Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation

AbstractIn this paper we investigate a problem of the identification of an unknown source from one supplementary temperature measurement at a given instant of time for the transient heat equation. Under an a priori condition we answer the question concerning the best possible accuracy for the problem. The Fourier regularization method is utilized for solving the problem, and its convergent rate is analyzed. Numerical results are presented to illustrate the accuracy and efficiency of the method

An inverse stability result for non compactly supported potentials by one arbitrary lateral Neumann observation

In this paper we investigate the inverse problem of determining the time independent scalar potential of the dynamic Schr\"odinger equation in an infinite cylindrical domain, from partial measurement of the solution on the boundary. Namely, if the potential is known in a neighborhood of the boundary of the spatial domain, we prove that it can be logarithmic stably determined in the whole waveguide from a single observation of the solution on any arbitrary strip-shaped subset of the boundary

Stochastic cosmic ray sources and the TeV break in the all-electron spectrum

Despite significant progress over more than 100 years, no accelerator has been unambiguously identified as the source of the locally measured flux of cosmic rays. High-energy electrons and positrons are of particular importance in the search for nearby sources as radiative energy losses constrain their propagation to distances of about 1 kpc around 1 TeV. At the highest energies, the spectrum is therefore dominated and shaped by only a few sources whose properties can be inferred from the fine structure of the spectrum at energies currently accessed by experiments like AMS-02, CALET, DAMPE, Fermi-LAT, H.E.S.S. and ISS-CREAM. We present a stochastic model of the Galactic all-electron flux and evaluate its compatibility with the measurement recently presented by the H.E.S.S. collaboration. To this end, we have MC generated a large sample of the all-electron flux from an ensemble of random distributions of sources. We confirm the non-Gaussian nature of the probability density of fluxes at individual energies previously reported in analytical computations. For the first time, we also consider the correlations between the fluxes at different energies, treating the binned spectrum as a random vector and parametrising its joint distribution with the help of a pair-copula construction. We show that the spectral break observed in the all-electron spectrum by H.E.S.S. and DAMPE is statistically compatible with a distribution of astrophysical sources like supernova remnants or pulsars, but requires a rate smaller than the canonical supernova rate. This important result provides an astrophysical interpretation of the spectrum at TeV energies and allows differentiating astrophysical source models from exotic explanations, like dark matter annihilation. We also critically assess the reliability of using catalogues of known sources to model the electron-positron flux.Comment: 30 pages, 12 figures; extended discussion; accepted for publication in JCA

Conditional Stability for an Inverse Problem of Determining a Space-Dependent Source Coefficient in the Advection-Dispersion Equation with Robin’s Boundary Condition

This paper deals with an inverse problem of determining the space-dependent source coefficient in one-dimensional advection-dispersion equation with Robin’s boundary condition. Data compatibility for the inverse problem is analyzed by which an admissible set for the unknown is set forth. Furthermore, with the help of an integral identity, a conditional Lipschitz stability is established by suitably controlling the solution of an adjoint problem

Stability estimate in an inverse problem for non-autonomous Schr\"odinger equations

We consider the inverse problem of determining the time dependent magnetic field of the Schr\"odinger equation in a bounded open subset of $R^n$, with $n \geq 1$, from a finite number of Neumann data, when the boundary measurement is taken on an appropriate open subset of the boundary. We prove the Lispchitz stability of the magnetic potential in the Coulomb gauge class by $n$ times changing initial value suitably

Identification of a multi-dimensional space-dependent heat source from boundary data

We investigate the linear but ill-posed inverse problem of determining a multi-dimensional space-dependent heat source in the parabolic heat equation from Cauchy boundary data. This model is important in practical applications where the distribution of internal sources is to be monitored and controlled with care and accuracy from non-invasive and non-intrusive boundary measurements only. The mathematical formulation ensures that a solution of the inverse problem is unique but the existence and stability are still issues to be dealt with. Even if a solution exists it is not stable with respect to small noise in the measured boundary data hence the inverse problem is still ill-posed. The Landweber method is developed in order to restore stability through iterative regularization. Furthermore, the conjugate gradient method is also developed in order to speed up the convergence. An alternating direction explicit finite-difference method is employed for discretising the well-posed problems resulting from these iterative procedures. Numerical results in two-dimensions are illustrated and discussed

Inverse problems of damped wave equations with Robin boundary conditions: an application to blood perfusion

Knowledge of the properties of biological tissues is essential in monitoring any abnormalities that may be forming and have a major impact on organs malfunctioning. Therefore, these disorders must be detected and treated early to save lives and improve the general health. Within the framework of thermal therapies, e.g. hyperthermia or cryoablation, the knowledge of the tissue temperature and of the blood perfusion rate are of utmost importance. Therefore, motivated by such a significant biomedical application, this paper investigates, for the first time, the uniqueness and stable reconstruction of the space-dependent (heterogeneous) perfusion coefficient in the thermal-wave hyperbolic model of bio-heat transfer from Cauchy boundary data using the powerful technique of Carleman estimates. Additional novelties consist in the consideration of Robin boundary conditions, as well as developing a mathematical analysis that leads to stronger stability estimates valid over a shorter time interval than usually reported in the literature of coefficient identification problems for hyperbolic partial differential equations. Numerically, the inverse coefficient problem is recast as a nonlinear least-squares minimization that is solved using the conjugate gradient method (CGM). Both exact and noisy data are inverted. To achieve stability, the CGM is stopped according to the discrepancy principle. Numerical results for a physical example are presented and discussed, showing the convergence, accuracy and stability of the inversion procedure

Solving a nonlinear inverse problem of identifying an unknown source term in a reaction-diffusion equation by adomian decomposition method

We consider the inverse problem of finding the nonlinear source for nonlinear Reaction-Diffusion equation whenever the initial and boundary condition are given. We investigate the numerical solution of this problem by using Adomian Decomposition Method (ADM). The approach of the proposed method is to approximate unknown coefficients by a nonlinear function whose coefficients are determined from the solution of minimization problem based on the overspecified data. Further, the Tikhonov regularization method is applied to deal with noisy input data and obtain a stable approximate solution. This method is tested for two examples. The results obtained show that the method is efficient and accurate. This study showed also, the speed of the convergent of ADM.Publisher's Versio

Wavelet-Galerkin Method for Identifying an Unknown Source Term in a Heat Equation

We consider the problem of identification of the unknown source in a heat equation. The problem is ill posed in the sense that the solution (if it exists) does not depend continuously on the data. Meyer wavelets have the property that their Fourier transform has compact support. Therefore, by expanding the data and the solution in the basis of the Meyer wavelets, high-frequency components can be filtered away. Under the additional assumptions concerning the smoothness of the solution, we discuss the stability and convergence of a wavelet-Galerkin method for the source identification problem. Numerical examples are presented to verify the efficiency and accuracy of the method