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We treat an inverse electrical conductivity problem which deals with the reconstruction of nonlinear electrical conductivity starting from boundary measurements in steady currents operations. In this framework, a key role is played by the Monotonicity Principle, which establishes a monotonic relation connecting the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). Monotonicity Principles are the foundation for a class of non-iterative and real-time imaging methods and algorithms. In this article, we prove that the monotonicity principle for the Dirichlet Energy in nonlinear problems holds under mild assumptions. Then, we show that apart from linear and p-Laplacian cases, it is impossible to transfer this monotonicity result from the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce a new boundary operator, identified as an average DtN operator.

We demonstrate the asymptotic analysis of a semi-linear optimal control problem posed on a smooth oscillating boundary domain in the present paper. We have considered a more general oscillating domain than the usual “pillar-type” domains. Consideration of such general domains will be useful in more realistic applications like circular domain with rugose boundary. We study the asymptotic behavior of the problem under consideration using a new generalized periodic unfolding operator. Further, we are studying the homogenization of a non-linear optimal control problem and such non-linear problems are limited in the literature despite the fact that they have enormous real-life applications. Among several other technical difficulties, the absence of a sufficient criteria for the optimal control is one of the most attention-grabbing issues in the current setting. We also obtain corrector results in this paper.

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In this paper, a noniterative reconstruction method for solving the inverse potential problem is proposed. The forward problem is governed by a modified Helmholtz equation. The inverse problem consists in the reconstruction of a set of anomalies embedded into a geometrical domain from partial or total boundary measurements of the associated potential. Since the inverse problem is written in the form of an ill‐posed boundary value problem, the idea is to rewrite it as a topology optimization problem. In particular, a shape functional measuring the misfit between the solution obtained from the model and the data taken from the boundary measurements is minimized with respect to a set of ball‐shaped anomalies by using the concept of topological derivatives. It means that the shape functional is expanded asymptotically and then truncated up to the desired order term. The resulting truncated expansion is trivially minimized with respect to the parameters under consideration that leads to a noniterative second order reconstruction algorithm. As a result, the reconstruction process becomes very robust with respect to the noisy data and independent of any initial guess. Finally, some numerical experiments are presented showing the capability of the proposed method in reconstructing multiple anomalies of different sizes and shapes by taking into account complete or partial boundary measurements.

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A multiparticle Brownian dynamics simulation algorithm with a Soddemann–Dünweg–Kremer potential that accounts for pairwise excluded volume interactions between both backbone monomers and associating groups (stickers) on a chain is used to describe the static behavior of associative polymer solutions, across a range of concentrations into the semidilute unentangled regime. Predictions for the fractions of stickers bound by intrachain and interchain associations, as a function of system parameters such as the number of stickers on a chain, the number of backbone monomers between stickers, the solvent quality, and monomer concentration, are obtained. A systematic comparison between simulation results and scaling relations predicted by the mean-field theory of Dobrynin [Macromolecules 37, 3881–3893 (2004)] is carried out. Different regimes of scaling behavior are identified by the theory depending on the monomer concentration, the density of stickers on a chain, and whether the solvent quality for the backbone monomers corresponds to θ or good solvent conditions. Simulation results validate the predictions of the mean-field theory across a wide range of parameter values in all the scaling regimes. The value of the des Cloizeaux exponent, θ2=1/3, proposed by Dobrynin for sticky polymer solutions, is shown to lead to a collapse of simulation data for all the scaling relations considered here. Three different signatures for the characterization of gelation are identified, with each leading to a different value of the concentration at the solgel transition. The Flory–Stockmayer expression relating the degree of interchain conversion at the solgel transition to the number of stickers on a chain, modified by Dobrynin to account for the presence of intrachain associations, is found to be validated by simulations for all three gelation signatures. Simulation results confirm the prediction of scaling theory for the gelation line that separates sol and gel phases, when the modified Flory–Stockmayer expression is used. Phase separation is found to occur with increasing concentration for systems in which the backbone monomers are under θ-solvent conditions and is shown to coincide with a breakdown in the predictions of scaling theory.