In this work we consider, in a Banach space framework, the iterative regularization of linear ill-posed problems. Our focus is on the recovery of solutions that have a logarithmic source representation. The mathematical framework on logarithms of operators is provided, and convergence rates for a class of iterative regularization schemes are deduced. This class includes classical stationary iteration methods like the Richardson iteration and the implicit method. In addition, parametric regularization methods, e.g.\xspace Lavrentiev’s method, are also covered, if the regularization parameters are taken from some countable set. The presentation includes both a priori and a posteriori parameter choice strategies.
We present a unified framework for estimating stochastic parameters in general mixed variational problems. This nonlinear inverse problem is formulated as a stochastic optimization problem using the output least-squares (OLS) objective, which minimizes the discrepancy between observed data and the computed solution. A key challenge in OLS-based formulations is the efficient computation of first- and second-order derivatives of the OLS functional, which depend on the sensitivities of the parameter-to-solution map—often costly and difficult to evaluate in stochastic settings. To address this, we develop a rigorous computational approach based on first- and second-order adjoint methods for inverse problems governed by stochastic saddle-point problems. Specifically, we propose a new first-order adjoint method for computing the gradient of the OLS objective and introduce two novel second-order adjoint methods for Hessian evaluation. The versatility of the framework is illustrated through applications to stochastic diffusion, fourth-order plate bending, linear elasticity, and nearly-incompressible linear elasticity. A stochastic Galerkin discretization framework is employed to enable efficient implementation of the proposed adjoint-based derivative computations. Numerical experiments demonstrate the accuracy and efficiency of the computational framework.
In this paper we show how a general framework of iterative regularization can be applied to solving nonlinear constrained optimization problems when system parameters must be recovered from incomplete noise contaminated data with unknown reporting rate, and the model constraining the underlying optimization problem does not have a closed form solution. To explore advantages and limitations of the regularized algorithms, we conduct a numerical study of the 1991-1997 Vibrio cholerae 01 epidemic in Peru. We use weekly spatio-temporal datasets of cholera cases to show that structural idenitfiability of system parameters does not always translate into stable estimation of these parameters from overlapping waves of noisy data. To face this challenge, we explore the possibility of introducing temperature fluctuations and human mobility into the model of cholera spread. We show that despite a large (in comparison) size of the solution space, our new network cholera model with temperature-dependent environmental transmission rate gives rise to very stable estimates of the unknown system parameters, likely benefiting from the combined calibration data set used for parameter estimation. This makes it possible to forecast the dynamics of future cholera cases with tenable certainty.
The purpose of this paper is to introduce the Tykhonov well-posedness (shortly well-posedness) by perturbations of inverse tensor variational inequalities. In addition, we obtain some metric characterizations of α \alpha -well-posedness by perturbations. Under suitable conditions, we show that the well-posedness by perturbations of an inverse tensor variational inequality is equivalent to the existence and uniqueness of its solution. Finally, we prove that the well-posedness by perturbations of an inverse tensor variational inequality is equivalent to the well-posedness by perturbations of a classical tensor variational inequality.
We consider nonlinear elliptic partial differential equations and variational inequalities posed on decomposed domains, where subdomains are coupled through nonlinear Robin interface conditions. Classical Robin–Robin domain decomposition methods provide efficient parallel solvers in the linear setting, but their extension to nonlinear problems introduces significant analytical challenges, particularly because the associated interface operators generally lack known Lipschitz constants. In this chapter, we develop a fully nonlinear Robin–Robin domain decomposition algorithm and establish a rigorous convergence result without requiring explicit bounds on the interface operator. The key idea is an adaptive relaxation strategy, where the relaxation parameter is automatically adjusted using only differences of successive iterates. This guarantees geometric convergence even when the unrelaxed iteration is not contractive. We present the functional setting, derive stability estimates for the nonlinear subdomain mapping, and prove convergence of the adaptive-relaxation iteration by a fixed-point argument. The local variational inequalities are solved using an augmented Lagrangian Uzawa method, making the approach suitable for obstacle-type and Signorini boundary conditions. Numerical examples illustrate the robustness of the algorithm in cases where classical Robin–Robin iterations fail to converge.
