Volume 32, Issue 5

We recover the conductivity σ at the boundary of a domain from a combination of Dirichlet and Neumann boundary data and generalized power/current density data at the boundary, from a single quite arbitrary set of data, in AET or CDII. The argument is elementary, algebraic and local. More generally, we consider the variable exponent p ⁢ ( ⋅ ) {p(\,\cdot\,)} -Laplacian as a forward model with the interior density data σ ⁢ | ∇ ⁡ u | q {\sigma|\nabla u|^{q}} , and find out that single measurement specifies the boundary conductivity when p - q ≥ 1 {p-q\geq 1} , and otherwise the measurement specifies two alternatives. We present heuristics for selecting between these alternatives. Both p and q may depend on the spatial variable x , but they are assumed to be a priori known. We illustrate the practical situations with numerical examples with the code available.

A problem of identification of the set of a thermal-conductivity coefficients for the nonlinear heat equation in the case of non-uniqueness is considered. Classes of inverse heat conduction problems (IHCP) with a non-unique solution are defined. Explicit descriptions of sets of thermal-conductivity coefficients for these classes are obtained. For solving the identification problem the functional identification approach is used. Unlike traditional methods, the proposed algorithm does not utilize approximations of the coefficient with a finite system of basis functions. The results of computational experiments are presented. It is shown that the functional identification approach makes it possible to numerically identify the non-uniqueness of the solution of the inverse problem of heat conduction.

We investigate a time-dependent inverse source problem for a parabolic partial differential equation with an integral constraint and subject to Neumann boundary conditions in a domain of R d \mathbb{R}^{d} , d ≥ 1 d\geq 1 . We prove the well-posedness as well as higher regularity of solutions in Hölder spaces. We then develop and implement an algorithm that we use to approximate solutions of the inverse problem by means of a finite element discretization in space. Due to instability in inverse problems, we apply Tikhonov regularization combined with the discrepancy principle for selecting the regularization parameter in order to obtain a stable reconstruction. Our numerical results show that the proposed scheme is an accurate technique for approximating solutions of this inverse problem.

In X-ray CT imaging, there are some cases where the obtained CT images have serious ring artifacts and noise, and these degraded CT images seriously affect the quality of clinical diagnosis. Thus, developing an effective method that can simultaneously suppress ring artifacts and noise is of great importance. Total variation (TV) is a famous prior regularization for image denoising in the image processing field, however, for degraded CT images, it can suppress the noise but fail to reduce the ring artifacts. To address this issue, the L 0 L_{0} smoothing filter is incorporated with TV prior for CT ring artifacts and noise removal problem where the problem is transformed into several optimization sub-problems which are iteratively solved. The experiments demonstrate that the ring artifacts and noise presented in the CT image can be effectively suppressed by the proposed method and meanwhile the detailed features such as edge structure can be well preserved. As the superiority of TV and L 0 L_{0} smoothing filters are fully utilized, the performance of the proposed method is better than the existing methods such as the TV-based method and L 0 L_{0} -based method.

The phrase solotone effect refers to a persistent irregular pattern within a set of eigenvalues. This effect arises when the underlying differential equations contain discontinuities. Models of the Earth contain such discontinuities and research into solotones was originally motivated by vibration problems in geophysics. More recently, solotone effects have been discovered within heat conduction, and within quantum wells. However, at present, only one-dimensional arrangements have been shown to give rise to a solotone effect. In this paper, we compute the eigenvalues of an inhomogeneous vibrating rectangular membrane. The resulting eigenspectra, when taken as a whole, fail to display a solotone effect. Nevertheless, we show that irregular patterns do exist within selected subsets of eigenvalues. Our analysis therefore provides the first example of a solotone effect in a spatial dimension greater than one. Furthermore, this discovery could be used to extend the solotone inverse method already implemented in one dimension.

Radiation parameterization schemes are crucial components of weather and climate models, but they are also known to be computationally intensive. In recent decades, researchers have attempted to emulate these schemes using neural networks, with more attention to convolutional neural networks. However, in this paper, we explore the potential of recurrent neural networks (RNNs) for predicting solar heating rates. Our architecture was trained and tested using long-term hindcast data from the Pechora Sea region, with the conventional RRTMG scheme serving as a shortwave parameterization. Our findings show that the RNN offers rapid learning, fast inference, and excellent data fitting. We also present preliminary results demonstrating the use of RNNs for operational weather forecasting, which achieved a significant speedup in parameterization and reduced the overall forecast time by 40 %.

This paper is concerned with discontinuous inverse problem generated by complex-valued weight Sturm–Liouville differential operator with λ-dependent boundary conditions. We establish some properties of spectral characteristic and prove that the potential on the whole interval can be uniquely determined by the Weyl-type function or two spectra.

