Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

The focus of this article is on conditional stability estimates for ill-posed inverse problems in partial differential equations. Conditional stability estimates have been obtained in related literature by a couple different methods. In this article, we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We provide the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capabilities of our method are illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.     

Saddlepoint approximations to P-values for comparison of density estimates

Summary  This article is concerned with computing approximate p-values for the maximum of the absolute difference between kernel density estimates. The approximations are based on treating the process of local extrema of the differences as a nonhomogeneous Poisson Process and estimating the corresponding local intensity function. The process of local extrema is characterized by the intensity function, which determines the rate of local extrema above a given threshold. A key idea of this article is to provide methods for more accurate estimation of the intensity function by using saddlepoint approximations for the joint density of the difference between kernel density estimates and using the first and second derivative of the difference. In this article, saddlepoint approximations are compared to gaussian approximations. Simulation results from saddlepoint approximations show consistently better agreement between empirical p-value and predetermined value with various bandwidths of kernel density estimates.     

Numerical Simulation for Convection-Diffusion Problem with Periodic Micro-Structure

In this article, the convection dominated convection-diffusion problems with the periodic micro-structure are discussed. A two-scale finite element scheme based on the homogenization technique for this kind of problems is provided. The error estimates between the exact solution and the approximation solution, of the homogenized equation or the two-scale finite element scheme are analyzed. It is shown that the scheme provided in this article is convergent for any fixed diffusion coefficient 5, and it may be convergent independent of δ under some conditions. The numerical results demonstrating the theoretical results are presented in this article.     

Efficient estimation of functional-coefficient regression models with different smoothing variables

In this article,a procedure for estimating the coefficient functions on the functional-coefficient regression models with different smoothing variables in different coefficient functions is defined.First step,by the local linear technique and the averaged method,the initial estimates of the coefficient functions are given.Second step,based on the initial estimates,the efficient estimates of the coefficient functions are proposed by a one-step back-fitting procedure.The efficient estimators share the same asymptotic normalities as the local linear estimators for the functional-coefficient models with a single smoothing variable in different functions.Two simulated examples show that the procedure is effective.     

Stochastic Nonlinear Equations of Schrödinger Type

In this article, the solution for a stochastic nonlinear equation of Schrödinger type, which is perturbed by an infinite dimensional Wiener process, is investigated. The existence of the solution is proved by using the Galerkin method. Moment estimates for the solution are also derived. Examples from physics are given in the final part of the article.     

A backward nonlinear heat equation: regularization with error estimates

In this article the authors consider a backward nonlinear heat equation. The uniqueness of the problem is proved and the problem is regularized by finite dimensional approximations. Error estimates in some particular cases are given.     

SOME BOUNDARY VALUE PROBLEMS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS OF SECOND ORDER

The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem.Firstly the formulation and estimates of solutions of the oblique derivative problem are given, and then by the above estimates and the method of parameter extension,the existence of solutions of the above problem is proved.In this article,the complex analytic method is used,namely the corresponding problem for degenerate elliptic complex equations of first order is firstly discussed,afterwards the above problem for the degenerate elliptic equations of second order is solved.     

Individual homogenization of nonlinear parabolic operators

In this article, we prove an individual homogenization result for a class of almost periodic nonlinear parabolic operators. The spatial and temporal heterogeneities are almost periodic functions in the sense of Besicovitch. The latter allows discontinuities and is suitable for many applications. First, we derive stability and comparison estimates for sequences of G-convergent nonlinear parabolic operators. Furthermore, using these estimates, the individual homogenization result is shown.     

Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations

In this article we study global-in-time Strichartz estimates for the Schrödinger evolution corresponding to long-range perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article [D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math. 130 (2008) 571-634] of the third author, where it is proved that local smoothing estimates imply Strichartz estimates. By [D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math. 130 (2008) 571-634] the local smoothing estimates are known to hold for small perturbations of the Laplacian. Here we consider the case of large perturbations in three increasingly favorable scenarios: (i) without non-trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term, (ii) with non-trapping assumptions we prove global estimates modulo a lower order spatially localized error term, and (iii) for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum.     

Existence and blow-up of solutions for a porous medium equation with nonlocal boundary condition

In this article, a porous medium equation with nonlocal boundary condition and a localized source is studied. The results of the existence of global solutions or blow-up of solutions are given. The blow-up rate estimates are also obtained under some conditions.     

