Forward Problem (forward + problem)

Distribution by Scientific Domains

Selected Abstracts

A review of the adjoint-state method for computing the gradient of a functional with geophysical applications

R.-E. Plessix
SUMMARY Estimating the model parameters from measured data generally consists of minimizing an error functional. A classic technique to solve a minimization problem is to successively determine the minimum of a series of linearized problems. This formulation requires the Fréchet derivatives (the Jacobian matrix), which can be expensive to compute. If the minimization is viewed as a non-linear optimization problem, only the gradient of the error functional is needed. This gradient can be computed without the Fréchet derivatives. In the 1970s, the adjoint-state method was developed to efficiently compute the gradient. It is now a well-known method in the numerical community for computing the gradient of a functional with respect to the model parameters when this functional depends on those model parameters through state variables, which are solutions of the forward problem. However, this method is less well understood in the geophysical community. The goal of this paper is to review the adjoint-state method. The idea is to define some adjoint-state variables that are solutions of a linear system. The adjoint-state variables are independent of the model parameter perturbations and in a way gather the perturbations with respect to the state variables. The adjoint-state method is efficient because only one extra linear system needs to be solved. Several applications are presented. When applied to the computation of the derivatives of the ray trajectories, the link with the propagator of the perturbed ray equation is established. [source]

A hybrid fast algorithm for first arrivals tomography

Manuela Mendes
ABSTRACT A hybrid algorithm, combining Monte-Carlo optimization with simultaneous iterative reconstructive technique (SIRT) tomography, is used to invert first arrival traveltimes from seismic data for building a velocity model. Stochastic algorithms may localize a point around the global minimum of the misfit function but are not suitable for identifying the precise solution. On the other hand, a tomographic model reconstruction, based on a local linearization, will only be successful if an initial model already close to the best solution is available. To overcome these problems, in the method proposed here, a first model obtained using a classical Monte Carlo-based optimization is used as a good initial guess for starting the local search with the SIRT tomographic reconstruction. In the forward problem, the first-break times are calculated by solving the eikonal equation through a velocity model with a fast finite-difference method instead of the traditional slow ray-tracing technique. In addition, for the SIRT tomography the seismic energy from sources to receivers is propagated by applying a fast Fresnel volume approach which when combined with turning rays can handle models with both positive and negative velocity gradients. The performance of this two-step optimization scheme has been tested on synthetic and field data for building a geologically plausible velocity model. This is an efficient and fast search mechanism, which permits insertion of geophysical, geological and geodynamic a priori constraints into the grid model and ray path is completed avoided. Extension of the technique to 3D data and also to the solution of ,static correction' problems is easily feasible. [source]

Towards non-linear inversion for characterization of time-lapse phenomena through numerical modelling

A. Abubakar
ABSTRACT We compare two geophysical survey measurements of the same type made at different times in order to characterize the change in the geological medium during the elapsed time. The aim of this study is to develop a strategy using a full non-linear inversion algorithm as the interpretation tool. In this way, not only the location and the form of the changes are recovered, but also the changes in the material parameters of the geological medium can be estimated. In order to solve this fully non-linear problem, the so-called ,multiplicative regularized contrast source inversion' (MR-CSI) method is employed. The unique property of this iterative method is that it does not solve the forward problem at each iterative step. This makes it possible to use the non-linear inversion algorithm for large-scale computation problems. The numerical results show that by taking into account the non-linear nature of the problem, interpretation of the time-lapse data can be significantly improved, compared with that obtained using linear inversion. [source]

Detection and quantification of flaws in structures by the extended finite element method and genetic algorithms

