The diagonalized contrast source inversion approach for elastic wave inversion

This study focuses on the inverse scattering of objects embedded in a homogeneous elastic background. The medium is probed by ultrasonic sources, and the scattered fields are observed along a receiver array. The goal is to retrieve the shape, location, and constitutive parameters of the objects through an inversion procedure. The problem is formulated using a vector integral equation. As is well-known, this inverse scattering problem is nonlinear and ill-posed. In a realistic configuration, this nonlinear inverse scattering problem involves a large number of unknowns, hence the application of full nonlinear inversion approaches such as Gauss-Newton or nonlinear gradient methods might not be feasible, even with present-day computer power. Hence, in this study we use the so-called diagonalized contrast source inversion (DCSI) method in which the nonlinear problem is approximately transformed into a number of linear problems. We will show that, by using a three-step procedure, the nonlinear inverse problem can be handled at the cost of solving three constrained linear inverse problems. The robustness and efficiency of this approach is illustrated using a number of synthetic examples.     

Intraframe image decoding based on a nonlinear variational approach

Despite advances in the domain of source coding, little recent work has been devoted to the problem of joint coding and decoding. In particular, to our knowledge, the design of decoders has been little investigated. In this article, we focus on decoding for intraframe images of video data. In our approach, we propose a new method for decoding by nonlinear filtering. Here, we break with the usual approach of perfect reconstruction filters at the decoder, and instead pose the reconstruction model as a minimization problem. The solution can be viewed as an inverse problem with the optimization of the transform/quantization/decoding structure formulated using a variational approach. We introduce sufficient conditions on the design of the decoder involving a priori assumptions on the solution and knowledge of the coder (transformation and quantization). We develop an optimization method for the reconstruction filters at the decoder to account for effects due to quantization noise. Experiments using this nonlinear inverse dynamic filtering demonstrate peak signal-to-noise ratio gains over standard linear inverse filtering as well as appreciable visual improvements. © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 369–380, 1998     

Inverse design of anti-reflection coatings using the nonlinear approximate inverse

We consider a multi-frequency inverse scattering problem arising in the design of anti-reflection coatings. These thin films are deposited onto photovoltaic solar cells to enhance their performance. The objective is to determine the space-dependent refractive index in an inhomogeneous optical layer from the reflection coefficients at the surface. The relevant model yields a boundary value problem for the one-dimensional (1D) Helmholtz equation, which we formulate as an equivalent integral equation. The resulting inverse problem is nonlinear and ill-posed. We consider a series expansion of the field depending on the order of nonlinearity of the model. The first-order solution is obtained by using the Born approximation which is valid for weak scattering. Stronger scatterers are sought by considering a nonlinearity of higher order. The mathematical and numerical framework is provided by the (noniterative) method of the approximate inverse (AI) for nonlinear inverse problems. Numerical results are presented to attest the efficiency and stability of the method.     

Application of evolution strategies for the solution of an inverse problem in near-field optics

We introduce an inversion procedure for the characterization of a nanostructure from near-field intensity data. The method proposed is based on heuristic arguments and makes use of evolution strategies for the solution of the inverse problem as a nonlinear constrained-optimization problem. By means of some examples we illustrate the performance of our inversion method. We also discuss its possibilities and potential applications.     

Image reconstruction in three-dimensional magnetostatic permeability tomography

Magnetostatic permeability tomography is an imaging technique that attempts to reconstruct the permeability distribution of an object using magnetostatic measurement data. The data for image reconstruction are external magnetic field measurements on the surface of the object due to an applied magnetostatic field. Theoretically, the normal and tangential components of the magnetic field in the surface uniquely define the internal isotropic permeability distributions. However, the inverse permeability problem is an ill-posed nonlinear problem. Regularization is needed for a stable solution. In this paper, we present a numerical method to solve the reconstruction problem in three dimensions using a regularized Gauss-Newton scheme. We have solved the forward problem using an edge finite-element method and we have employed an efficient technique to calculate the Jacobian matrix. The permeability of the object is assumed to be linear and isotropic. We present the reconstruction results for the permeability using synthetically generated data with additive noise.     

