We present a novel area-preservation mapping/flattening method using the optimal mass transport technique, based on the Monge-Brenier theory. Our optimal transport map approach is rigorous and solid in theory, efficient and parallel in computation, yet general for various applications. By comparison with the conventional Monge-Kantorovich approach, our method reduces the number of variables from O(n²) to O(n), and converts the optimal mass transport problem to a convex optimization problem, which can now be efficiently carried out by Newton's method. Furthermore, our framework includes the area weighting strategy that enables users to completely control and adjust the size of areas everywhere in an accurate and quantitative way. Our method significantly reduces the complexity of the problem, and improves the efficiency, flexibility and scalability during visualization. Our framework, by combining

We study the stability properties of explicit marching schemes for second-kind Volterra integral equations that arise when solving boundary value problems for the heat equation by means of potential theory. It is well known that explicit finite-difference or finite-element schemes for the heat equation are stable only if the time step 􏰃t is of the order O(􏰃x2), where 􏰃x is the finest spatial grid spacing. In contrast, for the Dirichlet and Neumann problems on the unit ball in all dimensions d ≥ 1, we show that the simplest Volterra marching scheme, i.e., the forward Euler scheme, is unconditionally stable. Our proof is based on an explicit spectral radius bound of the marching matrix, leading to an estimate that an L2-norm of the solution to the integral equation is bounded by cd T d /2 times the norm of the right-hand side. For the Robin problem on the half-space in any dimension, with constant Robin (heat transfer) coefficient κ, we exhibit a constant C such that the forward Euler scheme is stable if 􏰃t < C /κ 2 , independent of any spatial discretization. This relies on new lower bounds on the spectrum of real symmetric Toeplitz matrices defined by convex sequences. Finally, we show that the forward Euler scheme is unconditionally stable for the Dirichlet problem on any smooth convex domain in any dimension, in the L∞-norm.

Recent technical advances lead to the coupling of PET and MRI scanners, enabling one to acquire functional and anatomical data simultaneously. In this paper, we propose a tight frame based PET-MRI joint reconstruction model via the joint sparsity of tight frame coefficients. In addition, a nonconvex balanced approach is adopted to take the different regularities of PET and MRI images into account. To solve the nonconvex and nonsmooth model, a proximal alternating minimization algorithm is proposed, and the global convergence is present based on the Kurdyka--Łojasiewicz property. Finally, the numerical experiments show that our proposed models achieve better performance over the existing PET-MRI joint reconstruction models.

The Euler equation of compressible flows is solved by the finite volume method, where high order accuracy is achieved by the reconstruction of each component of upwind fluxes of a flux splitting using the biased averaging procedure. Compared to the solution reconstruction in Godunov-type methods, its implementation is simple and easy, and the computational complexity is relatively low. This approach is parameter-free and requires neither a Riemann solver nor field-by-field decomposition. The numerical results from both dynamic and steady state calculations demonstrate the accuracy and robustness of this approach. Some techniques for the acceleration of the convergence to the steady state are discussed, including multigrid and multistage Runge–Kutta time methods.

This paper discusses three basic issues related to the design of finite difference schemes for unsteadyviscous incompressible flows using vorticity formulations: the boundary condition for vorticity, an efficient time-stepping procedure, and the relation between these schemes and the ones based on velocity-pressure formulation. We show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier. We also show that if we couple a standard centered difference scheme with third- or fourth-order explicit Runge-Kutta methods, the resulting schemes have nocell Reynolds number constraints. For high Reynolds number flows, these schemes are stable under the CFL condition given by the convective terms. Finally, we show that the classical MAC scheme is the same as Thom's formula coupled with second-order centered differences in the interior, in the sense that one can define discrete vorticity in a natural way for the MAC scheme and get the same values as the ones computed from Thom's formula. We use this to derive an efficient fourth-order Runge-Kutta time discretization for the MAC scheme from the one for Thom's formula. We present numerical results for driven cavity flow at high Reynolds number (10 ).