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.

Developing open-source tools for reproducible inverse problems: the PyLops journey

2021 ◽  
Author(s):  
Matteo Ravasi ◽  
Carlos Alberto da Costa Filho ◽  
Ivan Vasconcelos ◽  
David Vargas
Keyword(s):  
Inverse Problems ◽  
Large Scale ◽  
Mathematical Formulation ◽  
Optimization Methods ◽  
Linear Operators ◽  
Gravity Potential ◽  
Nonlinear Inversion ◽  
Unified Framework ◽  
Computationally Intensive ◽  
High Level

<p>Inverse problems lie at the core of many geophysical algorithms, from earthquake and exploration seismology, all the way to electromagnetics and gravity potential methods.</p><p>In 2018, we open-sourced PyLops, a Python-based framework for large-scale inverse problems. By leveraging the concept of matrix-free linear operators – together with the efficiency of numerical libraries such as NumPy, SciPy, and Numba – PyLops solves computationally intensive inverse problems with high-level code that is highly readable and resembles the underlying mathematical formulation. While initially aimed at researchers, its parsimonious software design choices, large test suite, and thorough documentation render PyLops a reliable and scalable software package easy to run both locally and in the cloud.</p><p>Since its initial release, PyLops has incorporated several advancements in scientific computing leading to the creation of an entire ecosystem of tools: operators can now run on GPUs via CuPy, scale to distributed computing through Dask, and be seamlessly integrated into PyTorch’s autograd to facilitate research in machine-learning-aided inverse problems. Moreover, PyLops contains a large variety of inverse solvers including least-squares, sparsity-promoting algorithms, and proximal solvers highly-suited to convex, possibly nonsmooth problems. PyLops also contains sparsifying transforms (e.g., wavelets, curvelets, seislets) which can be used in conjunction with the solvers. By offering a diverse set of tools for inverse problems under one unified framework, it expedites the use of state-of-the-art optimization methods and compressive sensing techniques in the geoscience domain.</p><p>Beyond our initial expectations, the framework is currently used to solve problems beyond geoscience, including astrophysics and medical imaging. Likewise, it has inspired the development of the occamypy framework for nonlinear inversion in geophysics. In this talk, we share our experience in building such an ecosystem and offer further insights into the needs and interests of the EGU community to help guide future development as well as achieve wider adoption.</p>

scholarly journals Constructing Carrollian CFTs

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Nishant Gupta ◽  
Nemani V. Suryanarayana
Keyword(s):  
Scalar Fields ◽  
Higher Dimensions ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Relativistic Limit ◽  
Conformal Field ◽  
Local Symmetries ◽  
Derivatives Of ◽  
Special Case

Abstract We construct classical theories for scalar fields in arbitrary Carroll spacetimes that are invariant under Carrollian diffeomorphisms and Weyl transformations. When the local symmetries are gauge fixed these theories become Carrollian conformal field theories. We show that generically there are at least two types of such theories: one in which only time derivatives of the fields appear and the other in which both space and time derivatives appear. A classification of such scalar field theories in three (and higher) dimensions up to two derivative order is provided. We show that only a special case of our theories arises in the ultra-relativistic limit of a covariant parent theory.

Vibration attenuation of a two-link flexible arm carried by a translational stage

2018 ◽  
Vol 24 (23) ◽  
pp. 5650-5664 ◽  
Author(s):  
Shang–Teh Wu ◽  
Shan-Qun Tang ◽  
Kuan–Po Huang
Keyword(s):  
Feedback Loop ◽  
Control Method ◽  
Flexible Manipulator ◽  
Closed Loop System ◽  
Flexible Arm ◽  
Vibration Attenuation ◽  
Shaft Encoder ◽  
High Frequency Vibrations ◽  
Spatial Derivatives ◽  
Derivatives Of

This paper investigates the vibration control of a two-link flexible manipulator carried by a translational stage. The first and the second links are each driven by a stage motor and a joint motor. By treating the joint motor as a virtual spring, the two-link manipulator can be regarded as an integral flexible arm driven by the stage motor. A noncollocated controller is devised based on feedback from the deflection of the virtual spring, which can be measured by a shaft encoder. Stability of the closed-loop system is analyzed by examining the spatial derivatives of the modal functions. By including a bandpass filter in the feedback loop, residual vibrations can be attenuated without exciting high-frequency vibrations. The control method is simple to implement; its effectiveness is confirmed by simulation and experimental results.

The effect of microscale random Alfvén waves on the propagation of large-scale Alfvén waves

1983 ◽  
Vol 29 (2) ◽  
pp. 243-253 ◽  
Author(s):  
Tomikazu Namikawa ◽  
Hiromitsu Hamabata
Keyword(s):  
Large Scale ◽  
Ponderomotive Force ◽  
Collisionless Plasma ◽  
Alfven Waves ◽  
Velocity Fields ◽  
Velocity Shear ◽  
Alfvén Waves ◽  
Spectrum Function ◽  
Spatial Derivatives ◽  
Derivatives Of

The ponderomotive force generated by random Alfvén waves in a collisionless plasma is evaluated taking into account mean magnetic and velocity shear and is expressed as a series involving spatial derivatives of mean magnetic and velocity fields whose coefficients are associated with the helicity spectrum function of random velocity field. The effect of microscale random Alfvén waves through ponderomotive and mean electromotive forces generated by them on the propagation of large-scale Alfvén waves is also investigated.

