Abstract This work investigates both direct and inverse problems of the variable-exponent sub-diffusion model, which attracts increasing attentions in both practical applications and theoretical aspects. Based on the perturbation method, which transfers the original model to an equivalent but more tractable form, the uniqueness of solutions to the variable-exponent subdiffusion equation with homogeneous Dirichlet boundary condition are established from its partial information, which results in the uniqueness of the inverse space-dependent source problem from local internal observation or partial boundary flux data. Then, based on the variational identity connecting the inversion input data with the unknown source function, we propose a weak norm and prove the conditional stability for the inverse problem in this norm. The iterative thresholding algorithm and Nesterov iteration scheme are employed to numerically reconstruct the smooth and non-smooth sources, respectively. Numerical experiments are performed to investigate their effectiveness.

Abstract In this paper, we study the so-called clamped transmission eigenvalue problem. This is a new transmission eigenvalue problem that is derived from the scattering of an impenetrable clamped obstacle in a thin elastic plate. The scattering problem is modeled by a biharmonic wave operator given by the Kirchhoff–Love infinite plate problem in the frequency domain. These scattering problems have not been studied to the extent of other models. Unlike other transmission eigenvalue problems, the problem studied here is a system of homogeneous PDEs defined in all of R 2 . This provides unique analytical and computational difficulties when studying the clamped transmission eigenvalue problem. We are able to prove that there exist infinitely many real clamped transmission eigenvalues. This is done by studying the equivalent variational formulation. We also investigate the relationship of the clamped transmission eigenvalues to the Dirichlet and Neumann eigenvalues of the negative Laplacian for the bounded scattering obstacle.

Abstract We study the convergence rates of a non-stationary asymptotical regularization method for linear statistical inverse problems from indirect noisy measurements. The convergence rates are usually derived under some range-type source conditions, which are difficult to be verified rigorously in general, especially for the models governed by some partial differential equations. We shall propose a new strategy associated with conditional stability estimates to obtain the error bounds in terms of mean-squared error. To the best of our knowledge, it is the first time to apply conditional stability estimates for convergence rates in statistical inverse problems. As an application, an inverse source problem in a wave equation is considered and its conditional stability estimate is verified rigorously, and therefore the convergence rates are derived immediately. Several numerical experiments for the inverse source problem are also provided to demonstrate the accuracy and efficiency of the non-stationary asymptotical regularization method.

Abstract New techniques for monitoring greenhouse gas (GHG) emissions are urgently needed in multiple spatial scales, to find ways to effectively mitigate emissions and for sustainable policy-making. This paper focuses on a novel facility-scale monitoring technique which uses sequential multi-beam open-path laser spectroscopy measurements and wind field data to infer spatio-temporal distributions of GHG emission sources. The problem of mapping and quantifying the GHG source is considered as a non-stationary, three-dimensional (3D) tomography problem in which the GHG source is reconstructed simultaneously with the temporally evolving 3D concentration of the GHG. The tomographic reconstruction problem is highly rank deficient due to the practically realizable limited-angle setup of sensors. However, the proposed approach for the reconstruction takes advantage of the non-stationarity of the inverse problem—we model the temporal evolution of the 3D concentration distribution and the contribution of the source to it by a convection–diffusion model in which the wind field data is incorporated. The reconstruction problem is formulated in the framework of Bayesian state estimation, using laser spectroscopy observations and the evolution model. Particularly, Kalman smoother estimates are computed. In this paper, the 3D tomographic reconstruction of GHG concentration and source is tested both with numerical simulation studies and with experimental field data. The results demonstrate the feasibility of the approach for mapping and quantifying various types of GHG sources and for quantifying their uncertainties in multiple wind conditions.

Abstract In Learning Theory, the smoothness assumption on the target function (known as source condition) is a key factor in establishing theoretical convergence rates for an estimator. The existing general form of the source condition, as discussed in learning theory literature, has traditionally been restricted to a class of functions that can be expressed as a product of an operator monotone function and a Lipschitz continuous function. In this note, we remove these restrictions on the index function and establish optimal convergence rates for least-square regression over a Hilbert space with general regularization under a general source condition, thereby significantly broadening the scope of existing theoretical results.

Abstract In this work, we construct the Born and inverse Born approximation and series to recover two function-valued coefficients in the Helmholtz equation for inverse scattering problems from the scattering data at two different frequencies. An analysis of the convergence and approximation error of the proposed regularized inverse Born series is provided. The results show that the proposed series converges when the inverse Born approximations of the perturbations are sufficiently small. The preliminary numerical results show the capability of the proposed regularized inverse Born approximation and series for recovering the isotropic inhomogeneous media.

Abstract We consider the Schrödinger equation with a multipoint potential of the Bethe−Peierls−Thomas−Fermi type. We show that such a potential in dimension d = 2 or d = 3 is uniquely determined by its scattering amplitude at a fixed positive energy. Moreover, we show that there is no non-zero potential of this type with zero scattering amplitude at a fixed positive energy and a fixed incident direction. Nevertheless, we also show that a multipoint potential of this type is not uniquely determined by its scattering amplitude at a positive energy E and a fixed incident direction. Our proofs also contribute to the theory of inverse source problem for the Helmholtz equation with multipoint source.

Abstract This paper proposes a computational method to reconstruct both the birth and mortality coefficients in an age-structured population diffusive model. In mathematical oncology, solving this inverse problem is crucial for assessing the effectiveness of anti-cancer treatments and thus, gaining insights into the post-treatment dynamics of tumors. Through some linear and nonlinear transformations, the targeted inverse model is transformed into an auxiliary third-order nonlinear PDE. Subsequently, a coupled age-dependent quasi-linear parabolic PDE system is derived using the Fourier–Klibanov basis. The resulting PDE system is then approximated through the minimization of a cost functional, weighted by a suitable Carleman function. Ultimately, an analysis of the minimization problem is studied through a new Carleman estimate, and some computational results are presented to show how the proposed method works.

Abstract We propose a novel low-rank tensor completion framework that integrates nonlocal self-similarity and data-driven multidimensional transforms. The core of our method lies in extracting similar fully-bandwidth patches and assembling them into third-order sub-tensors. Each sub-tensor is then modeled as low-rank coding tensors under learnable multidimensional transforms, which adaptively captures intrinsic correlations across spatial, spectral, and temporal dimensions. To solve this nonconvex optimization problem, we design a multi-block proximal alternating minimization algorithm with guaranteed global convergence. Extensive experiments on multispectral images, videos, and color images demonstrate that our method achieves significant improvements over state-of-the-art approaches.

Abstract The inverse problem of identifying an unknown space-dependent potential coefficient in the parabolic equation is considered from the additional observation at the terminal time in this work. A novel conditional stability estimate is established for a large terminal time T with suitable assumptions on the input data. Then the potential coefficient and solution of the parabolic equation are parameterized by separate deep neural networks (DNNs), and a new loss function is proposed to reconstruct the unknown potential coefficient. The DNN approximations of the potential coefficient for both continuous and empirical loss functions are analyzed rigorously via utilizing analogous arguments for the conditional stability. Meanwhile, the error estimates are expressed explicitly by the noise level and neural network architectural parameters, which yields a prior rule for determining the number of observations and choosing the size of neural networks. Some numerical experiments are provided to illustrate the robustness of the approach against various noise levels of measured observation and the accuracy of the numerical solutions.