Richardson's leapfrog scheme is notoriously unconditionally unstable in well-posed, forward, linear dissipative evolution equations. Remarkably, that scheme can be stabilized, marched backward in time, and provide useful reconstructions in an interesting but limited class of ill-posed, time-reversed, 2D incompressible Navier–Stokes initial value problems. Stability is achieved by applying a compensating smoothing operator at each time step to quench the instability. Eventually, this leads to a distortion away from the true solution. This is the stabilization penalty. In many interesting cases, that penalty is sufficiently small to allow for useful results. Effective smoothing operators based on , with real p>2, can be efficiently synthesized using FFT algorithms. Similar stabilizing techniques were successfully applied in several other ill-posed evolution equations. The analysis of numerical stability is restricted to a related linear problem. However, as is found in leapfrog computations of well-posed meteorological and oceanic wave propagation problems, such linear stability is necessary but not sufficient in the presence of nonlinearities. Here, likewise, additional Robert–Asselin–Williams (RAW) time-domain filtering must be used to prevent characteristic leapfrog nonlinear instability unrelated to ill-posedness. Several 2D Navier–Stokes backward reconstruction examples are included, based on the stream function-vorticity formulation, and focusing on pixel images of recognizable objects. Such images, associated with non-smooth underlying intensity data, are used to create severely distorted data at time T>0. Successful backward recovery is shown to be possible at parameter values significantly exceeding expectations.

For many inverse parameter problems for partial differential equations in which the domain contains only well-separated objects, an asymptotic solution to the forward problem involving ‘polarization tensors’ exists. These are functions of the size and material contrast of inclusions, thereby describing the saturation component of the non-linearity. In this paper, we show how such an asymptotic series can be applied to non-linear least-squares reconstruction problems, by deriving an approximate diagonal Hessian matrix for the data misfit term. Often, the Hessian matrix can play a vital role in dealing with the non-linearity, generating good update directions which accelerate the solution towards a global minimum, but the computational cost can make direct calculation infeasible. Since the polarization tensor approximation assumes sufficient separation between inclusions, our approximate Hessian does not account for non-linearity in the form of lack of superposition in the inverse problem. It does, however, account for the non-linear saturation of the change in the data with increasing material contrast. We, therefore, propose to use it as an initial Hessian for quasi-Newton schemes. We present numerical experimentation into the accuracy and reconstruction performance of the approximate Hessian for the case of electrical impedance tomography, providing a proof of principle of the reconstruction scheme.

This paper is concerned with the inverse scattering problem which aims to determine the spatially distributed dielectric constant coefficient of the 2D Helmholtz equation from multifrequency backscatter data associated with a single direction of the incident plane wave. We propose a globally convergent convexification numerical algorithm to solve this nonlinear and ill-posed inverse problem. The key advantage of our method over conventional optimization approaches is that it does not require a good first guess about the solution. First, we eliminate the coefficient from the Helmholtz equation using a change of variables. Next, using a truncated expansion with respect to a special Fourier basis, we approximately reformulate the inverse problem as a system of quasilinear elliptic PDEs, which can be numerically solved by a weighted quasi-reversibility approach. The cost functional for the weighted quasi-reversibility method is constructed as a Tikhonov-like functional that involves a Carleman Weight Function. Our numerical study shows that, using a version of the gradient descent method, one can find the minimizer of this Tikhonov-like functional without any advanced a priori knowledge about it.

Lightning location is a significant issue in the protection of transmission lines, renewable energy sources, and electrical equipment. In this article, a new technique for the determination of lightning striking points is been proposed. This method is depending on measured values of lightning-induced voltage obtained from distribution power lines in the vicinity of the lightning channel. The proposed method considers lightning location as well as lightning current wave shape at the channel base. It was validated using a number of measured return stroke currents, and the outcomes are debated accordingly. All electromagnetic field components were taken into consideration in the proposed method, and lightning current wave shape was determined in contrast with the other, widely used methods. Evaluated lightning current wave shapes, based on local measured data, can be used to examine and improve existing protection schemes in renewable energy generation farms, transmission lines, and other power sites. Moreover, lightning maps and ground flash density can be obtained from the evaluated lighting location data to be used in lightning risk assessment studies.

