This paper studies the numerical analysis of a parameter identification problem governed by elliptic equations with power-type nonlinearity. We propose a numerical reconstruction via a suitable least-squares minimization problem based on piecewise linear finite elements. As one of our main novelties, we establish conditional stability estimates at the continuous level, which form the theoretical foundation of the present finite element analysis. Our stability analysis relies on tailored analytical tools, including Hardy-type inequalities, fractional Gagliardo-Nirenberg inequalities, and weighted spaces with singular distance weights. By invoking the achieved conditional stability together with the Carstensen quasi-interpolation operator and associated estimates in negative Sobolev spaces, we derive a priori error estimates for the proposed finite element approximation in terms of the mesh size, the regularization parameter, the noise level, and the nonlinearity exponent. Our results extend the recent stability and error estimates for the linear case by Jin et al. and sharpen their error estimates and convergence order under weaker regularity assumptions.

In this paper, the SQP method applied to a hyperbolic PDE-constrained optimization problem is considered. The model arises from the acoustic full waveform inversion in the time domain. The analysis is mainly challenging due to the involved hyperbolicity and second-order bilinear structure. This notorious character leads to an undesired effect of loss of regularity in the SQP method, calling for a substantial extension of developed parabolic techniques. We propose and analyze a novel strategy for the well-posedness and convergence analysis based on the use of a smooth-in-time initial condition, a tailored self-mapping operator, and a two-step estimation process along with Stampacchia's method for second-order wave equations. Our final theoretical result is the R-superlinear convergence of the SQP method.

We propose and analyze a class of linear regularization methods for inverse problems in Banach spaces based on the profound application of abstract functional calculus (AFC). Our main novelties include error estimates, convergence rates, and converse results for the proposed approach in the deterministic and random cases. These results are established based on certain AFC with scaling-invariant properties and a suitable AFC description of real interpolation spaces, which we design and explore for the first time in the context of inverse problems. The achieved results are applicable to prominent instances, including the Tikhonov, Showalter, and Lavrentiev regularization methods. In the final part of the paper, we report on some numerical tests conforming our theoretical findings.

. This paper explores a fully discrete approximation for a nonlinear hyperbolic PDE-constrained optimization problem (P) with applications in acoustic full waveform inversion. The optimization problem is primarily complicated by the hyperbolic character and the second-order bilinear structure in the governing wave equation. While the control parameter is discretized using the piecewise constant elements, the state discretization is realized through an auxiliary first-order system along with the leapfrog time-stepping method and continuous piecewise linear elements. The resulting fully discrete minimization problem (Ph) is shown to be well-defined. Furthermore, building upon a suitable CFL-condition, we prove stability and uniform convergence of the state discretization. Our final result is the strong convergence result for (Ph) in the following sense: Given a local minimizer nu of (P) satisfying a reasonable growth condition, there exists a sequence of local minimizers of (Ph) converging strongly towards nu.

This paper is devoted to the finite element analysis for the Bean–Kim model governed by the full 3D Maxwell equations. Describing type-II superconductivity at the macroscopic level, this model leads to a challenging coupled system consisting of the Faraday equation and a hyperbolic quasi-variational inequality (QVI) of the second kind with L 1 -type nonlinearity, that arises explicitly from the magnetic field dependency in the critical current. With the involved Maxwell coupling in the 3D H (curl)-setting, the hyperbolic QVI character poses the primary challenge in the numerical investigation. Two mixed finite element methods based on implicit Euler and leapfrog time-stepping are proposed. On the one hand, the implicit Euler method results in a nonstandard system of curl-curl elliptic QVI with a first-order curl-type nonlinearity. Though the well-posedness of this system is guaranteed, its numerical realization is not straightforward and requires the use of a two-stage iteration process of high computational complexity. On the other hand, by approximating the electric and magnetic fields at two different time step levels, the leapfrog method turns out to be more suitable as it naturally eliminates the notorious QVI structure. More importantly, utilizing suited subdifferential and optimization techniques, we are able to prove an efficiently computable explicit formula for its exact solution in terms of the electric field, which makes its numerical computation substantially more favorable than the Euler method. As further advantages, the leapfrog method applies to broad scenarios involving low regular data of bounded variation (BV) in time for both the applied current source and the temperature distribution. Through nonstandard technical arguments tailored to the BV data, our analysis proves the conditional stability and, eventually, the uniform convergence of the proposed leapfrog method. This paper is closed by 3D numerical tests showcasing the reasonable and efficient performance of the proposed numerical solution.