Inverse problem in optical diffusion tomography. III. Inversion formulas and singular-value decomposition

We continue our study of the inverse scattering problem for diffuse light. In particular, we derive inversion formulas for this problem that are based on the functional singular-value decomposition of the linearized forward-scattering operator in the slab, cylindrical, and spherical geometries. Computer simulations are used to illustrate our results in model systems.     

Inverse geometry heat transfer problem based on a radial basis functions geometry representation

We present a methodology for solving a non‐linear inverse geometry heat transfer problem where the observations are temperature measurements at points inside the object and the unknown is the geometry of the volume where the problem is defined. The representation of the geometry is based on radial basis functions (RBFs) and the non‐linear inverse problem is solved using the iteratively regularized Gauss–Newton method. In our work, we consider not only the problem with no geometry restrictions but also the bound‐constrained problem. The methodology is used for the industrial application of estimating the location of the 1150°C isotherm in a blast furnace hearth, based on measurements of the thermocouples located inside it. We validate the solution of the algorithm against simulated measurements with different levels of noise and study its behaviour on different regularization matrices. Finally, we analyse the error behaviour of the solution. Copyright © 2005 John Wiley & Sons, Ltd.     

A linear-encoding model explains the variability of the target morphology in regeneration

A fundamental assumption of today''s molecular genetics paradigm is that complex morphology emerges from the combined activity of low-level processes involving proteins and nucleic acids. An inherent characteristic of such nonlinear encodings is the difficulty of creating the genetic and epigenetic information that will produce a given self-assembling complex morphology. This ‘inverse problem’ is vital not only for understanding the evolution, development and regeneration of bodyplans, but also for synthetic biology efforts that seek to engineer biological shapes. Importantly, the regenerative mechanisms in deer antlers, planarian worms and fiddler crabs can solve an inverse problem: their target morphology can be altered specifically and stably by injuries in particular locations. Here, we discuss the class of models that use pre-specified morphological goal states and propose the existence of a linear encoding of the target morphology, making the inverse problem easy for these organisms to solve. Indeed, many model organisms such as Drosophila, hydra and Xenopus also develop according to nonlinear encodings producing linear encodings of their final morphologies. We propose the development of testable models of regeneration regulation that combine emergence with a top-down specification of shape by linear encodings of target morphology, driving transformative applications in biomedicine and synthetic bioengineering.     

Nondestructive elastostatic identification of unilateral cracks through BEM and neural networks

An inverse problem in nonlinear elastostatics is considered which concerns the identification of unilateral contact cracks by means of boundary measurements for given static loadings. Highly nonlinear structural behaviour like closed cracks can hardly be identified. In this case, the analysis of more than one loading cases is proposed and tested in this paper. The direct problem is modelled by using a direct multiregion boundary element formulation. The arising linear complementarity problem is solved explicitly by a pivoting (Lemke) technique. In view of the complexity of the inverse problem, a neural network based identification approach is adopted which uses feed-forward multilayer neural networks trained by back-propagation, error-driven supervised training. The applicability of the method is demonstrated by some numerical examples.     

利用线性抽样方法重建Dirichlet边界条件声波散射区域

线性抽样方法是考虑一个第一类线性积分方程中的参数从区域内部趋近散射区域边界时,该方程的解趋近于无穷大。本文在此基础上就Dirichlet边界条件的声波反散射问题,利用积分方程理论严格证明了线性抽样方法对其的可用性,具体数值例子表明该方法是有效的。     

Multi-model method for solving nonlinear transient inverse heat conduction problems

The multi-model inverse method for nonlinear inverse problems is established based on the multi-model control theory. First the model switching variable is chosen and several typical operating balance points in the workspace of the balance variable are selected. Then for each operating balance point the linear local model is established, and the local controller is designed for each linear local model. Finally, according to the instantaneous matching degree between the actual model and the local models, the inversion results of each local controller are weighted and synthesized to obtain the final inversion result. Numerical tests are implemented to solve the one-dimensional nonlinear inverse heat conduction problem by the multi-model inverse method associated with the dynamic matrix control (DMC) and DMC filter, respectively. Numerical results by the multi-model inverse method based on DMC demonstrate that the multi-model inverse method is a highly computationally efficient and accurate algorithm for inverse problems with complicated direct problems. Numerical results by the multi-model inverse method based on DMC filter show that the presented method can extend the applied field of the complicated linear inverse algorithms such as digital filter to the nonlinear inverse problems and it can obtain satisfactory inversion results.     

