Inverse Source Problem Research Articles

A Symmetry of Boundary Functions Method for Solving the Backward Time-Fractional Diffusion Problems

In the paper, we develop three new methods for estimating unknown initial temperature in a backward time-fractional diffusion problem, which is transformed to a space-dependent inverse source problem for a new variable in the first method. Then, the initial temperature can be recovered by solving a second-order boundary value problem. The boundary functions and a unique zero element constitute a group symmetry. We derive energetic boundary functions in the symmetry group as the bases to retrieve the source term as an unknown function of space and time. In the second method, the solution bases are energetic boundary functions, and then by collocating the governing equation we obtain the expansion coefficients for retrieving the entire solution and initial temperature. For the first two methods, boundary fluxes are over-specified to retrieve the initial condition. In the third method, we give two boundary conditions and a final time temperature to construct the bases in another symmetry group; the governing equation is collocated to a linear system to obtain the whole solution (initial temperature involved). These three methods are assessed and compared by numerical experiments.

Assessing the potential of using a virtual Veselago lens in quantitative microwave imaging

This study explores the potential of implementing the focusing properties of a virtual ideal Veselago lens within a standard free-space microwave imaging scenario. To achieve this, the virtual lens is introduced as an inhomogeneous numerical background for the inverse source problem. This numerical Vesealgo lens is incorporated into the incident and scattered field decomposition, resulting in a new data equation that involves the Veselago lens Green’s function. In addition to the contrast sources within the object-of-interest, the lens introduces virtual contrast sources along the lens boundaries that depend on the total tangential magnetic field. It is shown that a surface integral contribution that takes into account these surface contrast sources must be added to the collected free-space data before one can invert using the well-conditioned Veselago lens inversion operator. A preliminary investigation of the accuracy to which this surface integral contribution must be computed is performed using additive Gaussian noise. Results show that an error of less than one percent is required to achieve imaging performance similar to utilizing an actual Veselago lens. All results are performed within a 2D simulation environment.

New type of the unique continuation property for a fractional diffusion equation and an inverse source problem

In this work, a new type of the unique continuation property for time-fractional diffusion equations is studied. The proof is mainly based on the Laplace transform and the properties of Bessel functions. As an application, the uniqueness of the inverse problem in the simultaneous determination of spatially dependent source terms and fractional order from sparse boundary observation data is established.

Identifying the Heat Source in Radially Symmetry and Axis-Symmetry Problems

This paper solves the inverse source problem of heat conduction in which the source term only varies with time. The application of the discrete regularization method, a kind of effective radial symmetry and axisymmetric heat conduction problem source identification that does not involve the grid integral numerical method, is put forward. Taking the fundamental solution as the fundamental function, the classical Tikhonov regularization method combined with the L-curve criterion is used to select the appropriate regularization parameters, so the problem is transformed into a class of ill-conditioned linear algebraic equations to solve with an optimal solution. Several numerical examples of inverse source problems are given. Simultaneously, a few numerical examples of inverse source problems are given, and the effectiveness and superiority of the method is shown by the results.

Reconstruction of the time-dependent source in thermal grooving by surface diffusion

In hot polycrystalline materials, when a vertical flat grain boundary meets a horizontal surface the grain boundary forms a groove in the surface. Mathematically modelling features of such thermal grooving mechanism is therefore very important in characterizing polycrystalline materials composed of tiny grains intersecting an external free surface. With this aim in mind, we formulate and investigate a novel inverse problem of reconstructing the unknown time-dependent source term entering the fourth-order parabolic equation of thermal grooving by surface diffusion from a given integral observation. We formulate and prove in Theorems 2.3–2.7 that this linear inverse problem is well-posed. However, in practice, the ideal regularity of data under which the inverse source problem is stable is never satisfied due to the inherent non-smoothness of the measurement. Consequently, this leads to the inverse problem with raw data becoming ill-posed. In order to obtain accurate and stable solutions, we develop and compare two numerical methods, namely, a time-discrete method and an optimization method. We obtain error estimates and convergence rates for the time-discrete method. For the optimization method, an objective functional, which is proved to be Fréchet differentiable, is introduced and the conjugate gradient method (CGM), regularized by the discrepancy principle, is developed to compute the minimizer yielding the source term. The results of two numerical tests illustrate the performance of the two methods for both exact and noisy measured data.

