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The inverse problem of reconstructing two space-varying coefficients in a system of one-dimensional (1-d) time-dependent advection-diffusion-reaction (ADR) equations is considered. The ADR system can be used as a water quality model which describes the evolution of the biochemical oxygen demand (BOD) and dissolved oxygen (DO) in a river or stream. The coefficients to be reconstructed represents the effect of the deoxygenation and superficial reaeration processes on the DO and BOD concentration in water. Hölder stability estimates for the coefficients of interest are established using the Carleman estimate technique. [ABSTRACT FROM AUTHOR]

The problem of determining a term in the right-hand side of elliptic equations from an observation on a part of the boundary is investigated. The inverse problem is formulated as an operator equation and then stabilized by Tikhonov regularization method. The regularized problem is discretized based on Hinze's variational discretization concept and the regularization parameter is chosen guaranteeing that when noise level and the discretization mesh size tend to zero, the solution of the discretized regularized problem converges to the $ f^* $ f ∗ -minimum norm solution of the continuous inverse problem. Some numerical examples are presented for illustrating the performance of the proposed method. [ABSTRACT FROM AUTHOR]

Let X be a Banach space with norm ∥ ⋅ ∥ . Let A : D ⁢ (A) ⊂ X → X be an (possibly unbounded) operator that generates a uniformly bounded holomorphic semigroup. Suppose that ε > 0 and T > 0 are two given constants. The backward parabolic equation of finding a function u : [ 0 , T ] → X satisfying u t + A ⁢ u = 0 , 0 < t < T , ∥ u ⁢ (T) - φ ∥ ⩽ ε , for φ in X, is regularized by the generalized Sobolev equation u α ⁢ t + A α ⁢ u α = 0 , 0 < t < T , u α ⁢ (T) = φ , where 0 < α < 1 and A α = A ⁢ (I + α ⁢ A b ) - 1 with b ⩾ 1 . Error estimates of the method with respect to the noise level are proved. [ABSTRACT FROM AUTHOR]

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]

We investigate the problem of recovering the possibly both space and time-dependent forcing term along with the temperature in hyperbolic systems from many integral observations. In practice, these average weighted integral observations can be considered as generalized interior point measurements. This linear but ill-posed problem is solved using the Tikhonov regularization method in order to obtain the closest stable solution to a given a priori known initial estimate. We prove the Fréchet differentiability of the Tikhonov regularization functional and derive a formula for its gradient. This minimization problem is solved iteratively using the conjugate gradient method. The numerical discretization of the well-posed problems, that are: the direct, adjoint and sensitivity problems that need to be solved at each iteration is performed using finite-difference methods. Numerical results are presented and discussed for one and two-dimensional problems. [ABSTRACT FROM AUTHOR]

We study the problem of determining the initial condition in parabolic equations with time-dependent coefficients from integral observations which can be regarded as generalizations of point-wise interior observations. Our approach is new in the sense that for determining the initial condition we do not assume that the data available in the whole space domain at the final moment or in a subset of the space domain during a certain time interval, but some integral observations during a time interval. We propose a variational method in combination with Tikhonov regularization for solving the problem and then discretize it by finite difference splitting methods. The discretized minimization problem is solved by the conjugate gradient method and tested on computer to show its efficiency. Also as a by-product of the variational method, we propose a numerical scheme for estimating the degree of ill-posedness of the problem. [ABSTRACT FROM AUTHOR]

The finite difference method is applied to an optimal control problem for a system governed by a nonlinear Schrödinger equation with a complex coefficient. The optimal control problem is discretized by the finite difference method, the error estimate for the finite difference scheme is established and the convergence of difference approximations of the optimal control according to the functional is proved. [ABSTRACT FROM AUTHOR]