An Efficient Discrete Landweber Iteration for Nonlinear Problems

• Published in: International journal of applied and computational mathematics 2022-08, Vol.8 (4), Article 189
• Main Authors: Rajan, M. P., Jose, Jaise
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Applied mathematics; Approximation; Computational mathematics; Computational Science and Engineering; Convergence; Iterative methods; Mathematical analysis; Mathematical and Computational Physics; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Nonlinear equations; Nonlinearity; Nuclear Energy; Operations Research/Decision Theory; Operators (mathematics); Original Paper; Sparse matrices; Theoretical
• ISSN: 2349-5103; 2199-5796
• DOI: 10.1007/s40819-022-01390-6

The amount of discrete information required to compute the solution to various mathematical problems has been of great interest to the research community in the recent past. The projection schemes are one of the methods traditionally used for discretizing the problem. This paper presents a modified projection scheme for solving the nonlinear equation of the form G ( x ) = y in the context of the Landweber method x ~ k + 1 = x ~ k + G ′ ( x ~ k ) ∗ ( y ~ - G ( x ~ k ) ) , in which a finite-dimensional approximation A m ( x ~ k ) is used in place of Fréchet derivative G ′ ( x ~ k ) . This modified approximation A m ( x ~ k ) yields a sparse matrix structure and requires substantially fewer inner products compared to other known discretization schemes. We derive the convergence and convergence rate results under some nonlinearity conditions on the operator and an appropriate stopping rule. Finally, we illustrate the theoretical results by providing numerical examples.