problem in computer graphics and computational mathematics, especially in civil engineering including structural analysis of solid objects. In this chapter, we introduce a new method using harmonic functions for solving this problem. This method contains the following steps: (1) Partition the boundary surfaces of the 3D manifold based on sample points so that each sample point is on the edge of the partition. (2) Use gradually varied interpolation on the edges so that each point on the edge will be assigned a value. In addition, all values on the edge are gradually varied. (3) Use discrete harmonic functions to fit the unknown points, i.e. the points inside each partition patch. This solution of the fitting becomes the piecewise harmonic function.

In this paper, we study the problem of computing the effective diffusivity for a particle moving in chaotic and stochastic flows. In addition, we numerically investigate the residual diffusion phenomenon in chaotic advection. The residual diffusion refers to the nonzero effective (homogenized) diffusion in the limit of zero molecular diffusion as a result of chaotic mixing of the streamlines. In this limit, traditional numerical methods typically fail since the solutions of the advection-diffusion equations develop sharp gradients. Instead of solving the Fokker--Planck equation in the Eulerian formulation, we compute the motion of particles in the Lagrangian formulation, which is modeled by stochastic differential equations (SDEs). We propose an effective numerical integrator based on a splitting method to solve the corresponding SDEs in which the deterministic subproblem is symplectic preserving while the random subproblem