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We investigate the problem of learning a water quality model (BOD-DO model) from given data. Assuming that all parameters in the model are constants, we reformulate the problem as a system of linear equations for the unknown terms. Since in practice the system is often under-determined or over-determined and the observed data are noisy, we use an |$l^{1}$| -weighted regularization method to find a stable approximate solution. Then, Nesterov's algorithm is used to solve the regularized problem. Learning models with variable coefficients are also discussed. Numerical examples show that our approach works well with noisy data and has the ability to learn the BOD-DO model. [ABSTRACT FROM AUTHOR]

In this paper, we study a proximal method for the minimization problem arising from l0-regularization for nonlinear inverse problems. First of all, we prove the existence of solutions and give an optimality condition for solutions to the minimization problem. Then, we propose and prove the convergence of the proximal method for this minimization problem, which is controlled by step size conditions. Finally, we illustrate performance of the proximal method by applying it to a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of backward heat equations with time-dependent coefficient in the Banach space L p (R) , (1 < p < ∞). For this problem, we first prove the stability estimates of Hölder type. After that the Tikhonov-type regularization is applied to solve the problem. A priori and a posteriori parameter choice rules are investigated, which yield error estimates of Hölder type. Numerical implementations are presented to show the validity of the proposed scheme. [ABSTRACT FROM AUTHOR]

We investigate semismooth Newton and quasi-Newton methods for minimization problems arising from weighted ℓ1-regularization. We give proofs of the local convergence of these methods and show how their interpretation as active set methods leads to the development of efficient numerical implementations of these algorithms. We also propose and analyze Broyden updates for the semismooth quasi-Newton method. The efficiency of these methods is analyzed and compared with standard implementations. The paper concludes with some numerical examples that include both linear and nonlinear operator equations. [ABSTRACT FROM AUTHOR]