engleski

Greedy search of optimal approximate solutions

In this paper we develop a procedure to deal with a familyof parameter-dependent ill-posed problems, for which the exact solu-tion in general does not exist. The original problems are relaxed byconsidering corresponding approximate ones, whose optimal solutionsare well defined, where the optimality is determined by the minimalnorm requirement. The procedure is based upon greedy algorithms thatpreserve, at least asymptotically, Kolmogorov approximation rates. Inorder to provide a-priori estimates for the algorithm, a Tychonoff- typeregularization is applied, which adds an additional parameter to themodel. The theory is developed in an abstract theoretical frameworkthat allows its application to different kinds of problems. We presenta specific example that considers a family of ill-posed elliptic problems.The required general assumptions in this case translate to rather nat-ural uniform lower and upper bounds on coefficients of the considered operators.

optimal approximate solutions ; parametrized PDEs ; greedyalgorithms

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