scholarly journals 3D magnetotelluric inversion using a limited-memory quasi-Newton optimization

Geophysics ◽  
2009 ◽  
Vol 74 (3) ◽  
pp. F45-F57 ◽  
Author(s):  
Dmitry Avdeev ◽  
Anna Avdeeva
Keyword(s):  
Penalty Function ◽  
Model Space ◽  
Nonlinear Inversion ◽  
Underground Structures ◽  
Usual Model ◽  
Limited Memory ◽  
Simple Bounds ◽  
Convergence Performance ◽  
Modeling Code ◽  
Quasi Newton

The limited-memory quasi-Newton method with simple bounds is used to develop a novel, fully 3D magnetotelluric (MT) inversion technique. This nonlinear inversion is based on iterative minimization of a classical Tikhonov regularized penalty function. However, instead of the usual model space of log resistivities, the approach iterates in a model space with simple bounds imposed on the conductivities of the 3D target. The method requires storage proportional to [Formula: see text], where [Formula: see text] is the number of conductivities to be recovered and [Formula: see text] is the number of correction pairs (practically, only a few). These requirements are much less than those imposed by other Newton methods, which usually require storage proportional to [Formula: see text] or [Formula: see text], where [Formula: see text] is the number of data to be inverted. The derivatives of the penalty function are calculated using an adjoint method based on electromagnetic field reciprocity. The inversion involves all four entries of the MT impedance matrix; the [Formula: see text] integral equation forward-modeling code is used as an engine for this inversion. Convergence, performance, and accuracy of the inversion are demonstrated on synthetic numerical examples. After investigating erratic resistivities in the upper part of the model obtained for one of the examples, we conclude that the standard Tikhonov regularization is not enough to provide consistently smooth underground structures. An additional regularization helps to overcome the problem.

scholarly journals Solving volterra integral equation via nonlinear programming

2016 ◽  
Vol 5 (4) ◽  
pp. 192
Author(s):  
Jamal Othman
Keyword(s):  
Integral Equation ◽  
Nonlinear Programming ◽  
Volterra Integral Equation ◽  
Least Square ◽  
Nonlinear Programming Problem ◽  
Programming Technique ◽  
New Approach ◽  
Nonlinear Volterra Integral Equation ◽  
Limited Memory ◽  
Quasi Newton

In this paper we propose an approach to find approximate solution to the nonlinear Volterra integral equation of the second type through a nonlinear programming technique by firstly converting the integral equation into a least square cost function as an objective function for an unconstrained nonlinear programming problem which solved by a nonlinear programming technique (The preconditioned limited- memory quasi-Newton conjugates, gradient method) and as far as we read this is a new approach in the ways of solving the nonlinear Volterra integral equation. We use Maple 11 software as a tool for performing the suggested steps in solving the examples.

scholarly journals Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion

Geophysics ◽  
2001 ◽  
Vol 66 (1) ◽  
pp. 174-187 ◽  
Author(s):  
William Rodi ◽  
Randall L. Mackie
Keyword(s):  
Inverse Problems ◽  
Objective Function ◽  
Complete Solution ◽  
Model Space ◽  
Conjugate Gradients ◽  
Nonlinear Inversion ◽  
Spatial Derivatives ◽  
Computationally Intensive ◽  
Derivatives Of ◽  
Special Case

We investigate a new algorithm for computing regularized solutions of the 2-D magnetotelluric inverse problem. The algorithm employs a nonlinear conjugate gradients (NLCG) scheme to minimize an objective function that penalizes data residuals and second spatial derivatives of resistivity. We compare this algorithm theoretically and numerically to two previous algorithms for constructing such “minimum‐structure” models: the Gauss‐Newton method, which solves a sequence of linearized inverse problems and has been the standard approach to nonlinear inversion in geophysics, and an algorithm due to Mackie and Madden, which solves a sequence of linearized inverse problems incompletely using a (linear) conjugate gradients technique. Numerical experiments involving synthetic and field data indicate that the two algorithms based on conjugate gradients (NLCG and Mackie‐Madden) are more efficient than the Gauss‐Newton algorithm in terms of both computer memory requirements and CPU time needed to find accurate solutions to problems of realistic size. This owes largely to the fact that the conjugate gradients‐based algorithms avoid two computationally intensive tasks that are performed at each step of a Gauss‐Newton iteration: calculation of the full Jacobian matrix of the forward modeling operator, and complete solution of a linear system on the model space. The numerical tests also show that the Mackie‐Madden algorithm reduces the objective function more quickly than our new NLCG algorithm in the early stages of minimization, but NLCG is more effective in the later computations. To help understand these results, we describe the Mackie‐Madden and new NLCG algorithms in detail and couch each as a special case of a more general conjugate gradients scheme for nonlinear inversion.

scholarly journals The Adjoint Method for the Inverse Problem of Option Pricing

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Shou-Lei Wang ◽  
Yu-Fei Yang ◽  
Yu-Hua Zeng
Keyword(s):  
Inverse Problem ◽  
Option Pricing ◽  
Implied Volatility ◽  
Adjoint Method ◽  
Numerical Examples ◽  
Limited Memory ◽  
Minimization Procedure ◽  
Volatility Function ◽  
The Cost ◽  
Quasi Newton

The estimation of implied volatility is a typical PDE inverse problem. In this paper, we propose theTV-L1model for identifying the implied volatility. The optimal volatility function is found by minimizing the cost functional measuring the discrepancy. The gradient is computed via the adjoint method which provides us with an exact value of the gradient needed for the minimization procedure. We use the limited memory quasi-Newton algorithm (L-BFGS) to find the optimal and numerical examples shows the effectiveness of the presented method.

scholarly journals Parameters identification: an application to the Richards equation

2002 ◽  
Vol Volume 1, 2002 ◽  
Author(s):  
Pierre Ngnepieba ◽  
François Xavier Le Dimet ◽  
Alexis Boukong ◽  
Gabriel Nguetseng
Keyword(s):  
Standard Technique ◽  
Unconstrained Minimization ◽  
Richards Equation ◽  
Optimal Parameters ◽  
Scale Parameters ◽  
Limited Memory ◽  
Richards Model ◽  
International Audience ◽  
The One ◽  
Quasi Newton

International audience Inverse modeling has become a standard technique for estimating hydrogeologic parameters. These parameters are usually inferred by minimizing the sum of the squared differences between the observed system state and the one calculed by a mathematical model. Since some hydrodynamics parameters in Richards model cannot be measured, they have to be tuned with respect to the observation and the output of the model. Optimal parameters are found by minimizing cost function and the unconstrained minimization algorithm of the quasi-Newton limited memory type is used. The inverse model allows computation of optimal scale parameters and model sensi-tivity. La modélisation inverse est devenue une approche fréquemment utilisée pour l'estimation des paramètres en hydrogéologie. Fondamentalement cette technique est basée sur les méthodes de contrôle optimal qui nécessitent des observations et un modèle pour le calcul des dérivées du premier ordre. Le modèle adjoint du modèle de Richards est construit pour obtenir le gradient de la fonction coût par rapport aux paramètres de contrôle. Les paramètres hydrodynamiques sont pris comme paramètres de contrôle; leurs valeurs optimales sont trouvées en minimisant la fonction coût ceci en utilisant un algorithme de minimisation de type descente quasi-Newton. Cette approche est utilisée pour l'identification des paramètres hydrodynamiques sur un modèle d'écou-lement souterrain en zone non saturée, ainsi que les études de sensibilité du modèle.

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