Related papers: Modified Tikhonov regularization for identifying s…
Fractional Tikhonov regularization methods have been recently proposed to reduce the oversmoothing property of the Tikhonov regularization in standard form, in order to preserve the details of the approximated solution. Their regularization…
This paper derives a new class of adaptive regularization parameter choice strategies that can be effectively and efficiently applied when regularizing large-scale linear inverse problems by combining standard Tikhonov regularization and…
We study the Dirichlet problem in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order, with bounded, complex-valued coefficients. Our main result gives a sharp condition…
The need to blend observational data and mathematical models arises in many applications and leads naturally to inverse problems. Parameters appearing in the model, such as constitutive tensors, initial conditions, boundary conditions, and…
For a Jordan domain with sufficiently smooth boundaries, the solution to the Dirichlet problem for second order skew-symmetric strongly elliptic system with constant coefficients and regular enough boundary data is constructed in the form…
We propose, analyze, and test new iterative solvers for large-scale systems of linear algebraic equations arising from the finite element discretization of reduced optimality systems defining the finite element approximations to the…
We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence type elliptic operators. The construction is applied in two settings. First, we show…
In this paper, we develop a numerical multiscale method to solve elliptic boundary value problems with heterogeneous diffusion coefficients and with singular source terms. When the diffusion coefficient is heterogeneous, this adds to the…
We study weighted Tikhonov regularization for large-scale linear discrete ill-posed problems with random noise. Under a polynomial upper-bound assumption on the generalized eigenvalues of the discrete forward operator, we derive stochastic…
We propose a new practical adaptive refinement strategy for $hp$-finite element approximations of elliptic problems. Following recent theoretical developments in polynomial-degree-robust a posteriori error analysis, we solve two types of…
We establish existence and regularity results for boundary value problems arising from the first variation of the Willmore energy in the graphical setting. Our focus lies on two-dimensional surfaces with fixed clamped boundary conditions,…
We address the initial source identification problem for the heat equation, a notably ill-posed inverse problem characterized by exponential instability. Departing from classical Tikhonov regularization, we propose a novel approach based on…
We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we…
In this paper, the multiple-source ellipsoidal localization problem based on acoustic energy measurements is investigated via set-membership estimation theory. When the probability density function of measurement noise is…
This work introduces a new cubic regularization method for nonconvex unconstrained multiobjective optimization problems. At each iteration of the method, a model associated with the cubic regularization of each component of the objective…
With the rapid growth of data, how to extract effective information from data is one of the most fundamental problems. In this paper, based on Tikhonov regularization, we propose an effective method for reconstructing the function and its…
Despite a variety of available techniques the issue of the proper regularization parameter choice for inverse problems still remains one of the biggest challenges. The main difficulty lies in constructing a rule, allowing to compute the…
In this paper we deal with a Dirichlet problem for an elliptic equation involving the $1$-Laplacian operator and a source term. We prove that, when the growth of the source is subcritical, there exist two bounded nontrivial solutions to our…
Source identification problems have multiple applications in engineering such as the identification of fissures in materials, determination of sources in electromagnetic fields or geophysical applications, detection of contaminant sources,…
We consider a sequence of elliptic partial differential equations (PDEs) with different but similar rapidly varying coefficients. Such sequences appear, for example, in splitting schemes for time-dependent problems (with one coefficient per…