Distortion cancellation of frequency converted pulses with simple linear signal processing and application to frequency modulation to amplitude modulation conversion in high power lasers

It is known that a linear filter may be easily compensated with its inverse transfer function. However, it was shown that this approach could also be valid even for such a complex nonlinear system as frequency conversion. As a matter of fact, it is possible to at least partly precompensate for distortions occurring within, or even downstream from, frequency conversion crystals with a simple linear optical filter set upstream. In this paper, we give the theoretical background and derive the optimum precompensation filter from simple analytical formulas even in the case of saturation. We first show the relevance of our approach for Gaussian pulses: the pulse may be short or not and chirped or not, and the same linear precompensation filter may be used as long as saturation is not reached. We then study the case of phase-modulated pulses, as can be found on high power lasers such as lasers for fusion. We show that previous experimental results are in perfect agreement with these calculations. Finally, justified by our simple analytical formulas, we present a rigorous parametrical study giving the distortion reduction for any second and third harmonic generation system in the case of phase-modulated pulses.     

A Sliding Mode Control Algorithm for Solving an Ill-posed Positive Linear System

For the numerical solution of an ill-posed positive linear system we combine the methods from invariant manifold theory and sliding mode control theory, developing an affine nonlinear dynamical system with a positive control force and with the residual vector as being a gain vector. This system is proven asymptotically stable to the zero residual vector by using an argument from the Lyapunov stability theory. We find that the system fast tends to the sliding surface and then moves with a sliding mode, such that the resultant sliding mode control algorithm (SMCA) is robust against large noise and stable to find the numerical solution of an ill-posed linear system. It is interesting that even under a random noise with an intensity 10^-5 we can obtain a quite accurate solution of the linear Hilbert problem with dimension n = 500. For this highly ill-conditioned problem the number of iterations is still smaller than 100. Numerical tests, including the inverse problems of backward heat conduction problem and Cauchy problems, confirm that the present SMCA has superior computational efficiency and accuracy even for a highly ill-conditioned linear equations system under a large noise.     

Solution of convergence problem in ultrasound inverse scattering tomography

When the product of contrast and size of an object, which is to be reconstructed by using the ultrasound inverse scattering tomography algorithm, is large, it is well known that those algorithms fail to converge to a unique global minimum. In order to solve this well known and difficult convergence problem, in this paper we present a new method, which converges to the true solution, for obtaining the scattering potential without using the Born or Rytov approximation. This method converts the nonlinear nature of the problem into a linear one. Through computer simulations we will show the validity of the new approach for high contrast two-dimensional scattering objects which are insonified by an incident ultrasound plane wave. Numerical results show that the reconstruction error is very small for circularly symmetric two-dimensional cylindrical objects whose refractive indices range from small to even sufficiently large values for which the previous inverse scattering algorithms fail to converge.     

Ultrasound-modulated optical tomography: recovery of amplitude of vibration in the insonified region from boundary measurement of light correlation

We address a certain inverse problem in ultrasound-modulated optical tomography: the recovery of the amplitude of vibration of scatterers [p(r)] in the ultrasound focal volume in a diffusive object from boundary measurement of the modulation depth (M) of the amplitude autocorrelation of light [φ(r,τ)] traversing through it. Since M is dependent on the stiffness of the material, this is the precursor to elasticity imaging. The propagation of φ(r,τ) is described by a diffusion equation from which we have derived a nonlinear perturbation equation connecting p(r) and refractive index modulation [Δn(r)] in the region of interest to M measured on the boundary. The nonlinear perturbation equation and its approximate linear counterpart are solved for the recovery of p(r). The numerical results reveal regions of different stiffness, proving that the present method recovers p(r) with reasonable quantitative accuracy and spatial resolution.     

