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Inverse problems are more challenging when only partial data are available in general. In this paper, we propose a two-step approach combining the extended sampling method and the ensemble Kalman filter to reconstruct an elastic rigid obstacle using partial data. In the first step, the approximate location of the unknown obstacle is obtained by the extended sampling method. In the second step, the ensemble Kalman filter is employed to reconstruct the shape. The location obtained in the first step guides the construction of the initial particles of the ensemble Kalman filter, which is critical to the performance of the second step. Both steps are based on the same physical model and use the same scattering data. Numerical examples are shown to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper focuses on considering two inverse source problems (ISPs) for a multi-term time-fractional evolution equation with an involution term, interpolating the heat and wave equations. The fractional derivatives are defined in Caputo's sense. The ISPs are proved to be ill-posed in the sense of Hadamard. Recovering a space dependent source term from over-specified data given at some time constitute the first ISP, while in the second ISP determination of a time dependent component of the source term is considered when over-specified condition of integral type is given. The solution of ISPs are constructed by using Fourier's method. The time-dependent components of the solutions are presented in terms of the multinomial Mittag-Leffler function. Under certain conditions, the solutions of ISPs for the multi-term time-fractional evolution equation are shown to be classical solutions. In addition, some particular examples are formulated to illustrate the obtained results for the ISPs. [ABSTRACT FROM AUTHOR]

We propose a regularization method based on the iterative conjugate gradient method for the solution of a Cauchy problem for the wave equation in one dimension. This linear but ill-posed Cauchy problem consists of finding the displacement and flux on a hostile and inaccessible part of the medium boundary from Cauchy data measurements of the same quantities on the remaining friendly and accessible part of the boundary. This inverse boundary value problem is recast as a least-squares minimization problem that is solved by using the conjugate gradient method whose iterations are stopped according to the discrepancy principle for obtaining stable reconstructions. The objective functional associated is proved Fréchet differentiable and a formula for its gradient is derived. The well-posed direct, adjoint and sensitivity problems present in the conjugate gradient method are solved by using a finite-difference method. Two numerical examples to illustrate the accuracy and stability of the proposed numerical procedure are thoroughly presented and discussed. [ABSTRACT FROM AUTHOR]

In this paper, the inverse problem for identifying the initial value of time-fractional diffusion wave equation on spherically symmetric region is considered. The exact solution of this problem is obtained by using the method of separating variables and the property the Mittag–Leffler functions. This problem is ill-posed, i.e. the solution(if exists) does not depend on the measurable data. Three different kinds landweber iterative methods are used to solve this problem. Under the priori and the posteriori regularization parameters choice rules, the error estimates between the exact solution and the regularization solutions are obtained. Several numerical examples are given to prove the effectiveness of these regularization methods. [ABSTRACT FROM AUTHOR]

The total variation minimization is proposed for use in recovering the mass density of a fluid medium from back-scattered acoustic waves by the dynamical version of the boundary control method. This may be of particular interest to underwater acoustic imaging or ultrasound tomography. In particular, an analogue of the regularized mean curvature flow equation is proposed and developed to obtain numerical solution of the system of linear equations with the ill-conditioned Gram matrix. To demonstrate the computational effectiveness of the proposed numerical technique, we conduct a series of numerical experiments with the inverse problem for a 1 + 1 dimensional acoustic wave equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a problem of recovering a space-dependent source for a time fractional diffusion wave equation by the fractional Landweber method. The inverse problem has been transformed into an integral equation by using the final measured data. We use the fractional Landweber regularization method for overcoming the ill-posedness. We discuss an a-priori regularization parameter choice rule and an a-posteriori regularization parameter choice rule, and we also prove the conditional stability and convergence rates for the inverse problem. Numerical experiments for four examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is devoted to solve an inverse time-dependent source problem for a time-fractional diffusion-wave equation. The source function in the time-fractional diffusion-wave equation is assumed as a multiplication of a temporal function and a spatial function. We try to identify the time-dependent source term according to an additional solute concentration distribution measured at a point on the boundary or an inner point in the solution domain. Firstly, we discuss the continuity of the weak solution for the direct problem. Then, we transform the inverse problem into a first kind of Volterra integral equation and show its ill-posedness. We use a generalized Tikhonov regularization to solve the Volterra integral equation. The generalized cross-validation choice rule is applied to find a suitable regularization parameter. Lastly, we test three examples for the inverse time-dependent source problem and show the effectiveness of the proposed regularization method. [ABSTRACT FROM AUTHOR]

