The multiscale structure of materials has a strong effect on their effective properties. Many good multiscale mathematical models that can incorporate fine-scale effects of the choice of microstructure and accurately represent the physical phenomena at observable spatial scales are available in the literature. Sophisticated experimental devices can explore nowadays a large number of spatial scales, but important scales are technically still not yet available, i.e. the desired localized information cannot be distinguished via direct measurements from the neighboring local spatial fluctuations.
The challenge is to develop mathematical techniques able to use efficiently multiscale models together with experimental information collected from observable scales to extract essential information from spatial scales that cannot be reached experimentally.
The special issue focuses mainly on multiscale aspects of numerical and mathematical analysis of inverse problems. It includes eight papers dealing either with inverse questions, analysis of multiscale deterministic and/or stochastic scenarios or with a suitable combination of these. Special attention is devoted to the well-posedness and approximability of single-scale and multiscale inverse problems.
An efficient chromatography theory requires that competitive adsorption isotherms must be designed and then used to explore situations where experimental trial-and-error approaches are too complex and expensive. In the study, A Kohn-Vogelius formulation for the model-free adsorption isotherm estimation problem in chromatography, G. Lin, Y. Zhang, X. Cheng, M. Gulliksson, P. Forssén and T. Fornstedt develop a Kohn-Vogelius formulation for the reconstruction of the adsorption isotherm. The well-posedness and convergence of the proposed inverse method are studied. Using a problem-adapted adjoint, the authors derive a convergence rate under weak realistic conditions. Regularization parameter selection methods are also discussed. Based on the adjoint technique, a numerical algorithm for solving the proposed optimization problem is developed. Numerical tests for both synthetic and real-world problems are given to show the efficiency and feasibility of the proposed regularization method.
In the work A note on the derivation of filter regularization operators for nonlinear evolution equations, V. D. Nguyen, H. T. Nguyen, A. K. Vo and V. A. Vo study a general theory for filter regularized operators in Hilbert spaces to handle nonlinear evolution equations (including the Cauchy problem for nonlinear elliptic equations and for the backward-in-time nonlinear parabolic equations). The approximations are justified in terms of error estimates.
The paper The maximum surplus before ruin for two classes of perturbed risk model by W. Jiang and C. Ma brings the reader’s attention on the distribution of the maximum surplus before ruin in a perturbed risk model with two independent classes of risks, in which both of the two inter–claim times have phase-type distributions. The authors obtain the integro-differential equations for the distribution of the maximum surplus before ruin. Explicit expressions are derived if the two classes claim amount distributions belong to the rational family.
When modeling surface diffusion and reactions in heterogeneous catalysts, or in processes involved in surface crystal formation, the balance evolution equations often need to be posed on periodically-structured Riemannian manifiolds. In Applications of periodic unfolding on manifolds, S. Dobberschütz considers the homogenization of an elliptic model problem posed on such manifold. By passing to the homgenization limit, he obtains a generalization of the expected structure of the upscaled elliptic equation and corresponding cell problems. By constructing an equivalence relation of atlases, the invariance of the limit problem with respect to this equivalence relation can be shown. Particularly, this implies that the homogenization limit is independent of change of coordinates or scalings of the reference cell.
The setting proposed by the paper Regularization of initial inverse problem for strongly damped wave equation by H. T. Nguyen, V. A. Vo and H. C. Nguyen is suitable to handle the final value problem for damped elastic-like operators. To obtain a stable numerical approximation, the authors suggest a new well-behaving nonlinear regularized problem. Choosing a suitable Gevrey class of functions, they show that the regularized solutions converge uniformly (in time) to the solution to the original hyperbolic problem.
K. Kumazaki together with T. Aiki, N. Sato and Y. Murase propose a Multiscale model for moisture transport with adsorption phenomenon in concrete materials. To understand the way moisture moves through cementitious materials, one has to describe mathematically the relationship between the relative humidity and the degree of saturation of the material. In this setting, the model consists of a diffusion equation for the relative humidity posed in a macroscopic domain suitably coupled with free boundary problems describing the degree of saturation in infinitely many micro domains. The aim is to establish the local solvability of the multiscale model.
C. Schillings and A. M. Stuart present in Convergence analysis of the ensemble Kalman filter for inverse problems: the noisy case an analysis of ensemble Kalman inversion, based on the continuous time limit of the algorithm. They establish well-posedness and convergence results for a fixed ensemble size. The focus is on linear inverse problems where a complete theoretical analysis is possible.
The dynamics of a gas–liquid mixture filling a porous media with distributed microstructures is the topic of the paper Well-posedness and inverse Robin estimate for a multiscale elliptic/parabolic problem by M. Lind, A. Muntean and O. Richardson. Besides ensuring the solvability of the direct problem, the authors prove a local stability estimate for the inverse micro–macro Robin problem, potentially useful in identifying quantitatively a micro–macro interfacial Robin transfer coefficient given microscopic measurements on accessible fixed interfaces. To tackle the solvability issue, the authors rely on two-scale energy estimates and two-scale regularity/compactness cast in the Schauder’s fixed point theorem. Regularity and scaling arguments are then used to ensuring the suitable Fréchet differentiability of the solution and the structure of the inverse stability estimate.
Department of Mathematics and Computer Science, Karlstad University, Sweden
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