A STRONG POSITIVITY PROPERTY AND A RELATED INVERSE SOURCE PROBLEM FOR MULTI-TERM TIME-FRACTIONAL DIFFUSION EQUATIONS

In this article,we consider the diffusion equation with multi-term time-fractional derivatives.We first derive,by a subordination principle for the solution,that the solution is positive when the initial value is non-negative.As an application,we prove the uniqueness of solution to an inverse problem of determination of the temporally varying source term by integral type information in a subdomain.Finally,several numerical experiments are presented to show the accuracy and efficiency of the algorithm.

Fundamental solution method for inverse source problem of plate equation

The elastic plate vibration model is studied under the external force. The size of the source term by the given mode of the source and some observations from the body of the plate is determined over a time interval, which is referred to be an inverse source problem of a plate equation. The uniqueness theorem for this problem is stated, and the fundamental solution to the plate equation is derived. In the case that the plate is driven by the harmonic load, the fundamental solution method （FSM） and the Tikhonov regularization technique axe used to calculate the source term. Numerical experiments of the Euler-Bernoulli beam and the Kirchhoff-Love plate show that the FSM can work well for practical use, no matter the source term is smooth or piecewise.

EXPONENTIAL TIKHONOV REGULARIZATION METHOD FOR SOLVING AN INVERSE SOURCE PROBLEM OF TIME FRACTIONAL DIFFUSION EQUATION

In this paper,we mainly study an inverse source problem of time fractional diffusion equation in a bounded domain with an over-specified terminal condition at a fixed time.A novel regularization method,which we call the exponential Tikhonov regularization method with a parameter γ,is proposed to solve the inverse source problem,and the corresponding convergence analysis is given under a-priori and a-posteriori regularization parameter choice rules.Whenγis less than or equal to zero,the optimal convergence rate can be achieved and it is independent of the value of γ.However,when γ is greater than zero,the optimal convergence rate depends on the value of γ which is related to the regularity of the unknown source.Finally,numerical experiments are conducted for showing the effectiveness of the proposed exponential regularization method.

An Inverse Source Problem with Sparsity Constraint for the Time-Fractional Diffusion Equation

In this paper,an inverse source problem for the time-fractional diffusion equation is investigated.The observational data is on the final time and the source term is assumed to be temporally independent and with a sparse structure.Here the sparsity is understood with respect to the pixel basis,i.e.,the source has a small support.By an elastic-net regularization method,this inverse source problem is formulated into an optimization problem and a semismooth Newton(SSN)algorithm is developed to solve it.A discretization strategy is applied in the numerical realization.Several one and two dimensional numerical examples illustrate the efficiency of the proposed method.

Two-Dimensional Parabolic Inverse Source Problem with Final Overdetermination in Reproducing Kernel Space

A new method of the reproducing kernel Hilbert space is applied to a twodimensional parabolic inverse source problem with the final overdetermination. The exact and approximate solutions are both obtained in a reproducing kernel space. The approximate solution and its partial derivatives are proved to converge to the exact solution and its partial derivatives, respectively. A technique is proposed to improve some existing methods. Numerical results show that the method is of high precision, and confirm the robustness of our method for reconstructing source parameter.

Determination of pollution point source in parabolic system model

This paper considers an inverse problem for a partial differential equation to identify a pollution point source in a watershed. The mathematical model of the problem is a weakly coupled system of two linear parabolic equations for the concentrations u（x, t） and v（x, t） with an unknown point source F（x, t） = A（ t）δ（x- s） related to the concentration u（x, t）, where s is the point source location and A（t） is the amplitude of the pollution point source. Assuming that source F becomes inactive after time T＊, it is proved that it can be uniquely determined by the indirect measurements { v（0, t）, v（ a, t）, v（ b, t）, v（ l, t）, 0 〈 t ≤ T, T＊ 〈 T}, and, thus, the local Lipschitz stability for this inverse source problem is obtained. Based on the proof of its uniqueness, an inversion scheme is presented to determine the point source. Finally, two numerical examples are given to show the feasibility of the inversion scheme.

Gas emission source term estimation with 1-step nonlinear partial swarm optimization-Tikhonov regularization hybrid method

Source term identification is very important for the contaminant gas emission event. Thus, it is necessary to study the source parameter estimation method with high computation efficiency, high estimation accuracy and reasonable confidence interval. Tikhonov regularization method is a potential good tool to identify the source parameters. However, it is invalid for nonlinear inverse problem like gas emission process. 2-step nonlinear and linear PSO （partial swarm optimization）-Tikhonov regularization method proposed previously have estimated the emission source parameters successfully. But there are still some problems in computation efficiency and confidence interval. Hence, a new 1-step nonlinear method combined Tikhonov regularizafion and PSO algorithm with nonlinear forward dispersion model was proposed. First, the method was tested with simulation and experiment cases. The test results showed that 1-step nonlinear hybrid method is able to estimate multiple source parameters with reasonable confidence interval. Then, the estimation performances of different methods were compared with different cases. The estimation values with 1-step nonlinear method were close to that with 2-step nonlinear and linear PSO-Tikhonov regularization method, 1-step nonlinear method even performs better than other two methods in some cases, especially for source strength and downwind distance estimation. Compared with 2-step nonlinear method, 1-step method has higher computation efficiency. On the other hand, the confidence intervals with the method proposed in this paper seem more reasonable than that with other two methods. Finally, single PSO algorithm was compared with 1-step nonlinear PSO-Tikhonov hybrid regularization method. The results showed that the skill scores of 1-step nonlinear hybrid method to estimate source parameters were close to that of single PSO method and even better in some cases. One more important property of 1-step nonlinear PSO-Tikhonov regularization method is its reasonable confidence interval, which is not obtained by single PSO algorithm. Therefore, 1-step nonlinear hybrid regularization method proposed in this paper is a potential good method to estimate contaminant gas emission source term.

Radiogenic Source Identification for the Helium Production-Diffusion Equation

Knowledge of helium diffusion kinetics is critical for materials in which helium measurements are made,particulary for thermochronology.In most cases the helium ages were younger than expected,an observation attributes to diffusive loss of helium and the ejection of high energy alpha particles.Therefore it is important to accurately calculate the distribution of the source term within a sample.In this paper,the prediction of the helium concentrations as function of a spatially variable source term are considered.Both the forward and inverse solutions are presented.Under the assumption of radially symmetric geometry,an analytical solution is deduced based on the eigenfunction expansion.Two regularization methods,the Tikhonov regularization and the spectral cutoff regularization,are considered to obtain the regularized solution.Error estimates with optimal convergence order are shown between the exact solution and the regularized solution.Numerical examples are presented to illustrate the validity and effectiveness of the proposed methods.

SOURCE TERM IDENTIFICATION WITH DISCONTINUOUS DUAL RECIPROCITY A PPROXIM ATION AND QUASI-NEWTON METHOD FROM BOUNDARY OBSERVATIONS

This paper deals with discontinuous dual reciprocity boundary element method for solving an inverse source problem.The aim of this work is to determine the source term in elliptic equations for nonhomogenous anisotropic media,where some additional boundary measurements are required.An equivalent formulation to the primary inverse problem is established based on the minimization of a functional cost,where a regularization term is employed to eliminate the oscillations of the noisy data.Moreover,an efficient algorithm is presented and tested for some numerical examples.