An efficient iterative quasi-reversibility method for the inverse source problem of time-fractional diffusion equations

This paper is devoted to recovering the source term for a time-fractional diffusion equation from additional temperature data at fixed time t = 1 . We discuss a uniqueness result of the direct problem and the ill-posedness of the inverse problem, and then apply the iterative quasi-reversibility regularization method to solve the inverse problem. Finally, some one-dimensional and two-dimensional numerical examples are given to verify the effectiveness and feasibility of the proposed method.

On determining and breaking the gauge class in inverse problems for reaction-diffusion equations

Abstract We investigate an inverse boundary value problem of determination of a nonlinear law for reaction-diffusion processes, which are modeled by general form semilinear parabolic equations. We do not assume that any solutions to these equations are known a priori, in which case the problem has a well-known gauge symmetry. We determine, under additional assumptions, the semilinear term up to this symmetry in a time-dependent anisotropic case modeled on Riemannian manifolds, and for partial data measurements on ${\mathbb R}^n$ . Moreover, we present cases where it is possible to exploit the nonlinear interaction to break the gauge symmetry. This leads to full determination results of the nonlinear term. As an application, we show that it is possible to give a full resolution to classes of inverse source problems of determining a source term and nonlinear terms simultaneously. This is in strict contrast to inverse source problems for corresponding linear equations, which always have the gauge symmetry. We also consider a Carleman estimate with boundary terms based on intrinsic properties of parabolic equations.

Inverse source problem for multi-term time-fractional diffusion equation with nonlocal boundary conditions

<abstract><p>In this paper, we consider an inverse source problem with nonlocal boundary conditions for the heat equation involving multi-term time-fractional derivatives. We determine a source term independent of the space variable, and the temperature distribution from the energy measurement. We reduce the solution of the inverse problem to finding solutions to two problems. The well-posedness of each problem is shown using the generalized Fourier method.</p></abstract>

Error estimation to the direct sampling method for the inverse acoustic source problem with multi-frequency data

Error estimation to the direct sampling method for the inverse acoustic source problem with multi-frequency data

A stochastic Galerkin method for the direct and inverse random source problems of the Helmholtz equation

A stochastic Galerkin method for the direct and inverse random source problems of the Helmholtz equation

An inverse source problem for a pseudoparabolic equation with memory

<abstract><p>This paper is devoted to investigating the well-posedness, as well as performing the numerical analysis, of an inverse source problem for linear pseudoparabolic equations with a memory term. The investigated inverse problem involves determining a right-hand side that depends on the spatial variable under the given observation at a final time along with the solution function. Under suitable assumptions on the problem data, the existence, uniqueness and stability of a strong generalized solution of the studied inverse problem are obtained. In addition, the pseudoparabolic problem is discretized using extended cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this problem is effectively solved using the MATLAB subroutine <italic>lsqnonlin</italic>. Both exact and noisy data are inverted. Numerical results for a benchmark test example are presented and discussed. Moreover, the von Neumann stability analysis is also discussed.</p></abstract>

An inverse source problem for convective Brinkman-Forchheimer equations with the final overdetermination

An inverse source problem for convective Brinkman-Forchheimer equations with the final overdetermination

Lipschitz stability for determination of states and inverse source problem for the mean field game equations

Lipschitz stability for determination of states and inverse source problem for the mean field game equations

Stability estimates for the inverse source problem with passive measurements

Stability estimates for the inverse source problem with passive measurements

Stability for the inverse source problem in a two-layered medium separated by rough interface

Stability for the inverse source problem in a two-layered medium separated by rough interface

Jointly determining the point sources and obstacle from Cauchy data

A numerical method is developed for recovering both the source locations and the obstacle from the scattered Cauchy data of the time-harmonic acoustic field. First of all, the incident and scattered components are decomposed from the coupled Cauchy data by the representation of the single-layer potentials and the solution to the resulting linear integral system. As a consequence of this decomposition, the original problem of joint inversion is reformulated into two decoupled subproblems: an inverse source problem and an inverse obstacle scattering problem. Then, two sampling-type schemes are proposed to recover the shape of the obstacle and the source locations, respectively. The sampling methods rely on the specific indicator functions defined on target-oriented probing domains of circular shape. The error estimates of the decoupling procedure are established and the asymptotic behaviors of the indicator functions are analyzed. Extensive numerical experiments are also conducted to verify the performance of the sampling schemes.

