scholarly journals Multifrequency Topological Derivative Approach to Inverse Scattering Problems in Attenuating Media

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1702
Author(s):  
Ana Carpio ◽  
María-Luisa Rapún
Keyword(s):  
Inverse Problem ◽  
Forward Problem ◽  
Topological Derivative ◽  
Similar Data ◽  
Scattering Problems ◽  
Whole Space ◽  
One Step ◽  
Recorded Data ◽  
Main Components ◽  
The Waves

Detecting objects hidden in a medium is an inverse problem. Given data recorded at detectors when sources emit waves that interact with the medium, we aim to find objects that would generate similar data in the presence of the same waves. In opposition, the associated forward problem describes the evolution of the waves in the presence of known objects. This gives a symmetry relation: if the true objects (the desired solution of the inverse problem) were considered for solving the forward problem, then the recorded data should be returned. In this paper, we develop a topological derivative-based multifrequency iterative algorithm to reconstruct objects buried in attenuating media with limited aperture data. We demonstrate the method on half-space configurations, which can be related to problems set in the whole space by symmetry. One-step implementations of the algorithm provide initial approximations, which are improved in a few iterations. We can locate object components of sizes smaller than the main components, or buried deeper inside. However, attenuation effects hinder object detection depending on the size and depth for fixed ranges of frequencies.

Framework for Flaw Detection: Application to Dynamic Crack Detection in Flat Membranes

2008 ◽  
Author(s):  
Daniel Rabinovich ◽  
Dan Givoli ◽  
Shmuel Vigdergauz
Keyword(s):  
Inverse Problem ◽  
Crack Detection ◽  
Flexible Structures ◽  
Flaw Detection ◽  
Forward Problem ◽  
Dynamic Crack ◽  
Computational Framework ◽  
Time Harmonic ◽  
Present Context ◽  
Flat Membranes

A computational framework is developed for the detection of flaws in flexible structures. The framework is based on posing the detection problem as an inverse problem, which requires the solution of many forward problems. Each forward problem is associated with a known flaw; an appropriate cost functional evaluates the quality of each candidate flaw based on the solution of the corresponding forward problem. On the higher level, the inverse problem is solved by a global optimization algorithm. The performance of the computational framework is evaluated by considering the detectability of various types of flaws. In the present context detectability is defined by introducing a measure of the distance between the sought flaw and trial flaws in the space of the parameters characterizing the configuration of the flaw. The framework is applied to crack detection in flat membranes subjected to time-harmonic and transient excitations. The detectability of cracks is compared for these two cases.

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.

ON THE ADIABATICITY OF ACOUSTIC PROPAGATION THROUGH NONGRADUAL OCEAN STRUCTURES

2001 ◽  
Vol 09 (02) ◽  
pp. 359-365 ◽  
Author(s):  
E. C. SHANG ◽  
Y. Y. WANG ◽  
T. F. GAO
Keyword(s):  
Inverse Problem ◽  
Shallow Water ◽  
Acoustic Field ◽  
Solitary Waves ◽  
Sound Propagation ◽  
Acoustic Propagation ◽  
Forward Problem ◽  
Internal Solitary Waves ◽  
New Criterion

To assess the adiabaticity of sound propagation in the ocean is very important for acoustic field calculating (forward problem) and tomographic retrieving(inverse problem). Most of the criterion in the literature is too restrictive, specially for the nongradual ocean structures. A new criterion of adiabaticity is suggested in this paper. It works for nongradual ocean structures such as front and internal solitary waves in shallow-water.

Past, present, and future of geophysical inversion—A new millennium analysis

Geophysics ◽  
2001 ◽  
Vol 66 (1) ◽  
pp. 21-24 ◽  
Author(s):  
Sven Treitel ◽  
Larry Lines
Keyword(s):  
Inverse Problem ◽  
Review Article ◽  
Data Sets ◽  
Inversion Process ◽  
Geophysical Inversion ◽  
Field Recordings ◽  
Theory To Practice ◽  
Rough Model ◽  
Recorded Data

Geophysicists have been working on solutions to the inverse problem since the dawn of our profession. An interpreter infers subsurface properties on the basis of observed data sets, such as seismograms or potential field recordings. A rough model of the process that produces the recorded data resides within the interpreter’s brain; the interpreter then uses this rough mental model to reconstruct subsurface properties from the observed data. In modern parlance, the inference of subsurface properties from observed data is identified with the solution of a so‐called “inverse problem.” In contrast, the “forward problem” consists of the determination of the data that would be recorded for a given subsurface configuration and under the assumption that given laws of physics hold. Until the early 1960s, geophysical inversion was carried out almost exclusively within the geophysicist’s brain. Since then, we have learned to make the geophysical inversion process much more quantitative and versatile by recourse to a growing body of theory, along with the computer power to reduce this theory to practice. We should point out the obvious, however, namely that no theory and no computer algorithm can presumably replace the ultimate arbiter who decides whether the results of an inversion make sense or nonsense: the geophysical interpreter. Perhaps our descendants writing a future third Millennium review article can report that a machine has been solving the inverse problem without a human arbiter. For the time being, however, what might be called “unsupervised geophysical inversion” remains but a dream.

