Adaptive Wavelet Methods for Linear and Nonlinear Least-Squares Problems

• Published in: Foundations of computational mathematics Vol. 14; no. 2; pp. 237 - 283
• Main Author: Stevenson, Rob
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.04.2014; Springer; Springer Nature B.V
• Subjects: Applications of Mathematics; Approximation; Asymptotic properties; Boundary value problems; Computer Science; Convergence (Mathematics); Differential equations, Linear; Differential equations, Nonlinear; Economics; Galerkin methods; Least squares; Least squares method; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematical research; Mathematics; Mathematics and Statistics; Matrix Theory; Nonlinear equations; Numerical Analysis; Operators; Optimization; Wavelet; Wavelet transforms; Optimal convergence rates; 65N12; Galerkin discretization; 41A25; Asymptotically optimal computational complexity; 42C40; Adaptive wavelet methods; 47J25; 65J15; 65T60; 65N30; Least-squares formulations of boundary value problems
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-013-9184-6

The adaptive wavelet Galerkin method for solving linear, elliptic operator equations introduced by Cohen et al. (Math Comp 70:27–75, 2001) is extended to nonlinear equations and is shown to converge with optimal rates without coarsening. Moreover, when an appropriate scheme is available for the approximate evaluation of residuals, the method is shown to have asymptotically optimal computational complexity. The application of this method to solving least-squares formulations of operator equations G ( u ) = 0 , where G : H → K ′ , is studied. 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.