Volume 29, Issue 2

We consider a Cahn–Hilliard equation in a bounded domain Ω in ℝn{\mathbb{R}^{n}} over a time interval (0,T){(0,T)} and discuss the backward problem in time of determining intermediate data u⁢(x,θ){u(x,\theta)}, θ∈(0,T){\theta\in(0,T)}, x∈Ω{x\in\Omega} from the measurement of the final data u⁢(x,T){u(x,T)}, x∈Ω{x\in\Omega}. Under suitable a priori boundness assumptions on the solutions u⁢(x,t){u(x,t)}, we prove a conditional stability estimate for the semilinear Cahn–Hilliard equation ∥u⁢(⋅,θ)∥L2⁢(Ω)≤C⁢∥u⁢(⋅,T)∥H2⁢(Ω)κ0,\lVert u(\,\cdot\,,\theta)\rVert_{L^{2}(\Omega)}\leq C\lVert u(\,\cdot\,,T)% \rVert_{H^{2}(\Omega)}^{\kappa_{0}}, and a conditional stability estimate for the linear Cahn–Hilliard equation ∥u⁢(⋅,θ)∥Hβ⁢(Ω)≤C⁢∥u⁢(⋅,T)∥H2⁢(Ω)κ1,\lVert u(\,\cdot\,,\theta)\rVert_{H^{\beta}(\Omega)}\leq C\lVert u(\,\cdot\,,T% )\rVert_{H^{2}(\Omega)}^{\kappa_{1}}, where θ∈(0,T){\theta\in(0,T)}, β∈(0,4){\beta\in(0,4)} and κ0,κ1∈(0,1){\kappa_{0},\kappa_{1}\in(0,1)}. The proof is based on a Carleman estimate with the weight function e2⁢s⁢eλ⁢t{\mathrm{e}^{2s\mathrm{e}^{\lambda t}}} with large parameters s,λ∈ℝ+{s,\lambda\in\mathbb{R}^{+}}.

The criteria of the well-posedness is obtained for an inverse problem to a class of fractional order in the sense of Caputo degenerate evolution equations with a relatively bounded pair of operators and with the generalized Showalter–Sidorov initial conditions. It is formulated in terms of the relative spectrum of the pair and of the characteristic function of the problem. Sufficient conditions of the unique solvability are obtained for a similar problem with the Cauchy initial condition. For these purposes the unique solvability of the same inverse problem was studied for the equation with a bounded operator near an unknown function, which is solved with respect to the fractional derivative. General results are applied to the inverse problem research for the time fractional system of equations describing the dynamics of a viscoelastic fluid in the weakly degenerate and the strongly degenerate cases.

We study the solvability and stability for the problem of identifying parameter in a class of fractional differential variational inequalities. Our approach is based on a regularity analysis for fractional diffusion equations and fixed point techniques.

Recently, total variation (TV) regularization has become a standard technique for image recovery. The mean squared error (MSE) of the reconstruction can be reliably estimated by Stein’s unbiased risk estimate (SURE). In this work, we develop two recursive evaluations of SURE, based on Chambolle’s projection method (CPM) for TV denoising and alternating direction method of multipliers (ADMM) for TV deconvolution, respectively. In particular, from the proximal point perspective, we provide the convergence analysis for both iterative schemes and the corresponding Jacobian recursions, in terms of the solution distance, from which follows the convergence of noise evolution of Monte-Carlo simulation in practical computations. The theoretical analysis is supported by numerical examples.

Variable-order space-fractional diffusion equations provide very competitive modeling capabilities of challenging phenomena, including anomalously superdiffusive transport of solutes in heterogeneous porous media, long-range spatial interactions and other applications, as well as eliminating the nonphysical boundary layers of the solutions to their constant-order analogues. In this paper, we prove the uniqueness of determining the variable fractional order of the homogeneous Dirichlet boundary-value problem of the one-sided linear variable-order space-fractional diffusion equation with some observed values of the unknown solutions near the boundary of the spatial domain. We base on the analysis to develop a spectral-Galerkin Levenberg–Marquardt method and a finite difference Levenberg–Marquardt method to numerically invert the variable order. We carry out numerical experiments to investigate the numerical performance of these methods.

In the present paper, we devote our effort to a nonlinear inverse problem for recovering a time-dependent potential term in a multi-term time fractional diffusion equation from an additional measurement in the form of an integral over the space domain. First we study the existence, uniqueness, regularity and stability of the solution for the direct problem by using the fixed point theorem. And we obtain the uniqueness of the inverse time-dependent potential term problem. Numerically, we use the Levenberg–Marquardt method to find the approximate potential function. Four different examples are presented to show the feasibility and efficiency of the proposed method.

We are concerned with the inverse scattering problems associated with incomplete measurement data. It is a challenging topic of increasing importance that arise in many practical applications. Based on a prototypical working model, we propose a machine learning based inverse scattering scheme, which integrates a CNN (convolution neural network) for the data retrieval. The proposed method can effectively cope with the reconstruction under limited-aperture and/or phaseless far-field data. Numerical experiments verify the promising features of our new scheme.

In a Banach space, the inverse source problem for a fractional differential equation with Caputo–Dzhrbashyan derivative is considered. The initial and observation conditions are given by elements from D⁢(A){D(A)}, and the operator function on the right side is sufficiently smooth. Two types of the observation operator are considered: integral and at the final point. Under the assumptions that operator A is a generator of positive and compact semigroup the uniqueness, existence and stability of the solution are proved.

In this paper, we analyze the convergence rates for finite-dimensional variational regularization in Banach spaces by taking into account the noisy data and operator approximations. In particular, we determine the convergence rates by incorporating the smoothness concepts of Hölder stability estimates and the variational inequalities. Additionally, we discuss two ill-posed inverse problems to complement the abstract theory presented in our main results.

In this article, we are the first to formulate the direct and inverse problems of resistivity logging on determining the components of the electrical resistivity tensor of rocks from a set of high-frequency induction and lateral logging sounding measurements. Using a finite element approximation, high-order hierarchical basis functions, computationally efficient multilevel methods and a multistart algorithm with the DFO-LS local optimization method, we investigate the capability of reconstructing the horizontal and vertical resistivity components, as well as the tilt of the resistivity tensor principal axes with regard to the study of complex geological objects. A separate consideration is given to a realistic generalized geoelectric model of the unique hydrocarbon source with hard-to-recover reserves, the Bazhenov Formation.

The article proposes a modification of the approach to the analysis of inter-industry balance. Instead of linear models of inter-industry balance, based on the hypothesis of W. Leontief about the constancy of the cost standards of production factors, the article studies nonlinear models. For the case of production functions with constant elasticity of substitution (CES) an algorithm for solving the inverse problem is proposed, which allows to identify the model of nonlinear inter-industry balance based on the data of the symmetric input-output table. Based on the Young transform and Fennel duality, with the help of this model, we develop a technology for analyzing inter-industry relationships. The technology has been tested on the data of economic statistics of Russia.