On Regularization Methods for Inverse Problems of Dynamic Type

In this paper, we consider new regularization methods for linear inverse problems of dynamic type. These methods are based on dynamic programming techniques for linear quadratic optimal control problems. Two different approaches are followed: a continuous and a discrete one. We prove regularization properties and also obtain rates of convergence for the methods derived from both approaches. A numerical example concerning the dynamic EIT problem is used to illustrate the theoretical results.     

Semicontinuity of the Approximate Solution Sets of Multivalued Quasiequilibrium Problems

We consider two kinds of approximate solutions and approximate solution sets to multivalued quasiequilibrium problems. Sufficient conditions for the lower semicontinuity, Hausdorff lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity, and closedness of these approximate solution sets are established. Applications in approximate quasivariational inequalities, approximate fixed points, and approximate quasioptimization problems are provided.     

Fractional Steps Scheme to Approximate the Phase-Field Transition System with Non-Homogeneous Cauchy–Neumann Boundary Conditions

The paper presents a proof of the convergence for an iterative scheme of fractional steps type associated to the phase-field transition system (a nonlinear parabolic system) with non-homogeneous Cauchy–Neumann boundary conditions. The advantage of such method consists in simplifying the numerical computation necessary to be done in order to approximate the solution of a nonlinear parabolic system. On the basis of this approach, a numerical algorithm in the two dimensional case is introduced and an industrial implementation is made.     

A New Approach to Source Conditions in Regularization with General Residual Term

This article addresses Tikhonov-like regularization methods with convex penalty functionals for solving nonlinear ill-posed operator equations formulated in Banach or, more general, topological spaces. We present an approach for proving convergence rates that combines advantages of approximate source conditions and variational inequalities. Precisely, our technique provides both a wide range of convergence rates and the capability to handle general and not necessarily convex residual terms as well as nonsmooth operators. Initially formulated for topological spaces, the approach is extensively discussed for Banach and Hilbert space situations, showing that it generalizes some well-known convergence rates results.     

A Posteriori Error Estimates of Mixed Methods for Parabolic Optimal Control Problems

In this article, we shall give a brief review on the fully discrete mixed finite element method for general optimal control problems governed by parabolic equations. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. Furthermore, we derive a posteriori error estimates for the finite element approximation solutions of optimal control problems. Some numerical examples are given to demonstrate our theoretical results.     

Superconvergence of H 1-Galerkin Mixed Finite Element Methods for Second-Order Elliptic Equations

We derive superconvergence result for H ¹-Galerkin mixed finite element method for second-order elliptic equations over rectangular partitions. Compared to standard mixed finite element procedure, the method is not subject to the Ladyzhenskaya–Bab?ska–Brezzi (LBB) condition and the approximating finite element spaces are allowed to be of different polynomial degrees. Superconvergence estimate of order 𝒪(h ^k+3), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas elements, is established for the flux via a postprocessing technique.     

椭圆边界上的自然积分算子及各向异性外问题的耦合算法

1.引 言为求解微分方程的外边值问题常需要引进人工边界（见[1-4]）,对人工边界外部区域作自然边界归化得到的自然积分方程即Dirichlet-Neumann映射,正是人工边界上的准确的边界条件（见[2-6]）,这是一类非局部边界条件.自然积分算子即Dirichlet-Neumann算子,     

The semiclassical modified nonlinear Schrödinger equation I: Modulation theory and spectral analysis

We study an integrable modification of the focusing nonlinear Schrödinger equation from the point of view of semiclassical asymptotics. In particular, (i) we establish several important consequences of the mixed-type limiting quasilinear system including the existence of maps that embed the limiting forms of both the focusing and defocusing nonlinear Schrödinger equations into the framework of a single limiting system for the modified equation, (ii) we obtain bounds for the location of the discrete spectrum for the associated spectral problem that are particularly suited to the semiclassical limit and that generalize known results for the spectrum of the nonselfadjoint Zakharov-Shabat spectral problem, and (iii) we present a multiparameter family of initial data for which we solve the associated spectral problem in terms of special functions for all values of the semiclassical scaling parameter. We view our results as part of a broader project to analyze the semiclassical limit of the modified nonlinear Schrödinger equation via the noncommutative steepest descent procedure of Deift and Zhou, and we also present a selfcontained development of a Riemann-Hilbert problem of inverse scattering that differs from those given in the literature and that is well adapted to semiclassical asymptotics.