This paper proposes and analyzes an optimal control approach for acoustic full waveform inversion (FWI) to reconstruct the speed wave parameter entering the hyperbolic acoustic PDE model in the coefficient of the second-order time derivative of the acoustic pressure. We develop a novel technique accounting for an auxiliary first-order system. In contrast to the original state equation, the underlying control parameter appears in the auxiliary system not only as the coefficient of the time derivative but also as the initial data under the image of the solution operator for a specific elliptic problem. On this basis, we construct an adjoint state explicitly using the corresponding dual semigroup, leading to necessary optimality conditions with a low adjoint regularity such that no Sobolev smoothing effect occurs in the optimal solution. The final part of the paper is devoted to the second-order analysis of the optimal control approach. We study the two- and three-dimensional case where the reconstruction is only considered in an open set strictly contained in the hold-all domain. For this case, under additional regularity and compatibility assumptions on the data, we are able to prove second-order optimality conditions yielding local optimality in an L-2-neighborhood. A numerical test based on a synthetic configuration with nonsmooth data is provided, illustrating the performance of the control approach.

This paper examines the analysis and optimal control of an H(curl)-quasilinear first kind variational inequality with a bilateral vector curl-constraint, stemming from the ferromagnetic shielding phenomenon. We propose a tailored regularization approach based on the Helmholtz decomposition and a reduction of the first-order constraint to the zeroth-order one in combination with a smoothed Yosida penalization. In this way, a suitable family of approximating quasilinear variational equalities is obtained. The corresponding limiting analysis not only leads to a well-posedness result for the variational inequality but also reveals its dual formulation. Thereafter, as a second novelty, we prove a regularity result for the dual multiplier by means of the Lp-Helmholtz decomposition in a careful combination with elliptic regularity results for Dirichlet and Neumann problems. The last part of this paper is devoted to the analysis of the corresponding optimal control, which is mainly complicated by the involving H(curl)-quasilinearity, the bilateral vector curl-constraint, and the nonsmoothness. On the basis of the proposed regularization, as the final novelty, we derive necessary optimality conditions, including a characterization of the limiting dual multiplier through curl-projection and cut-off type arguments.

This paper analyzes the mathematical modeling of the transient eddy current approximation in the Maxwell obstacle problem. Here, the medium is assumed to be solely open, containing conducting and non-conducting materials with certain properties of anisotropy and non-smoothness. The proposed evolutionary PDE model preserves the Faraday law and excludes the displacement current from the governing Ampère–Maxwell variational inequality (VI). Our study strives to justify this model and delivers two main results: Global well-posedness of the model and its quantitative precision by uniform a priori estimates. The latter result yields an explicit bound for the smallness condition on the ratio between the electric permittivity and the electric conductivity in the region where the displacement current is disregarded. Below this threshold, the eddy current solution provides the desired reasonable approximation and justifies the proposed model.

This paper proposes and examines a finite element method (FEM) for a Maxwell obstacle problem in electric shielding. The model is given by a coupled system comprising the Faraday equation and an evolutionary variational inequality (VI) of Ampère--Maxwell-type. Based on the leapfrog (Yee) time-stepping and the Nédélec edge elements, we set up a fully discrete FEM where the obstacle is discretized in such a way that no additional nonlinear solver is required for the computation of the discrete VI. While the $L^2$-stability is achieved for the discrete solutions and the associated difference quotients, the scheme only guarantees the $L^1$-stability for the discrete magnetic curl field in the obstacle region. The lack of the global $L^2$-stability for the magnetic curl field is justified by the low regularity issue in Maxwell obstacle problems and turns to be the main challenge in the convergence analysis. Our convergence proof consists of two main stages. First, exploiting the $L^1$-stability in the obstacle region, we derive a convergence result towards a weaker system involving smooth feasible test functions. In the second step, we recover the original system by enlarging the feasible test function set through a specific constraint preserving mollification process in the spirit of Ern and Guermond [Comput. Methods Appl. Math., 16 (2016), pp. 51--75]. This paper is closed by three-dimensional numerical results of the proposed FEM confirming the theoretical convergence result and, in particular, the Faraday shielding effect.