New estimates in the four-dimensional divisor problem with applications

In this article we consider three problems: 1. The asymptotic behaviour of the quadratic moment of the exponential divisor function. 2. The distribution of powerful integers of type 4. 3. The average number of direct factors of a finite Abelian group. We prove new estimates for the error terms in the asymptotic representations. For this purpose new estimates in the general four-dimensional divisor problem are needed.     

两类由生物降解模型产生的反应扩散方程组解的渐进性

本文主要研究了两类由生物降解模型产生的非线性反应扩散方程组解的渐进性质．利用上下解方法,LP估计以及半群理论,我们证明了解的全局存在性,一致有界性,解的渐进性和解的衰减速率估计．     

Least squares methods for Volterra equations and generalizations

In this article least squares approximations to Volterra integral equations are considered, both with exact integration and with quadrature. Optimal error estimates are derived, and it is shown that the same order of convergence is obtained in both cases with only modest requirements on the quadrature rule used in the latter. The most important practical setting for least squares is the case of convolution kernels, and these are also studied in this article. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 679–693, 1998     

Interior error estimate for periodic homogenization

The aim of this Note is to give interior error estimates for problems in periodic homogenization, by using the periodic unfolding method. The interior error estimates are obtained by transposition without any supplementary hypothesis of regularity on correctors. This error is of order ?. To cite this article: G. Griso, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Weighted Norm Inequalities for Marcinkiewicz Integrals with Non-Smooth Kernels on Spaces of Homogeneous Type

In this article, we obtain some weighted estimates for Marcinkiewicz integrals with non-smooth kernels on spaces of homogeneous type. The weight $\omega$ considered here belongs to the Muckenhoupt’s class $A_∞.$ Moreover, weighted estimates for commutators of BMO functions and Marcinkiewicz integrals are also given.     

Crank‐Nicolson‐Galerkin finite element scheme for nonlocal coupled parabolic problem using the Newton's method

In this article, a finite element scheme based on the Newton's method is proposed to approximate the solution of a nonlocal coupled system of parabolic problem. The Crank‐Nicolson method is used for time discretization. Well‐posedness of the problem is discussed at continuous and discrete levels. We derive a priori error estimates for both semidiscrete and fully discrete formulations. Results based on usual finite element method are provided to confirm the theoretical estimates.     

Computation of alternated reachability tubes for linear control systems under uncertainty

The problem of constructing the reachability domain for linear control system in a presence of geometrically bounded unknown disturbance is considered. A high actuality of the problem for engineering applications requires an efficient calculation technique for the reach sets in a class of closed-loop control. The technique suggested in the article is based on the ellipsoidal approximations developed by A.B. Kurzhansky for the alternated reachability domains. In the article these estimates are complemented with an adaptive regularization to guarantee the continuability of the ellipsoidal estimates. The quadratic structure of the regularization combines well with an ellipsoidal nature of the estimates thus making it possible to adjust the existing ellipsoidal estimation schema in a transparent fashion for achieving the continuable and non-singular estimates via adaptive choice of regularization parameters.     

Superconvergent estimates of conforming finite element method for nonlinear time‐dependent Joule heating equations

In this article, we study the superconvergence analysis of conforming bilinear finite element method (FEM) for nonlinear Joule heating equations. Based on the rigorous estimates together with high accuracy analysis of this element, mean value technique and interpolation postprocessing approach, the superclose and superconvergent estimates about the related variables in H¹‐norm are derived for semidiscrete and a linearized backward Euler fully discrete schemes, which extends the results of optimal estimates obtained for conforming FEMs in the previous literature. At last, a numerical experiment is performed to verify the theoretical analysis.     

Error Estimates for a Class of Elliptic Optimal Control Problems

In this article, functional type a posteriori error estimates are presented for a certain class of optimal control problems with elliptic partial differential equation constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in the late 1990s are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible control and the optimal control is introduced. This error quantity can be estimated from both sides using the estimates for the cost functional value. The theoretical results are confirmed by numerical tests.     

Analysis of jump processes with nondegenerate jumping kernels

We prove regularity estimates for functions which are harmonic with respect to certain jump processes. The aim of this article is to extend the method of Bass–Levin (2002) [3] and Bogdan–Sztonyk (2005) [6] to more general processes. Furthermore, we establish a new version of the Harnack inequality that implies regularity estimates for corresponding harmonic functions.