Abstract. A new approach to the construction of finite-difference methods is presented. It is shown how the multi-point differentiators can generate regularizing algorithms with a stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iteratively regularized scheme for solving the numerical differentiation problem in the form of Volterra integral equation is developed. 1.

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

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.

The goal of this paper is to develop a general approach to solution of ill-posed nonlinear problems in a Hilbert space based on continuous processes with a regularization procedure. To avoid the illposed inversion of the Fréchet derivative operator a regularizing oneparametric family of operators is introduced. Under certain assumptions on the regularizing family a general convergence theorem is proved. The proof is based on a lemma describing asymptotic behavior of solutions of a new nonlinear integral inequality. Then the applicability of the theorem to the continuous analogs of the Newton, Gauss-Newton and simple iteration methods is demonstrated.

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Our basic model is a noncooperative multi-player game in which the governments of neighboring countries trade emission reductions. We prove the existence of a market equilibrium (combining properties of Pareto and Nash equilibria) and study algorithms of searching a market equilibrium. The algorithms are interpreted as repeated auctions in which the auctioneer has no information on countries' costs and benefits and every government has no information on the costs and benefits of other countries. In each round of the auction, the auctioneer o#ers individual prices for emission reductions and observes countries' best replies. We consider several auctioneer's policies and provide conditions that guarantee approaching a market equilibrium. From a game-theoretical point of view, the repeated auction describes a process of learning in a noncooperative repeated game with incomplete information. -- iii -- Contents 1 Market equilibrium, Nash equilibrium and Pareto maximum 3 2 Existence of mar...

We present a new version of the almost optimal Circumscribed Ellipsoid Algorithm (CEA) for approximating fixed points of nonexpanding Lipschitz functions. We utilize the absolute and residual error criteria with respect to the second norm. The numerical results confirm that the CEA algorithm is much more efficient than the simple iteration algorithm whenever the Lipschitz constant is close to 1. We extend the applicability of the CEA algorithm to larger classes of functions that may be globally expanding, however are nonexpanding/contracting in the direction of fixed points. We also develop an efficient hyper-bisection/secant hybrid method for combustion chemistry fixed point problems.

Interim Reports on work of the International Institute for Applied Systems Analysis receive only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organizations supporting the work. –ii– In this paper the firm is analyzed and modeled as a set of different subcoalitions (agents) each with their own objectives. It examines how the goals can be conflicting and in turn how this influences the payoff structure of the subcoalitions given that they follow ‘simple’ decision rules, i.e. rules of thumb. This implies that the subcoalitions act in a boundedly rational way. To see how these decision making procedures evolve we make use of an (evolutionary) dynamic game theoretical framework. Consequently, the main aim is to address the issue of modeling the dynamic and adaptive nature of the subcoalitions.

A review of the authors’ results is given. Several methods are discussed for solving nonlinear equations F (u) =f, whereF 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 versions of the DSM include a regularized Newton-type method, a gradient-type method, and a simple iteration method. Apriori 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.

A version of the Dynamical Systems Method for solving ill-conditioned linear algebraic systems is studied in this paper. An a priori and a posteriori stopping rules are justified. Algorithms for computing the solution in the case when SVD of the lefthand side matrix is available is presented. Numerical results show that when SVD of the left-hand side matrix is available or not computationally expensive to obtain the new method can be considered as an alternative to the Variational Regularization or the truncated singular value decomposition method. Keywords. Ill-conditioned, DSM, Variational Regularization 1