Non-destructive identification of residual stresses in steel under thermal loadings

We discuss computational aspects of the inverse and ill-posed problem of identifying residual stresses in steel structures under thermal loading. This corresponds to an inverse source problem in linear thermo-elasticity. The studies aim in investigating whether thermal loadings for the excitation of structures are sufficient in order to detect reliably inherent residual stresses. These stresses may result from the construction process or later thermal or mechanical treatment of the structure-like welding. By answering the raised question positively, our method provides an important basis for successful thermal straightenings. The quality of the solution of the inverse problem depends on a series of parameters, like material parameters, noise in the measurements, and the experimental setup. We numerically study the effects of these parameters and quantify the uncertainties in the results of the inverse problems by means of Sobol indices.     

Improved Logistic Regression Algorithm Based on Kernel Density Estimation for Multi-Classification with Non-Equilibrium Samples

Logistic regression is often used to solve linear binary classification problems such as machine vision, speech recognition, and handwriting recognition. However, it usually fails to solve certain nonlinear multi-classification problem, such as problem with non-equilibrium samples. Many scholars have proposed some methods, such as neural network, least square support vector machine, AdaBoost meta-algorithm, etc. These methods essentially belong to machine learning categories. In this work, based on the probability theory and statistical principle, we propose an improved logistic regression algorithm based on kernel density estimation for solving nonlinear multi-classification. We have compared our approach with other methods using non-equilibrium samples, the results show that our approach guarantees sample integrity and achieves superior classification.     

A multi-commodity supply chain design problem

We consider a multi-commodity supply chain design problem in which we need to determine where to locate facilities and how to allocate customers to facilities so as to minimize total costs. The cost associated with each facility exhibits economies of scale. We show that this problem can be formulated as a nonlinear integer program and propose a Lagrangian-relaxation solution algorithm. By exploiting the structure of the problem, we find a low-order polynomial algorithm for the nonlinear integer program that must be solved in solving the Lagrangian relaxation subproblems. We also compare our approach with an existing algorithm.     

Rotationally invariant pattern recognition by use of linear and nonlinear cascaded filters

We discuss the merits of using single-layer (linear and nonlinear) and multiple-layer (nonlinear) filters for rotationally invariant and noise-tolerant pattern recognition. The capability of each approach is considered with reference to a two-class, rotation-invariant, character recognition problem. The minimum average correlation energy (MACE) filter is a linear filter that is generally accepted to be optimal for detecting signals that are free from noise. Here it is found that an optimized MACE filter cannot differentiate between the characters E and F in a rotation-invariant manner. We have found, however, that this task is possible when a single optimized linear filter is used to achieve the required response when a nonlinear threshold function is included after the filter. We show that this structure can be cascaded to form a multiple-layer, cascaded filter and that the capability of such a system is enhanced by its increased noise tolerance in the character recognition problem. Finally, we show the capability of a two-layer cascade as a means to detect different species of bacteria in images obtained from a phase-contrast microscope.     

Fourier-based magnetic induction tomography for mapping resistivity

Magnetic induction tomography is used as an experimental tool for mapping the passive electromagnetic properties of conductors, with the potential for imaging biological tissues. Our numerical approach to solving the inverse problem is to obtain a Fourier expansion of the resistivity and the stream functions of the magnetic fields and eddy current density. Thus, we are able to solve the inverse problem of determining the resistivity from the applied and measured magnetic fields for a two-dimensional conducting plane. When we add noise to the measured magnetic field, we find the fidelity of the measured to the true resistivity is quite robust for increasing levels of noise and increasing distances of the applied and measured field coils from the conducting plane, when properly filtered. We conclude that Fourier methods provide a reliable alternative for solving the inverse problem.