On the Regularization-Homotopy Analysis Method for Linear and Nonlinear Fredholm Integral Equations of the First Kind

Fredholm integral equations of the first kind are considered by applying regularization method and the homotopy analysis method. This kind of integral equations are considered as an ill-posed problem and for this reason needs an effective method in solving them. This method first transforms a given Fredholm integral equation of the first kind to the second kind by the regularization method and then solves the transformed equation using homotopy analysis method. Approximation of the solution will be of much concern since it is not always the case to get the solution to converge and the existence of the solution is not always guaranteed as this kind of Fredholm integral equation is not well-posed

On sequential multiscale inversion and data assimilation

Multiscale approaches are very popular for example for solving partial differential equations and in many applied fields dealing with phenomena which take place on different levels of detail. The broad idea of a multiscale approach is to decompose your problem into different scales or levels and to use these decompositions either for constructing appropriate approximations or to solve smaller problems on each of these levels, leading to increased stability or increased efficiency. The idea of sequential multiscale is to first solve the problem in a large-scale subspace and then successively move to finer scale spaces. Our goal is to analyse the sequential multiscale approach applied to an inversion or state estimation problem. We work in a generic setup given by a Hilbert space environment. We work out the analysis both for an unregularized and a regularized sequential multiscale inversion. In general the sequential multiscale approach is not equivalent to a full solution, but we show that under appropriate assumptions we obtain convergence of an iterative sequential multiscale version of the method. For the regularized case we develop a strategy to appropriately adapt the regularization when an iterative approach is taken. We demonstrate the validity of the iterative sequential multiscale approach by testing the method on an integral equation as it appears for atmospheric temperature retrieval from infrared satellite radiances

A New HPM for Integral Equations

Homotopy perturbation method is an effective method for obtaining exact solutions of integral equations. However, it might perform poorly on ill-posed integral equations. In this paper, we introduce a new version of the homotopy perturbation method that efficiently solves ill-posed integral equations. Finally, several numerical examples, including a system of integral equations, are presented to demonstrate the efficiency of the new method

Applications of the homotopy analysis method to optimal control problems

Traditionally, trajectory optimization for aerospace applications has been performed using either direct or indirect methods. Indirect methods produce highly accurate solutions but suer from a small convergence region, requiring initial guesses close to the optimal solution. In past two decades, a new series of analytical approximation methods have been used for solving systems of dierential equations and boundary value problems. The Homotopy Analysis Method (HAM) is one such method which has been used to solve typical boundary value problems in nance, science, and engineering. In this investigation, a methodology is created to solve indirect trajectory optimization problems using the Homotopy Analysis Method. Use of the auxiliary convergence control parameter to widen the convergence region and increase the rate of convergence have been demonstrated on multiple optimal control problems. The guaranteed convergence and the ease of selecting the initial guess for trajectory optimization problems makes the method of high signicance. It has been demonstrated that initial guesses for the optimal control problem can be generated using a simple approach based on only the initial boundary conditions. The approach has been demonstrated on the Zermelo\u27s problem and two cases of a 2D ascent problem. It has been established that for free nal-time boundary value problems, nding the convergence region is much harder as compared to xed nal-time cases. To validate the approach, results are compared with those obtained using the MATLAB\u27s bvp4c function. A number of new challenges are discovered and listed during the process

Modeling and inversion of seismic data using multiple scattering, renormalization and homotopy methods

Seismic scattering theory plays an important role in seismic forward modeling and is the theoretical foundation for various seismic imaging methods. Full waveform inversion is a powerful technique for obtaining a high-resolution model of the subsurface. One objective of this thesis is to develop convergent scattering series solutions of the Lippmann-Schwinger equation in strongly scattering media using renormalization and homotopy methods. Other objectives of this thesis are to develop efficient full waveform inversion methods of time-lapse seismic data and, to investigate uncertainty quantification in full waveform inversion for anisotropic elastic media based on integral equation approaches and the iterated extended Kalman filter. The conventional Born scattering series is obtained by expanding the Lippmann-Schwinger equation in terms of an iterative solution based on perturbation theory. Such an expansion assumes weak scattering and may have the problems of convergence in strongly scattering media. This thesis presents two scattering series, referred to as convergent Born series (CBS) and homotopy analysis method (HAM) scattering series for frequency-domain seismic wave modeling. For the convergent Born series, a physical interpretation from the renormalization prospective is given. The homotopy scattering series is derived by using homotopy analysis method, which is based on a convergence control parameter $h$ and a convergence control operator $H$ that one can use to ensure convergence for strongly scattering media. The homotopy scattering scattering series solutions of the Lippmann-Schwinger equation, which is convergent in strongly scattering media. The homotopy scattering series is a kind of unified scattering series theory that includes the conventional and convergent Born series as special cases. The Fast Fourier Transform (FFT) is employed for efficient implementation of matrix-vector multiplication for the convergent Born series and the homotopy scattering series. This thesis presents homotopy methods for ray based seismic modeling in strongly anisotropic media. To overcome several limitations of small perturbations and weak anisotropy in obtaining the traveltime approximations in anisotropic media by expanding the anisotropic eikonal equation in terms of the anisotropic parameters and the elliptically anisotropic eikonal equation based on perturbation theory, this study applies the homotopy analysis method to the eikonal equation. Then this thesis presents a retrieved zero-order deformation equation that creates a map from the anisotropic eikonal equation to a linearized partial differential equation system. The new traveltime approximations are derived by using the linear and nonlinear operators in the retrieved zero-order deformation equation. Flexibility on variable anisotropy parameters is naturally incorporated into the linear differential equations, allowing a medium of arbitrarily anisotropy. This thesis investigates efficient target-oriented inversion strategies for improving full waveform inversion of time-lapse seismic data based on extending the distorted Born iterative T-matrix inverse scattering to a local inversion of a small region of interest (e. g. reservoir under production). The target-oriented approach is more efficient for inverting the monitor data. The target-oriented inversion strategy requires properly specifying the wavefield extrapolation operators in the integral equation formulation. By employing the T-matrix and the Gaussian beam based Green’s function, the wavefield extrapolation for the time-lapse inversion is performed in the baseline model from the survey surface to the target region. I demonstrate the method by presenting numerical examples illustrating the sequential and double difference strategies. To quantify the uncertainty and multiparameter trade-off in the full waveform inversion for anisotropic elastic media, this study applies the iterated extended Kalman filter to anisotropic elastic full waveform inversion based on the integral equation method. The sensitivity matrix is an explicit representation with Green’s functions based on the nonlinear inverse scattering theory. Taking the similarity of sequential strategy between the multi-scale frequency domain full waveform inversion and data assimilation with an iterated extended Kalman filter, this study applies the explicit representation of sensitivity matrix to the the framework of Bayesian inference and then estimate the uncertainties in the full waveform inversion. This thesis gives results of numerical tests with examples for anisotropic elastic media. They show that the proposed Bayesian inversion method can provide reasonable reconstruction results for the elastic coefficients of the stiffness tensor and the framework is suitable for accessing the uncertainties and analysis of parameter trade-offs