This report is concerned with the mathematical and numerical analysis of Bean’s critical state model for type-II superconductivity in combination with the Maxwell equations. We review three different approaches to prove the well-posedness of the resulting nonsmooth problem. At first, we discuss a direct proof and an equivalent representation by means of hyperbolic variational inequalities. Thereafter, we present a well-posedness result for a general class of hyperbolic Maxwell variational inequalities and show the well-posedness of the Bean-Maxwell system the other way around. Although this result allows a flexible choice of the nonlinearity, it is not suitable for the case where the nonlinearity depends explicitly on the time-variable. Therefore, we conclude the analysis by discussing a fully discrete approximation of the underlying variational inequality with temperature effects. Moreover, this method is the foundation for the numerical algorithm since it includes a strong convergence result and a priori error estimates. We close this report by presenting 3D numerical experiments for type-II superconductors.

This paper is devoted to the theoretical and numerical study of an optimal design problem in high-temperature superconductivity (HTS). The shape optimization problem is to find an optimal superconductor shape which minimizes a certain cost functional under a given target on the electric field over a specific domain of interest. For the governing PDE-model, we consider an elliptic curl-curl variational inequality (VI) of the second kind with an L1-type nonlinearity. In particular, the non-smooth VI character and the involved H(curl)-structure make the corresponding shape sensitivity analysis challenging. To tackle the non-smoothness, a penalized dual VI formulation is proposed, leading to the G{\^a}teaux differentiability of the corresponding dual variable mapping. This property allows us to derive the distributed shape derivative of the cost functional through rigorous shape calculus on the basis of the averaged adjoint method. The developed shape derivative turns out to be uniformly stable with respect to the penalization parameter, and strong convergence of the penalized problem is guaranteed. Based on the achieved theoretical findings, we propose 3D numerical solutions, realized using a level set algorithm and a Newton method with the Nedelec edge element discretization. Numerical results indicate a favourable and efficient performance of the proposed approach for a specific HTS application in superconducting shielding.

This paper examines optimal control problems governed by elliptic variational inequalities of the second kind with bounded and unbounded operators. To tackle the bounded case, we employ the polyhedricity of the test set appearing in the dual formulation of the governing variational inequality. Based thereon, we are able to prove the directional differentiability of the associated solution operator, which leads to a strong stationary optimality system. The second part of the paper deals with the unbounded case. Due to the non-smoothness of the variational inequality and the unboundedness of the governing elliptic operator, the directional differentiability of the solution operator becomes difficult to handle. Our strategy is to apply the Yosida approximation to the unbounded operator, while the non-smoothness of the variational inequality is still preserved. Based on the developed strong stationary result for the bounded case, we are able to derive optimality conditions for the unbounded case by passing to the limit in the Yosida approximation. Finally, we apply the developed results to Maxwell-type variational inequalities arising in superconductivity.

Velocity models presenting sharp interfaces are highly relevant in seismic imaging, e.g., for imaging the subsurface of the Earth in the presence of salt bodies. In order to mitigate the oversmoothing of classical regularization strategies such as the Tikhonov regularization, we propose a shape optimization approach for the sharp-interface reconstruction in time-domain acoustic full waveform inversion. Our main result includes the shape differentiability of the cost functional measuring the misfit between observed and predicted data. In particular, it reveals the expression of the distributed shape derivative in tensor form, built on a Lagrangian-type approach and regularity results for the wave equation with discontinuous coefficients. Based on the achieved distributed shape derivative and the level set method, we propose a numerical approach and present several numerical tests supporting our approach.

We propose and analyze variational source conditions (VSC) for the Tikhonov regularization method with Lp-norm penalties for a general ill-posed operator equation in a Banach space. Our analysis is based on the use of the celebrated Littlewood-Paley theory and the concept of (Rademacher) R-boundedness. On the basis of these two analytical tools, we validate the proposed VSC under a conditional stability estimate and a regularity requirement of the true solution in terms of Triebel-Lizorkin-type spaces. In the final part of the paper, the developed theory is applied to an inverse elliptic problem with measure data for the reconstruction of possibly unbounded diffusion coefficients in the Lp-setting. By means of VSC, convergence rates for the associated Tikhonov regularization with Lp-norm penalties are obtained.