Haim Waisman
Abstract This paper investigates the extended finite element method (XFEM)-GA detection algorithm proposed by Rabinovich et al. (Int. J. Numer. Meth. Engng 2007; 71(9):1051,1080; Int. J. Numer. Meth. Engng 2009; 77(3):337,359) on elastostatic problems with different types of flaws. This algorithm is designed for non-destructive assessment of structural components. Trial flaws are modeled using the XFEM as the forward problem and genetic algorithms (GAs) are employed as the optimization method to converge to the true flaw location and size. The main advantage of the approach is that XFEM alleviates the need for re-meshing the domain at every new iteration of the inverse solution process and GAs have proven to be robust and efficient optimization techniques in particular for this type of problems. In this paper the XFEM-GA methodology is applied to elastostatic problems where flaws are considered as straight cracks, circular holes and non-regular-shaped holes. Measurements are obtained from strain sensors that are attached to the surface of the structure at specific locations and provide the target solution to the GA. The results show convergence robustness and accuracy provided that a sufficient number of sensors are employed and sufficiently large flaws are considered. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Analysis of the bounded variation and the G regularization for nonlinear inverse problems

I. Cimrák
Abstract We analyze the energy method for inverse problems. We study the unconstrained minimization of the energy functional consisting of a least-square fidelity term and two other regularization terms being the seminorm in the BV space and the norm in the G space. We consider a coercive (non)linear operator modelling the forward problem. We establish the uniqueness and stability results for the minimization problems. The stability is studied with respect to the perturbations in the data, in the operator, as well as in the regularization parameters. We settle convergence results for the general minimization schemes. Copyright © 2009 John Wiley & Sons, Ltd. [source]

The asymptotic behaviour of weak solutions to the forward problem of electrical impedance tomography on unbounded three-dimensional domains

Michael Lukaschewitsch
Abstract The forward problem of electrical impedance tomography on unbounded domains can be studied by introducing appropriate function spaces for this setting. In this paper we derive the point-wise asymptotic behaviour of weak solutions to this problem in the three-dimensional case. Copyright © 2008 John Wiley & Sons, Ltd. [source]

On the convergence of the finite integration technique for the anisotropic boundary value problem of magnetic tomography

Roland Potthast
The reconstruction of a current distribution from measurements of the magnetic field is an important problem of current research in inverse problems. Here, we study an appropriate solution to the forward problem, i.e. the calculation of a current distribution given some resistance or conductivity distribution, respectively, and prescribed boundary currents. We briefly describe the well-known solution of the continuous problem, then employ the finite integration technique as developed by Weiland et al. since 1977 for the solution of the problem. Since this method can be physically realized it offers the possibility to develop special tests in the area of inverse problems. Our main point is to provide a new and rigorous study of convergence for the boundary value problem under consideration. In particular, we will show how the arguments which are used in the proof of the continuous case can be carried over to study the finite-dimensional numerical scheme. Finally, we will describe a program package which has been developed for the numerical implementation of the scheme using Matlab. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Light-curve inversions with truncated least-squares principal components: Tests and application to HD 291095 = V1355 Orionis

I.S. Savanov
Abstract We present a new inversion code that reconstructs the stellar surface spot configuration from the light curve of a rotating star. Our code employs a method that uses the truncated least-squares estimation of the inverse problem's objects principal components. We use spot filling factors as the unknown objects. Various test cases that represent a rapidly-rotating K subgiant are used for the forward problem. Tests are then performed to recover the artificial input map and include data errors and input-parameter errors. We demonstrate the robustness of the solution to false input parameters like photospheric temperature, spot temperature, gravity, inclination, unspotted brightness and different spot distributions and we also demonstrate the insensitivity of the solution to spot latitude. Tests with spots peppered over the entire stellar surface or with phase gaps do not produce fake active longitudes. The code is then applied to ten years of V and I -band light curve data of the spotted sub-giant HD291095. A total of 22 light curves is presented. We find that for most of the time its spots were grouped around two active longitudes separated on average by 180°. Switches of the dominant active region between these two longitudes likely occurred about every 3.15±0.23 years while the amplitude modulation of the brightness occurred with a possible period of 3.0±0.15 years. For the first time, we found evidence that the times of the activity flips coincide with times of minimum light as well as minimum photometric amplitude, i.e. maximum spottedness. From a comparison with simultaneous Doppler images we conclude that the activity flips likely take place near the rotational pole of the star. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]