A survey of information-based complexity

AbstractWe survey some recent results in information-based complexity. We focus on the worst case setting and also indicate some average case results

On the existence of optimal affine methods for approximating linear functionals

AbstractThe existence of an optimal affine method using linear information is established for the approximation of a linear functional on a convex set. This is a generalization of a result of S. A. Smolyak (“On Optimal Restoration of Functions and Functionals of Them,” Candidate Dissertation, Moscow State University, 1965)

The Application of Approximation and Complexity Theory Methods to the Solution of Computer Vision Problems

We survey aspects of approximation and complexity theory and their application to the numerous computer vision problems that require an approximate solution because only partial information is available. We consider ill-posed computer vision problems and the methods that can be employed towards reformulating them as well-posed. We are particularly interested in the surface reconstruction problem that is encountered in the construction of the 2 1/2-D sketch, and which has been formulated and solved using different methods. We apply regularization theory, information-based complexity, and other methods to the solution of this problem. Finally, the shape from shadows problem is formulated and the optimal error algorithm is constructed and analyzed

The Application of Approximation and Complexity Theory Methods to the Solution of Computer Vision Problems

We survey aspects of approximation and complexity theory and their application to the numerous computer vision problems that require an approximate solution because only partial information is available. We consider ill-posed computer vision problems and the methods that can be employed towards reformulating them as well-posed. We are particularly interested in the surface reconstruction problem that is encountered in the construction of the 2 1/2-D sketch, and which has been formulated and solved using different methods. We apply regularization theory, information-based complexity, and other methods to the solution of this problem. Finally, the shape from shadows problem is formulated and the optimal error algorithm is constructed and analyzed

Optimal solution of nonlinear equations

Journal ArticleWe survey recent worst case complexity results for the solution of nonlinear equations. Notes on worst and average case analysis of iterative algorithms and a bibliography of the subject are also included

Optimal solution of ordinary differential equations

AbstractWe survey some recent optimality results for the numerical solution of initial value problems for ODE. We assume that information used by an algorithm about a right-hand-side function is partial. Two settings of information-based complexity are considered: the worst case and asymptotic. Upper and lower bounds on the error are presented for three types of information: standard, linear, and nonlinear continuous. In both settings, minimum error algorithms are exhibited