This paper deals with the determination of an initial condition in degenerate hyperbolic equation from final observations. With the aim of reducing the execution time, this inverse problem is solved using an approach based on double regularization: a Tikhonov's regularization and regularization in equation by viscose-elasticity. So, we obtain a sequence of weak solutions of degenerate linear viscose-elastic problems. Firstly, we prove the existence and uniqueness of each term of this sequence. Secondly, we prove the convergence of this sequence to the weak solution of the initial problem. Also we present some numerical experiments to show the performance of this approach. [ABSTRACT FROM AUTHOR]

In this paper, inverse source problem for the wave equation is considered in one dimension with overdetermination condition. To solve this problem, a version of homotopy analysis method, called weighted homotopy analysis method, is introduced and applied. To show accuracy and reliability of the mentioned method, three numerical examples are given. [ABSTRACT FROM AUTHOR]

In this work we derive surface integral formulae for the frequency dependent power operator in arbitrary geometries. This operator is combined with layer representations and measured pressure (or normal velocity) near-field holograms to construct a non-negative intensity distribution quantifying the areas of the vibrating structure that produce radiation to the far-field. It is demonstrated that the sound producing regions of the structure are clearly revealed using the derived formulae and that the spatial resolution is limited to a half-wavelength. We validate our methodology using numerically generated data that corresponds to the excitation of an elastic spherical shell by a point force. [ABSTRACT FROM AUTHOR]

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the inverse time-dependent force function in the wave equation on regular and irregular domains. The SMRPI is developed for identifying the force function which satisfies in the wave equation subject to the integral overspecification over a portion of the spatial domain or to the overspecification at a point in the spatial domain. This method is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. Since the problem is known to be ill-posed, Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system. Three numerical examples are tested to show that numerical results are accurate for exact data and stable with noisy data. [ABSTRACT FROM PUBLISHER]

We present the new method Coupled Multiscenario Full Waveform Inversion (CMFWI) for the solution of the inversion problem arising in seismic applications. As in the case of classical Full Waveform Inversion (FWI), the proposed method is based on seismic reflection signals and it tries to recover the subsoil velocity profile by minimizing a misfit functional between recorded and computed data. CMFWI combines data generated by shooting one source at a time, and shares the effect of this signal with the signals associated with the other sources. Moreover, CMFWI is a new approach with respect to FWI employed with the same sources shot together. We numerically show that CMFWI performs better than FWI. In particular, this comparison focuses on different types of boundary conditions, wave sources, initial guesses of the velocity profile and signal-to-noise ratio. [ABSTRACT FROM AUTHOR]

The numerical computation of ill-posed, nonlinear, multidimensional initial value problems presents considerable difficulties. Conventional stepwise marching schemes for such problems, whether explicit or implicit, are necessarily unconditionally unstable and result in explosive noise amplification. Following previous work on backward parabolic equations, this paper develops and analyses a stabilized explicit marching scheme for ill-posed time-reversed viscous wave equations. The method uses easily synthesized linear smoothing operators at each time step to quench the instability. Smoothing operators based on positive real powerspof the negative Laplacian are helpful, andcan be realized efficiently on rectangular domains using FFT algorithms. The stabilized explicit scheme is unconditionally stable, marching forward or backward in time, and can be applied to nonlinear viscous wave equations by simply lagging the nonlinearity at the previous time step. However, the smoothing operation at each step leads to a distortion away from the true solution. This is thestabilization penalty. It is shown that in many problems of interest, that distortion is often small enough to allow for useful results. In the canonical case of linear autonomous selfadjoint time-reversed viscous wave equations, with solutions satisfying prescribed bounds, it is proved that the stabilized explicit scheme leads to an error estimate differing from thebest-possibleestimate, only by the stabilization penalty. The procedure is a valuable complement to the well-known quasi-reversibility method. As illustrative examples, the paper uses fictitiously blurredpixel images, obtained using sharp images as initial values in linear or nonlinear well-posed,forwardviscous wave equations. Such images are associated with highly irregular underlying data intensity surfaces that can severely challenge reconstruction procedures. Deblurring these images proceeds by applying the stabilized explicit scheme on the corresponding ill-posed,time-reversedequation. Instructive computational experiments demonstrate the capabilities of the method on 2D rectangular regions. [ABSTRACT FROM PUBLISHER]

A non-linear wave equation with an unknown time-convolution kernel is considered. The missing kernel is recovered from an additional (space) integral measurement. The global in time existence, uniqueness as well as the regularity of a solution is addressed. A new numerical algorithm based on Rothe’s method is designed and error estimates are derived. [ABSTRACT FROM PUBLISHER]

We investigate the uniqueness of a solution for an inverse problem of determining a spacewise-dependent source in some non-linear parabolic and hyperbolic problems. The material coefficients appearing in the governing equations may depend on both space and time. The aim is to identify a spacewise-dependent source from the usual initial and boundary conditions and the final-time over-determination. We formulate conditions on coefficients which guarantee the uniqueness of a solution of the inverse problem. [ABSTRACT FROM PUBLISHER]

