Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings
نویسندگان
چکیده
We study the optimal approximation of the solution of an operator equation A(u) = f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic PDEs that are given by an isomorphism A : Hs 0(Ω) → H−s(Ω), where s > 0 and Ω is an arbitrary bounded Lipschitz domain in Rd. Here we prove that approximation by linear mappings is as good as the best n-term approximation with respect to an optimal Riesz basis. We discuss why nonlinear approximation still might be very important for the approximation of elliptic problems. Our results are concerned with approximations and their errors, not with their numerical realization. AMS subject classification: 41A25, 41A46, 41A65, 42C40, 65C99
منابع مشابه
Optimal approximation of elliptic problems by linear and nonlinear mappings I
We study the optimal approximation of the solution of an operator equation A(u) = f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic PDEs that are given by an isomorphism A :...
متن کاملOptimal approximation of elliptic problems by linear and nonlinear mappings II
We study the optimal approximation of the solution of an operator equation A(u) = f by four types of mappings: a) linear mappings of rank n; b) n-term approximation with respect to a Riesz basis; c) approximation based on linear information about the right hand side f ; d) continuous mappings. We consider worst case errors, where f is an element of the unit ball of a Sobolev or Besov space Br q...
متن کاملOptimal approximation of elliptic problems by linear and nonlinear mappings IV: Errors in L2 and other norms
We study the optimal approximation of the solution of an operator equation A(u) = f by linear and different types of nonlinear mappings. In our earlier papers we only considered the error with respect to a certain Hs-norm where s was given by the operator since we assumed that A : Hs 0(Ω) → H−s(Ω) is an isomorphism. The most typical case here is s = 1. It is well known that for certain regular ...
متن کاملA Numerical Approach for Fractional Optimal Control Problems by Using Ritz Approximation
In this article, Ritz approximation have been employed to obtain the numerical solutions of a class of the fractional optimal control problems based on the Caputo fractional derivative. Using polynomial basis functions, we obtain a system of nonlinear algebraic equations. This nonlinear system of equation is solved and the coefficients of basis polynomial are derived. The convergence of the num...
متن کاملSolving linear and nonlinear optimal control problem using modified adomian decomposition method
First Riccati equation with matrix variable coefficients, arising in optimal and robust control approach, is considered. An analytical approximation of the solution of nonlinear differential Riccati equation is investigated using the Adomian decomposition method. An application in optimal control is presented. The solution in different order of approximations and different methods of approximat...
متن کامل