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In this paper we consider numerical approximations of a constraint minimization problem, where the object function is a quadratic Dirichlet functional for vector fields and the interior constraint is given by a convex function. The solutions of this problem are usually referred to as harmonic maps. The solution is characterized by a nonlinear saddle point problem, and the corresponding linearized problem is well-posed near strict local minima. The main contribution of the present paper is to establish a corresponding result for a proper finite element discretization in the case of two space dimensions. Iterative schemes of Newton type for the discrete nonlinear saddle point problems are investigated, and mesh independent preconditioners for the iterative methods are proposed. [ABSTRACT FROM AUTHOR]

The spectral viscosity approximate solution of convex Hamilton­Jacobi equations with periodic boundary conditions is studied. It is proved in this paper that the approximation and its gradient remain uniformly bounded, formally spectral accurate, and converge to the unique viscosity solution. The L1-convergence rate of the order 1 - ε∀ε > 0 is obtained. [ABSTRACT FROM AUTHOR]

We present an unconditionally energy stable finite-difference scheme for the phase field crystal equation. The method is based on a convex splitting of a discrete energy and is semi-implicit. The equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable, regardless of time step size. We present local-in-time error estimates that ensure the convergence of the scheme. While this paper is primarily concerned with the phase field crystal equation, most of the theoretical results hold for the related Swift-Hohenberg equation as well. [ABSTRACT FROM AUTHOR]

In this paper, we derive some improved a posteriori error estimates for finite element approximation of Neumann boundary control problems. We first establish local upper a posteriori error estimates for both the state and the control approximation of the general convex boundary control problems. We then derive local upper and lower a posteriori error estimates for a class of control problems that frequently appear in applications. [ABSTRACT FROM AUTHOR]

This paper is concerned with optimization or minimization problems that are governed by operator equations, such as partial differential or integral equations, and thus are naturally formulated in an infinite dimensional function space V. We first construct a prototype algorithm of steepest descent type in V and prove its convergence. By using a Riesz basis in V we can transform the minimization problem into an equivalent one posed in a sequence space of type $\ell_p$. We convert the prototype algorithm into an adaptive method in $\ell_p$. This algorithm is shown to be convergent under mild conditions on the parameters that appear in the algorithm. Under more restrictive assumptions we are also able to establish the rate of convergence of our algorithm and prove that the work/accuracy balance is asymptotically optimal. Finally, we give two particular examples. [ABSTRACT FROM AUTHOR]

Strong-stability-preserving Runge-Kutta (SSPRK) methods are a specific type of time discretization method that have been widely used for the time evolution of hyperbolic partial differential equations (PDEs). Under a suitable stepsize restriction, these methods share a desirable nonlinear stability property with the underlying PDE, e.g., stability with respect to total variation, the maximum norm, or other convex functionals. This is of particular interest when the solution exhibits shock-like or other nonsmooth behavior. Many results are known for SSPRK methods with nonnegative coefficients. However, it has recently been shown that such methods cannot exist with order greater than 4. In this paper, we give a systematic treatment of explicit SSPRK methods with general (i.e., possibly negative) coefficients up to order 5. In particular, we show how to optimally treat negative coefficients (corresponding to a change in the upwind direction of the spatial discretization) in the context of effective CFL coefficient maximization and provide proofs of optimality of some explicit SSPRK methods of orders 1 to 4. We also give the first known explicit fifth-order SSPRK schemes and show their effectiveness in practice versus more well-known fifth-order explicit Runge-Kutta schemes. [ABSTRACT FROM AUTHOR]

In this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted W1,1 space. Convergence rate for the derivative of the approximate solutions is established under the assumption that a weak pointwise-error estimate is given. In other words, we are able to convert weak pointwise-error estimates to optimal error bounds in a weighted W1,1 space. For convex conservation laws, the assumption of a weak pointwise-error estimate is verified by Tadmor [SIAM J. Numer. Anal., 28 (1991), pp. 891–906]. Therefore, one immediate application of our W1,1 - convergence theory is that for convex conservation laws we indeed have W1,1 -error bounds for the approximate solutions to conservation laws. Furthermore, the O(∈)-pointwise-error estimates of Tadmor and Tang [SIAM J. Numer. Anal., 36 (1999), pp. 1739–1758] are recovered by the use of the W1,1-convergence result. [ABSTRACT FROM AUTHOR]