The inverse Sturm–Liouville problem with smooth potentials is considered. The high-order estimate of the entire function associated with two Sturm–Liouville problems is established. Applying this estimate expression to inverse Sturm–Liouville problems, we proved that the conclusion in [L. Amour, J. Faupin and T. Raoux, Inverse spectral results for Schrödinger operators on the unit interval with partial information given on the potentials, J. Math. Phys. 50 2009, 3, Article ID 033505] remains true for more general case.

This paper deals with differential pencils possessing a term depending on the unknown function with a fixed argument. We deduce the so called main equation together with its fine structure for the spectral problem. Then, according to the boundary conditions and the position of argument, we describe two cases: degenerate and non-degenerate. For these two cases, the uniqueness of inverse spectral problem is studied and a constructive procedure for reconstructing the potentials along with necessary and sufficient conditions of the inverse problem solvability are obtained.

We consider the composition of operators with non-closed range in Hilbert spaces and how the nature of ill-posedness is affected by their composition. Specifically, we study the Hausdorff-, Cesàro-, integration operator, and their adjoints, as well as some combinations of those. For the composition of the Hausdorff- and the Cesàro-operator, we give estimates of the decay of the corresponding singular values. As a curiosity, this provides also an example of two practically relevant non-compact operators, for which their composition is compact. Furthermore, we characterize those operators for which a composition with a non-compact operator gives a compact one.

We consider a nonlinear coefficient inverse problem of reconstructing the density and the memory matrix of a viscoelastic medium by probing the medium with a family of wave fields excited by moment tensor point sources. A spatially non-overdetermined formulation is investigated, in which the manifolds of point sources and detectors do not coincide and have a total dimension equal to three. The requirements for these manifolds are established to ensure the unique solvability of the studied inverse problem. The results are achieved by reducing the problem to a chain of connected systems of linear integral equations of the M. M. Lavrentiev type.

A controlled system of differential equations under the action of an unknown disturbance is considered. The problem discussed in the paper consists in constructing algorithms for forming a control that provides the realization of a prescribed motion for any admissible disturbance. Namely these algorithms should provide the closeness in the metric of the space of differentiable functions of a phase trajectory of a given controlled system and some etalon trajectory of an analogous system functioning when any outer actions are absent. As the space of admissible disturbances, we take the space of measurable square integrable (with respect to the Euclidean norm) functions. The cases of inaccurate measurements of phase trajectories of both systems at all times and at discrete times are under study. Two computer oriented algorithms for solving the problem are designed. The algorithms are based on the (well-known in the theory of guaranteed control) method of extremal shift. In the process, its local (at each time of control correction) regularization is performed by the method of smoothing functional (the Tikhonov method). In addition, estimates for algorithm’s convergence rate are presented.

Overdetermined boundary value problems and the minimal operators generated by them are extremely important in the description of regular boundary value problems for differential equations, and are also widely used in the study of local properties of solutions. In addition, for inverse problems of mathematical physics arising from applications, when determining unknown data, it is necessary to study problems with overdetermined boundary conditions, which is reflected in the study of problems, including for hyperbolic equations and systems, arising in physics, geophysics, seismic tomography, geoelectrics, electrodynamics, medicine, ecology, economics and many other practical areas. Thus, the study of overdetermined boundary value problems is of both theoretical and applied interest. In this paper, a criterion for the regular solvability of the overdetermined Cauchy problem for the Gellerstedt equation and the minimal differential operator generated by it in a hyperbolic domain is established, as which both the case of a characteristic triangle and the case of a more general domain with fairly general assumptions about the boundary of the domain are considered. Due to overdetermined boundary conditions, the problem under consideration will be ill-posed in the general case, therefore, for its regular solvability, additional conditions must be imposed on the initial data. In other words, we have considered the inverse problem: to determine what requirements the initial data of the problem, in particular the right part of the Gellerstedt equation, should meet, in question, so that the overdetermined Cauchy problem is regularly solvable. The proof is based on the Gellerstedt potential, the properties of solutions of the Goursat problem in the characteristic triangle, and the properties of special functions.

We present an approach to describe the group behavior of the high-frequency traders in the stock market based on Mean Field Games and optimal control theory. The problem is formalized as a system of coupled PDEs: Kolmogorov–Fokker–Planck, evolving forward in time with initial condition, and Hamilton–Jacobi–Bellman, evolving backwards in time with terminal condition. Under certain assumptions, the system of PDEs can be reduced to Riccati-type ODEs. We solve the inverse problem to describe the behavior of the high-frequency traders during the Chinese stock market crisis in 2015.