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Adaptive wavelet methods for linear and nonlinear least-squares problems. (English) Zbl 1303.65043

An adaptive wavelet Galerkin method for solving linear elliptic operator equations was introduced by A. Cohen et al. [Math. Comput. 70, No. 233, 27–75 (2001; Zbl 0980.65130)]. In this paper, the adaptive (wavelet) Galerkin method is extended to nonlinear equations. Let \(H\) be a real Hilbert space with the dual \(H'\). Let \(\Psi\) be a Riesz basis of \(H\), where \(\Psi\) is a suitable wavelet basis in applications to a variational formulation of a partial differential equation. Further let \(F:\, {\mathrm{dom}}(F) \to H'\) with \({\mathrm{dom}}(F)\subset H\) be a nonaffine mapping. In the first part of this paper, the author presents a new proof for an application of the adaptive Galerkin method to the nonlinear problem \(F(u) =0\), where the Fréchet derivative \(DF(u)\) is elliptic. Without coarsening, the author presents an adaptive method that produces a sequence of (approximate) Galerkin solutions. It is shown that this sequence converges to the solution \(u\) of \(F(u) =0\) at the best possible rate and, under additional assumptions of \(F\) and \(\Psi\), with optimal computational cost.
In the second part of this paper, the author applies the adaptive wavelet Galerkin method to least squares problems. Let \(G:\,{\mathrm{dom}}(G) \to K'\) with \({\mathrm{dom}}(G)\subset H\) be a nonaffine mapping, where \(H\) and \(K\) are possibly different, real Hilbert spaces and where \(DG(u)\) is a bounded invertible linear mapping. For solving the nonlinear equation \(G(u) =0\), the author uses the least squares functional \(Q(v) = \frac{1}{2}\, \|G(v)\|^2\). For formulations of partial differential equations as first order least-squares systems, a valid approximate residual evaluation is developed that is easy to implement and quantitatively efficient. In a numerical example, the adaptive wavelet Galerkin method is applied to a least squares formulation of a nonlinear ordinary differential equation of first order.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65T60 Numerical methods for wavelets
47J25 Iterative procedures involving nonlinear operators
35J60 Nonlinear elliptic equations

Citations:

Zbl 0980.65130

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References:

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