This paper develops a Tikhonov regularization theory for nonlinear ill-posed operator equations in Banach spaces. As the main challenge, we consider the so-called oversmoothing state in the sense that the Tikhonov penalization is not able to capture the true solution regularity and leads to the infinite penalty value in the solution. We establish a vast extension of the Hilbertian convergence theory through the use of invertible sectorial operators from the holomorphic functional calculus and the prominent theory of interpolation scales in Banach spaces. Applications of the proposed theory involving $\ell^1$, Bessel potential spaces, and Besov spaces are discussed.

This paper is devoted to the mathematical modeling and analysis of a hyperbolic Maxwell quasi-variational inequality (QVI) for the Bean-Kim superconductivity model with temperature and magnetic field dependence in the critical current. Our analysis relies on local (resp. global) boundedness and local (resp. global) Lipschitz continuity assumptions on the critical current with respect to the temperature (resp. magnetic field). Emerging from the Euler time discretization, we analyze the corresponding H(curl)-elliptic QVI and prove its existence using a fixed-point argument in combination with techniques from variational inequalities and Maxwell’s equations. Based on the existence result for the H(curl)-elliptic QVI, we examine the stability and convergence of the Euler scheme, which serve as our fundament for the global well-posedness of the governing hyperbolic Maxwell QVI.

This article examines a linear-quadratic elliptic optimal control problem in which the cost functional and the state equation involve a highly oscillatory periodic coefficient Aε. The small parameter ε > 0 denotes the periodicity length. We propose a high-order effective control problem with constant coefficients that provides an approximation of the original one with error O(εM), where M ∈ ℕ is as large as one likes. Our analysis relies on a Bloch wave expansion of the optimal solution and is performed in two steps. In the first step, we expand the lowest Bloch eigenvalue in a Taylor series to obtain a high-order effective optimal control problem. In the second step, the original and the effective problem are rewritten in terms of the Bloch and the Fourier transform, respectively. This allows for a direct comparison of the optimal control problems via the corresponding variational inequalities, leading to our main theoretical result on the high-oder approximation.

An inverse problem of reconstructing the magnetic reluctivity in a quasilinear magnetostatic Maxwell system is studied. To overcome the ill-posedness of the inverse problem, we propose and investigate two regularisations posed as constrained minimisation problems. The first uses the total variation (perimeter) regularisation, and the second makes use of the phase field regularisation. Existence of minimisers, sequential stability with respect to data perturbation, and consistency as the regularisation parameters tending to zero are rigorously analysed. Under some regularity assumption, we infer a relation between the regularisation parameters that allows one to recover a solution to the original inverse problem from the phase field regularised problem. The second focus of the paper is set on the first-order analysis of both regularisation approaches. For the phase field approach, two types of optimality systems are derived through a weak directional differentiability result and the domain variation technique of shape calculus. As a final result, we show the convergence of the optimality conditions obtained from shape calculus, leading to a necessary optimality system for the total variation inverse problem.

This paper is concerned with the analysis of an adaptive edge element method for solving elliptic $curl$-$curl$ variational inequalities of second kind. We derive a posteriori error estimators based on a special combination of the Moreau--Yosida regularization and Nédélec's edge elements of first family. With the help of these a posteriori error estimators, an AFEM algorithm is proposed and studied. We are able to establish both the reliability and the efficiency of these estimators, by means of a special linear auxiliary problem involving the discrete Moreau--Yosida-regularized dual formulation, along with a local regular decomposition for $H(curl)$-functions and the bubble functions. Furthermore, we demonstrate the strong convergence of the sequence of the edge element solutions generated by the adaptive algorithm toward the solution of a limiting problem, by first achieving the convergence of the maximal error indicator and the residual corresponding to the sequence of the adaptive edge element solutions, under a reasonable condition on the regularization parameter in terms of the adaptive mesh size. Three-dimensional numerical experiments are presented to verify the robustness and effectiveness of the adaptive algorithm when it is applied to a problem arising from the type-II (high-temperature) superconductivity.