Introductions to articles dealing with applied mathematics that were published within the issue are presented, including "How to Transform One Convex Function Into Another," by Heinz Bauschke, Yves Lucet, and Michael Trienis, and "The QR Algorithm Revisited," by David S. Watkins.

Computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of computational convex analysis, which focuses on the numerical computation of fundamental transforms arising from convex analysis. Current models use symbolic, numeric, and hybrid symbolic-numeric algorithms. Our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing (computing the distance transform, the generalized distance transform, and mathematical morphology operators), partial differential equations (solving Hamilton--Jacobi equations and using differential equations numerical schemes to compute the convex envelope), max-plus algebra (computing the equivalent of the fast Fourier transform), multifractal analysis, etc. The fields of applications include, among others, computer vision, robot navigation, thermodynamics, electrical networks, medical imaging, and network communication [ABSTRACT FROM AUTHOR]

A number of works concerning rigorous convergence theory for adaptive finite element methods (AFEMs) for controlling global energy errors have appeared in recent years. However, many practical situations demand AFEMs designed to efficiently compute quantities which depend on the unknown solution only on some subset of the overall computational domain. In this work we prove convergence of an AFEM for controlling local energy errors. The first step in our convergence proof is the construction of novel a posteriori error estimates for controlling a weighted local energy error. This weighted local energy notion admits versions of standard ingredients for proving convergence of AFEMs such as quasi-orthogonality and error contraction, but modulo "pollution terms" which use weaker norms to measure effects of global solution properties on the local energy error. We then prove several convergence results for AFEMs based on various marking strategies, including a contraction result in the case of convex polyhedral domains. [ABSTRACT FROM AUTHOR]

Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on ℝ, and suppose that the cost c(x, y) of matching two points x, y satisfies the Monge condition: c(x1, y1) + c(x2, y2) < c(x1, y2) + c(x2, y1) whenever x1 < x2 and y1 < y2. We introduce a notion of locally optimal transport plan, motivated by the weak KAM (Aubry-Mather) theory, and show that all locally optimal transport plans are conjugate to shifts and that the cost of a locally optimal transport plan is a convex function of a shift parameter. This theory is applied to a transportation problem arising in image processing: for two sets of point masses on the circle, both of which have the same total mass, find an optimal transport plan with respect to a given cost function c satisfying the Monge condition. In the circular case the sorting strategy fails to provide a unique candidate solution, and a naive approach requires a quadratic number of operations. For the case of N real-valued point masses we present an O(N∣ log ϵ∣) algorithm that approximates the optimal cost within ϵ; when all masses are integer multiples of 1/M, the algorithm gives an exact solution in O(N log M) operations. [ABSTRACT FROM AUTHOR]

The proximal average operator provides a parametric family of convex functions that continuously transform one convex function into another even when the domains of the two functions do not intersect. We prove that the proximal average operator is a homotopy with respect to the epi-topology. study its properties, and present several explicit formulas for specific classes of functions. The parametric family inherits desirable properties such as differentiability and strict convexity from the given functions. The results illustrate the powerful tools available in convex and variational analysis from both a theoretical and a computational point of view. [ABSTRACT FROM AUTHOR]

We present a collection of fourteen conjectures and open problems in the fields of nonlinear analysis and optimization. These problems can be classified into three groups: problems of pure mathematical interest, problems motivated by scientific computing and applications, and problems whose solutions are known but for which we would like to know better proofs. For each problem we provide a succinct presentation, a list of appropriate references, and a view of the state of the art of the subject. [ABSTRACT FROM AUTHOR]