This article deals with ultrasonic transmission through multilayer systems. The equivalent ultrasonic impedance of such systems is based on the transmission line built as a convenient model to reduce the computational time for numerical evaluation. Also, the one dimensional wave equation is considered as the full model for the propagation of a plane wave inside the material. From numerical simulations, we obtained signals useful to assess the equivalent model. Results clearly depict coherence between the models since there are full agreements in the computed velocity and stress at each interface.The identification of the parameters involved in the model, characteristic acoustic impedance and time of flight, related to the layer densities and propagation velocities, is posed as a non-linear parameter estimation problem. An ultrasonic transmission is set up and the complete waveform is recorded after travelling through the layers. The minimization of a least squares objective function defined to measure the misfit between the real and predicted waveforms, in frequency, was successfully performed and the estimated parameters give useful information for the mechanical characterization of the sample. [ABSTRACT FROM PUBLISHER]

The objective of this article is to develop a computational method, which combines an efficient high-order finite difference method for solving wave equation, auto differentiation tools and globally convergent optimization algorithms to estimate the unknown coefficient in a 1-D wave equation. The forward problem is solved by efficient and higher order finite difference methods while the gradient of the cost functional with respect to the coefficient is calculated by an adjoint code which is generated by an auto differentiation tool, such as Tangent linear and Adjoint Model Compiler. Three numerical examples are conducted to demonstrate the effectiveness, robustness and accuracy of the proposed computational method. [ABSTRACT FROM AUTHOR]

We discuss a total wavefield-based inversion approach for the reconstruction of the material profile of heterogeneous semi-infinite domains, directly in the time domain, based on surficial measurements of the domain's response to prescribed wave illumination. The ability to recover the in-depth profile of moduli/wave velocities typically associated with geotechnical site characterization applications is of particular interest. We address four key issues associated with the wavefield-based inversion: (a) to limit the semi-infinite extent of the heterogeneous physical domain, a perfectly-matched-layer (PML) is introduced at the truncation interface; (b) to account for the introduction of the PML, we use a mixed unsplit-field PML formulation for the coupled PML-regular-domain problem; (c) to tackle the inversion, we adopt a partial-differential-equation (PDE)-constrained optimization framework that formally leads to a classic Karush-Kuhn-Tucker (KKT) system comprising the initial-value state, final-value adjoint and time-independent control problems; and (d) to narrow the feasibility space and alleviate the inherent solution multiplicity, we discuss Tikhonov and Total Variation regularization schemes, endowed with a regularization factor continuation algorithm. We also limit the total observation time to optimally account for the domain's heterogeneity during inversion iterations. We report on the theory and results that lead efficiently to the reconstruction of both sharp and smooth profiles in one dimension. [ABSTRACT FROM AUTHOR]

The identification of the space-dependent coefficient in two-dimensional Helmholtz-type equations from a complete knowledge of the field modelled by these equations and its normal derivative on the boundary and additional internal measurements of the field is investigated. This problem is approached by combining a regularized dual reciprocity boundary element method (DRBEM) with a constrained non-linear minimisation technique. The optimal regularization parameter is chosen according to Morozov's discrepancy principle. The accuracy, convergence and stability of the proposed numerical method are carefully analysed and a sensitivity analysis with respect to the initial guess for the minimisation process is also performed. The numerical results obtained show that the combined regularized DRBEM-constrained non-linear minimisation technique is accurate, convergent, stable and robust. [ABSTRACT FROM AUTHOR]

This article considers the problem of estimating the velocity in a two-dimensional acoustic wave equation. In order to overcome the defects of local convergence of conventional methods, a general homotopy inversion strategy is proposed for the continuous inverse model. On the basis of this strategy, a practical algorithm is constructed by introducing Tikhonov regularization method for the nonlinear operator equation which is obtained from the discretization of the continuous inverse model. Numerical simulations show that the algorithm is a fast and widely convergent inversion method. [ABSTRACT FROM AUTHOR]

This article considers the problem of estimating the velocity in a two-dimensional acoustic wave equation. The wavelet analysis is introduced and a wavelet multiscale method is constructed, based on the idea of hierarchical approximation. The inverse problem is decomposed into a sequence of inverse problems which rely on the scale variables and are solved successively according to the size of scale from the longest to the shortest. And in every inversion, at different scales, the regularized Gauss-Newton method is used, which is stable and fast, until the parameter of the primary inverse problem is found. The results of numerical simulations indicate that the method is a widely convergent optimization method (in some cases it may be global), and exhibits the advantages of conventional methods on computational efficiency and precision. [ABSTRACT FROM AUTHOR]