In this paper, we study an inverse transmission scattering problem of a time-harmonic acoustic wave from the viewpoint of Bayesian statistics. In Bayesian inversion, the solution of the inverse problem is the posterior distribution of the unknown parameters conditioned on the observational data. The shape of the scatterer will be reconstructed from full-aperture and limited-aperture far-field measurement data. We first prove a well-posedness result for the posterior distribution in the sense of the Hellinger metric. Then, we employ the Markov chain Monte Carlo method based on the preconditioned Crank-Nicolson algorithm to extract the posterior distribution information. Numerical results are given to demonstrate the effectiveness of the proposed method.

In this research work, a crack diagnosis method for beam-column structures is proposed considering axial load effects through experimental data. The proposed method is employed for the detection of damage locations including single and multiple damage scenarios considering four cases of simply supported beam-column. The results show that the locations of single and multiple damage scenarios can be well recognized in low axial loads considering random noise effect. However, in the vicinity of the critical load, healthy and damaged static data are uncommon. On the other hand, increasing the axial load, especially when it reaches to the critical load, has a negative effect on the static responses that is precisely considered by the proposed damaged index.

Two efficient methods are presented to detect multiple cracks in 2D elastic bodies, based on the insights from Extended Finite Element. Adetection mesh is assigned to the cracked body and the responses are measured at the nodes. A finite element model with the same mesh is used to represent the uncracked state of the physical body. In the first method which is called Crack Detection based on Residual Error (CDRE), the residual error norm is calculated based on the uncracked body stiffness matrix and the cracked body responses. The contour of the error norm would show the crack pattern; the method is computationally efficient. In the second method that is coined as Crack Detection based on Stiffness Residual (CDSR), the crack locations are found based on the difference between the stiffness matrix of the cracked body and the uncracked body. The stiffness matrix of the cracked body is found by solving a dynamic inverse problem based on a modified Tikhonov regularization. The efficiency and accuracy of the identification method are enhanced by predicting the crack pattern by the CDRE method. Several examples are presented to show the accuracy and robustness of the methods in the presence of high noise levels.

In this paper, we present a method of choice of an adaptative regularization parameter for data corrupted by Poisson noise based on a bilevel approach. The forward operator considered is nonlinear. The existence and unicity of the smoothed lower level problem, the differentiability properties of the constraint, and the adjoint method used to calculate the gradient of the reduced functional are studied in detail. The variance of the KL functional for Poisson noise is also investigated. The method is applied to the spectral CT inverse problem. Better reconstruction results are obtained with the bilevel method of choice than with a scalar regularization parameter.

This paper presents a novel approach for detecting damage in a cable-stayed bridge based on adaptive signal processing and feature extraction. The Empirical Mode Decomposition (EMD) technique is employed in this research to obtain the Intrinsic Mode Functions (IMFs). Several features in time and frequency domains are extracted from IMFs as damage-sensitive features. Next, to reduce false alarms, several spectral parameters are introduced as effective features. A Support Vector Machine (SVM) is employed as a classifier to compare the detection ability of the proposed and conventional features. The second part of this research employs the application of several feature selection techniques that are based on the filter, wrapper, and embedded approaches to obtain the most effective IMFs and then combine them to create a subset that carries the most significant information about the structural damage. To investigate the capability of the proposed approach, vibration acceleration data obtained from a real structure, the Yonghe bridge, is employed. Results indicate that the proposed approach can identify damage with high accuracy.

Inverse problems are of importance in many fields of science and engineering. Electromagnetic inversion deals with estimating information contained in electromagnetic measurements. The inversion scheme needs to be designed properly to compensate for Gibbs oscillations effects in the solution, and hence give better validation for the estimated quantities. In this paper an inversion methodology based on simulated annealing is presented that has the ability to extract information about electrical conductivity and dielectric permittivity of a vertically stratified medium using the scattered electric field. Furthermore, Gibbs phenomenon and its oscillation effect on the inversion solution have been studied, and an efficient approach has been developed to render more accurate estimations. Results of implementing the proposed approach and its resolution compared with the original methodology are presented.