The Uniform Flexure of an Incomplete Tore

1949 ◽  
Vol 2 (4) ◽  
pp. 469
Author(s):  
W Freiberger ◽  
RCT Smith
Keyword(s):  
General Solution ◽  
Cross Section ◽  
Harmonic Functions ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Elasticity Problems ◽  
Three Dimensional Elasticity ◽  
Derivatives Of ◽  
Special Case

In this paper we discuss the flexure of an incomplete tore in the plane of its circular centre-line. We reduce the problem to the determination of two harmonic functions, subject to boundary conditions on the surface of the tore which involve the first two derivatives of the functions. We point out the relation of this solution to the general solution of three-dimensional elasticity problems. The special case of a narrow rectangular cross-section is solved exactly in Appendix II.

scholarly journals Hexagonal Grid Computation of the Derivatives of the Solution to the Heat Equation by Using Fourth-Order Accurate Two-Stage Implicit Methods

2021 ◽  
Vol 5 (4) ◽  
pp. 203
Author(s):  
Suzan Cival Buranay ◽  
Nouman Arshad ◽  
Ahmed Hersi Matan
Keyword(s):  
Heat Equation ◽  
Fourth Order ◽  
Boundary Value ◽  
Time Variable ◽  
Implicit Methods ◽  
First Order ◽  
Mixed Derivatives ◽  
Hexagonal Grids ◽  
Spatial Derivatives ◽  
Derivatives Of

We give fourth-order accurate implicit methods for the computation of the first-order spatial derivatives and second-order mixed derivatives involving the time derivative of the solution of first type boundary value problem of two dimensional heat equation. The methods are constructed based on two stages: At the first stage of the methods, the solution and its derivative with respect to time variable are approximated by using the implicit scheme in Buranay and Arshad in 2020. Therefore, Oh4+τ of convergence on constructed hexagonal grids is obtained that the step sizes in the space variables x1, x2 and in time variable are indicated by h, 32h and τ, respectively. Special difference boundary value problems on hexagonal grids are constructed at the second stages to approximate the first order spatial derivatives and the second order mixed derivatives of the solution. Further, Oh4+τ order of uniform convergence of these schemes are shown for r=ωτh2≥116,ω>0. Additionally, the methods are applied on two sample problems.

scholarly journals Adjoint Numerical Method for a Multiphysical Inverse Problem of Two-Phase Well Testing in Petroleum Reservoirs

2021 ◽  
Vol 2090 (1) ◽  
pp. 012139
Author(s):  
OA Shishkina ◽  
I M Indrupskiy
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Objective Function ◽  
Petroleum Reservoirs ◽  
Forward Problem ◽  
Adjoint Equations ◽  
Well Testing ◽  
Adjoint Methods ◽  
Two Phase ◽  
Gradient Calculation

Abstract Inverse problem solution is an integral part of data interpretation for well testing in petroleum reservoirs. In case of two-phase well tests with water injection, forward problem is based on the multiphase flow model in porous media and solved numerically. The inverse problem is based on a misfit or likelihood objective function. Adjoint methods have proved robust and efficient for gradient calculation of the objective function in this type of problems. However, if time-lapse electrical resistivity measurements during the well test are included in the objective function, both the forward and inverse problems become multiphysical, and straightforward application of the adjoint method is problematic. In this paper we present a novel adjoint algorithm for the inverse problems considered. It takes into account the structure of cross dependencies between flow and electrical equations and variables, as well as specifics of the equations (mixed parabolic-hyperbolic for flow and elliptic for electricity), numerical discretizations and grids, and measurements in the inverse problem. Decomposition is proposed for the adjoint problem which makes possible step-wise solution of the electric adjoint equations, like in the forward problem, after which a cross-term is computed and added to the right-hand side of the flow adjoint equations at this timestep. The overall procedure provides accurate gradient calculation for the multiphysical objective function while preserving robustness and efficiency of the adjoint methods. Example cases of the adjoint gradient calculation are presented and compared to straightforward difference-based gradient calculation in terms of accuracy and efficiency.

scholarly journals Border Reduction in Existence Problems of Harmonic Forms

1967 ◽  
Vol 29 ◽  
pp. 137-143 ◽  
Author(s):  
Mitsuru Nakai ◽  
Leo Sario
Keyword(s):  
Complete Solution ◽  
Harmonic Forms ◽  
Harmonic Field ◽  
Regular Region ◽  
Special Case

Given an arbitrary Riemannian n-space V let σ be a harmonic field in the complement V-V0 of a regular region V0. The problem of constructing in v a harmonic field ρ with the property was given a complete solution in [2]. The corresponding problem for harmonic forms σ, ρ remains open iri the general case. In the special case of locally flat spaces the construction can be carried out by replacing by the point norm [3].

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.

3-D electromagnetic modeling and nonlinear inversion

Geophysics ◽  
2000 ◽  
Vol 65 (3) ◽  
pp. 804-822 ◽  
Author(s):  
Ganquan Xie ◽  
Jianhua Li ◽  
Ernest L. Majer ◽  
Daxin Zuo ◽  
Michael L. Oristaglio
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Inverse Problems ◽  
Differential Equations ◽  
Field Data ◽  
Efficient Solution ◽  
Parallel Computer ◽  
Nonlinear Inversion ◽  
Electromagnetic Modeling ◽  
The Finite Element Method

We describe a new algorithm for 3-D electromagnetic inversion that uses global integral and local differential equations for both the forward and inverse problems. The coupled integral and differential equations are discretized by the finite element method and solved on a parallel computer using domain decomposition. The structure of the algorithm allows efficient solution of large 3-D inverse problems. Tests on both synthetic and field data show that the algorithm converges reliably and efficiently and gives high‐resolution conductivity images.

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