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

In this paper, we present a measure of distance in a second-order cone based on a class of continuously differentiable strictly convex functions on ₊₊. Since the distance function has some favorable properties similar to those of the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451464 [1992]), we refer to it as a quasi D-function. Then, a proximal-like algorithm using the quasi D-function is proposed and applied to the second-cone programming problem, which is to minimize a closed proper convex function with general second-order cone constraints. Like the proximal point algorithm using the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451464 [1992]; Chen and Teboulle in SIAM J. Optim. 3:538543 [1993]), under some mild assumptions we establish the global convergence of the algorithm expressed in terms of function values; we show that the sequence generated by the

In this paper we introduce a high-order discontinuous Galerkin method for two-dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The stream function is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability. The method is efficient for inviscid or high Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments.

We present a hybrid asymptotic/numerical method for the accurate computation of single- and double-layer heat potentials in two dimensions. It has been shown in previous work that simple quadrature schemes suffer from a phenomenon called “geometrically induced stiffness,” meaning that formally high-order accurate methods require excessively small time steps before the rapid convergence rate is observed. This can be overcome by analytic integration in time, requiring the evaluation of a collection of spatial boundary integral operators with non-physical, weakly singular kernels. In our hybrid scheme, we combine a local asymptotic approximation with the evaluation of a few boundary integral operators involving only Gaussian kernels, which are easily accelerated by a new version of the fast Gauss transform. This new scheme is robust, avoids geometrically induced stiffness, and is easy to use in the presence of moving geometries. Its extension to three dimensions is natural and straightforward, and should permit layer heat potentials to become flexible and powerful tools for modeling diffusion processes.

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