Abstract not found

This paper presents an abstract analysis of bounded variation (BV) methods for ill-posed operator equations Au = z. Let T (u) def = kAu \Gamma zk 2 + ffJ(u); where the penalty, or "regularization", parameter ff ? 0 and the functional J(u) is the BV norm or seminorm of u, also known as the total variation of u. Under mild restrictions on the operator A and the functional J(u), it is shown that the functional T (u) has a unique minimizer which is stable with respect to certain perturbations in the data z, the operator A, the parameter ff, and the functional J(u). In addition, convergence results are obtained which apply when these perturbations vanish and the regularization parameter is chosen appropriately.

Abstract. The problem of nding adapted solutions to systems of coupled linear forwardbackward stochastic di erential equations (FBSDEs, for short) is investigated. A necessary condition of solvability leads to a reduction of general linear FBSDEs to a special one. By some ideas from controllability in control theory, using some functional analysis, we obtain a necessary and su cient condition for the solvability of the linear FBSDEs with the processes Z (serves as a correction, see x1) being absent in the drift. Then a Riccati type equation for matrix-valued (not necessarily square) functions is derived using the idea of the Four-Step-Scheme (introduced in [11] for general FBSDEs). The solvability of such a Riccati type equation is studied which leads to a representation of adapted solutions to linear FBSDEs. Keywords. Linear forward-backward stochastic di erential equations, adapted solution, Riccati type equation. AMS Mathematics subject classi cation. 60H10.

We consider the positive solutions to the semilinear problem: An + %f ( I,) = 0, in B”. 1, = 0, on aB”. i where B ” is the unit ball in R”.)I 2 I. and I. is a positive parameter. It is well known that iffis a smooth function, then any positive solution to the equation is radially symmetric, and all solutions can be pdrameterized by their maximum values. We develop a unitied approach to obtain the exact multiplicity of the positive solutions for a wide class ol ’ nonlinear functions./1ll). and the precise shape of the global bifurcation diagrams are rigorously proved. Our technique combines the birurcation analysis. stability analysis, and topological methods. We show that the shape of the bil’urcdtion curve depends on the shape of the graph of (unction Jrr)/fr as well as the growth rate of,/: cj 1999 Academic Press 1.

This paper proves the estimate ku" uk L 1 (Q T ) C" 1=5 , where, for all " > 0, u" is the weak solution of (u" ) t + div(q f(u" )) ('(u")+"u") = 0 with initial and boundary conditions, u is the entropy weak solution of u t + div(qf(u)) ('(u)) = 0 with the same initial and boundary conditions, and C > 0 does not depend on ". The domain

2. Continuous methods for well posed problems 3. Discretization theorems for well-posed problems

The Proximal Point Algorithm (PPA) is a method for solving inclusions of the form 0 2 T (z) where T is a monotone operator on a Hilbert space. The algorithm is one of the most powerful and versatile solution techniques for solving variational inequalities, convex programs, and convex--concave mini--max problems. It possesses a robust convergence theory for very general problem classes and is the basis for a wide variety of decomposition methods called splitting methods. Yet, the classical PPA typically exhibits slow convergence in many applications. For this reason, acceleration methods for the PPA algorithm are of great practical importance. In this paper we propose a variable metric implementation of the proximal point algorithm. In essence, the method is a Newton--like scheme applied to the Moreau--Yosida resolvent of the operator T . In this article, we establish the global and linear convergence of the proposed method. In addition, we characterize the super--linear convergence of ...

The aim of this paper is to establish rigorous results on the large time behavior of nonlocal models for aggregation, including the possible presence of nonlinear diffusion terms modeling local repulsions. We show that, as expected from the practical motivation as well as from numerical simulations, one obtains concentrated densities (Dirac δ distributions) as stationary solutions and large time limits in the absence of diffusion. In addition, we provide a comparison for aggregation kernels with infinite respectively finite support. In the first case, there is a unique stationary solution corresponding to concentration at the center of mass, and all solutions of the evolution problem converge to the stationary solution for large time. The speed of convergence in this case is just determined by the behavior of the aggregation kernels at zero, yielding either algebraic or exponential decay or even finite time extinction. For kernels with finite support, we show that an infinite number of stationary solutions exist, and solutions of the evolution problem converge only in a set-valued sense to the set of stationary solutions, which we characterize in detail. Moreover, we also consider the behavior in the presence of nonlinear diffusion terms, the most interesting case being the one of small diffusion coefficients. Via the implicit function theorem we give a quite general proof of a rather natural assertion for such models, namely that there exist stationary solutions that have the form of a local peak around the center of mass. Our approach even yields the order of the size of the support in terms of the diffusion coefficients. All these results are obtained via a metric gradient flow formulation using the Wasserstein metric for probability measures, and are carried out in the case of a single spatial dimension for convenience.

A review of the authors’s results is given. Several methods are discussed for solving nonlinear equations F(u) = f, where F is a monotone operator in a Hilbert space, and noisy data are given in place of the exact data. A discrepancy principle for solving the equation is formulated and justified. Various versions of the Dynamical Systems Method (DSM) for solving the equation are formulated. These methods consist of a regularized Newton-type method, a gradient-type method, and a simple iteration method. A priori and a posteriori choices of stopping rules for these methods are proposed and justified. Convergence of the solutions, obtained by these methods, to the minimal norm solution to the equation F(u) = f is proved. Iterative schemes with a posteriori choices of stopping rule corresponding to the proposed DSM are formulated. Convergence of these iterative schemes to a solution to equation F(u) = f is justified. New nonlinear differential inequalities are derived and applied to a study of large-time behavior of solutions to evolution equations. Discrete versions of these inequalities are established.

results are given without proofs, sometimes short sketches are included to help the reader's intuition. Applications to (1.1), (1.2) are given with more details. We rst introduce basic notation and denitions. In the whole section Y is an ordered Banach space with norm k k and order cone Y+ . Recall that an order cone is a closed convex cone such that Y+ \ ( Y+ ) = f0g. We assume that Y is strongly ordered which means that int Y+ , the interior of Y+ , is nonempty. For x; y 2 Y we write x y x < y x y if y x 2 Y+ ; if x y and x 6= y ; if y x 2 int Y+ : The reversed signs are used in the usual way. Two points are said to be related (or ordered) if they are related by or . The notation A B (similarly for < and ) between two sets means that x y whenever x 2 A and y 2 B. A mapping F : D(F ) Y ! Y is said to be monotone if x; y 2 D(F ) and x y imply F (x) F (y). It is called strongly monotone if x; y 2 D(F ) and x < y imply F (x) F (y). A linear strongly mon...