Inverse Source Problem for the Subdiffusion Equation on a Metric Star Graph with Integral Overdetermination Condition

Inverse Source Problem for the Subdiffusion Equation on a Metric Star Graph with Integral Overdetermination Condition

Active control of electromagnetic fields in layered media

This article presents a numerical strategy for actively manipulating electromagnetic (EM) fields in layered media. In particular, we develop a scheme to characterize an EM source that will generate some predetermined field patterns in prescribed disjoint exterior regions in layered media. The proposed question of specifying such an EM source is not an inverse source problem (ISP) since the existence of a solution is not guaranteed. Moreover, our problem allows for the possibility of prescribing different EM fields in mutually disjoint exterior regions. This question involves a linear inverse problem that requires solving a severely ill-posed optimization problem (i.e. suffering from possible non-existence or non-uniqueness of a solution). The forward operator is defined by expressing the EM fields as a function of the current at the source using the layered media Green's function (LMGF), accounting for the physical parameters of the layered media. This results to integral equations that are then discretized using the method of moments (MoM), yielding an ill-posed system of linear equations. Unlike in ISPs, stability with respect to data is not an issue here since no data is measured. Rather, stability with respect to input current approximation is important. To get such stable solutions, we applied two regularization methods, namely, the truncated singular value decomposition (TSVD) method and the Tikhonov regularization method with the Morozov Discrepancy Principle. We performed several numerical simulations to support the theoretical framework and analyzes, and to demonstrate the accuracy and feasibility of the proposed numerical algorithms.

Conforming VEM for general second-order elliptic problems with rough data on polygonal meshes and its application to a Poisson inverse source problem

This paper focuses on the analysis of conforming virtual element methods (VEMs) for general second-order linear elliptic problems with rough source terms and applies it to a Poisson inverse source problem with rough measurements. For the forward problem, when the source term belongs to [Formula: see text], the right-hand side for the discrete approximation defined through polynomial projections is not meaningful even for standard conforming VEM. The modified discrete scheme in this paper introduces a novel companion operator in the context of conforming VEM and allows data in [Formula: see text]. This paper has three main contributions. The first contribution is the design of a conforming companion operator [Formula: see text] from the conforming virtual element space to the Sobolev space [Formula: see text], a modified virtual element scheme, and the a priori error estimate for the Poisson problem in the best-approximation form without data oscillations. The second contribution is the extension of the a priori analysis to general second-order elliptic problems with source term in [Formula: see text]. The third contribution is an application of the companion operator in a Poisson inverse source problem when the measurements belong to [Formula: see text]. Tikhonov’s regularization technique regularizes the ill-posed inverse problem, and the conforming VEM approximates the regularized problem given a finite measurement data. The inverse problem is also discretized using the conforming VEM and error estimates are established. Numerical tests on different polygonal meshes for general second-order problems, and for a Poisson inverse source problem with finite measurement data verify the theoretical results.

On the solvability of direct and inverse problems for a generalized diffusion equation

This paper delves into both direct and two inverse source problems associated with a diffusion equation featuring integral convolution over time, while considering non-classical boundary conditions. The inverse source problems are shown to exhibit ill-posed characteristics in accordance with Hadamard’s definition. A bi-orthogonal function system is employed to express series solutions for the inverse source problems. By imposing specific conditions on the provided data, we establish the existence of unique series solutions. Several special cases of the diffusion equation are presented, depending on the nature of the memory kernel. Furthermore, to illustrate the findings regarding inverse source problems, we provide specific examples.