Second Order Topological Derivative for the Inverse Problem in Diffusion with Retention

2015 ◽  
Author(s):  
Jairo Rocha de Faria ◽  
Ana Paula Pintado Wyse ◽  
Antônio José Boness Santos ◽  
Luiz Bevilacqua ◽  
Flavio Pietrobon Costa
Keyword(s):  
Inverse Problem ◽  
Second Order ◽  
Topological Derivative

Analysis of the Influence of Fixture Locator Errors on the Compliance of Work Part Features to Geometric Tolerance Specifications

2003 ◽  
Vol 125 (3) ◽  
pp. 609-616 ◽  
Author(s):  
Rodrigo A. Marin ◽  
Placid M. Ferreira
Keyword(s):  
Inverse Problem ◽  
Reference Frame ◽  
Forward Problem ◽  
Acceptable Level ◽  
Geometric Tolerance ◽  
Mechanical Part ◽  
Geometric Tolerances ◽  
Nc Program ◽  
Functional Features ◽  
Position And Orientation

A machining fixture controls position and orientation of datum references (used to define important functional features of the geometry of a mechanical part) relative the reference frame for an NC program. Inaccuracies in fixture’s location scheme result in a deviation of the work part from its nominal specified geometry. For a part to be acceptable this deviation must be within the limits allowed by the geometric tolerances specified. This paper addresses the problem of characterizing the acceptable level of inaccuracy in the location scheme so that the features machined on the part comply with the limits associated with its geometric tolerances. First we solve the “forward problem” that involves predicting the tolerance deviation resulting at a feature from a known set of errors on the locators. However, the paper concentrates on solving the “inverse” problem that involves establishing bounds on the errors of the locators to ensure that the limits specified by geometric tolerances at a feature are not violated.

The Fundamental Theorem of Linear Algebra

2011 ◽  
Author(s):  
Gidon Eshel
Keyword(s):  
Inverse Problem ◽  
Linear Algebra ◽  
Fundamental Theorem ◽  
Forward Problem ◽  
Pictorial Representation ◽  
Basic Ideas

This chapter summarizes pictorially some of the linear algebraic foundations discussed thus far by revisiting the fundamental theorem of linear algebra, the unifying view of matrices, vectors, and their interactions. To make the discussion helpful and informal yet rigorous, and to complement the slightly more formal introduction of the basic ideas given in an earlier chapter, here the theorem’s pictorial representation is emphasized. The discussions cover the forward problem, when A ɛ ℝM×N maps an x ɛ ℝN from A’s domain onto b ɛ ℝM in A’s range, how A transforms x into b; and the inverse problem, discussed in detail in chapter 9, section 9.4.1.

Two sides of the same inverse problem, in identifying the pore size distribution based on experiments with non-Newtonian fluids

2021 ◽  
Author(s):  
Martin Lanzendörfer
Keyword(s):  
Inverse Problem ◽  
Pore Size Distribution ◽  
Pore Size ◽  
Size Distribution ◽  
Forward Problem ◽  
Size Distributions ◽  
Pore Sizes ◽  
Pore Size Distributions ◽  
Flow Through ◽  
Effective Pore Size

<p>Following the capillary bundle concept, i.e. idealizing the flow in a saturated porous media in a given direction as the Hagen-Poiseuille flow through a number of tubular capillaries, one can very easily solve what we would call the <em>forward problem</em>: Given the number and geometry of the capillaries (in particular, given the pore size distribution), the rheology of the fluid and the hydraulic gradient, to determine the resulting flux. With a Newtonian fluid, the flux would follow the linear Darcy law and the porous media would then be represented by one constant only (the permeability), while materials with very different pore size distributions can have identical permeability. With a non-Newtonian fluid, however, the flux resulting from the forward problem (while still easy to solve) depends in a more complicated nonlinear way upon the pore sizes. This has allowed researchers to try to solve the much more complicated <em>inverse problem</em>: Given the fluxes corresponding to a set of non-Newtonian rheologies and/or hydraulic gradients, to identify the geometry of the capillaries (say, the effective pore size distribution).</p><p>The potential applications are many. However, the inverse problem is, as they usually are, much more complicated. We will try to comment on some of the challenges that hinder our way forward. Some sets of experimental data may not reveal any information about the pore sizes. Some data may lead to numerically ill-posed problems. Different effective pore size distributions correspond to the same data set. Some resulting pore sizes may be misleading. We do not know how the measurement error affects the inverse problem results. How to plan an optimal set of experiments? Not speaking about the important question, how are the observed effective pore sizes related to other notions of pore size distribution.</p><p>All of the above issues can be addressed (at least initially) with artificial data, obtained e.g. by solving the forward problem numerically or by computing the flow through other idealized pore geometries. Apart from illustrating the above issues, we focus on <em>two distinct aspects of the inverse problem</em>, that should be regarded separately. First: given the forward problem with <em>N</em> distinct pore sizes, how do different algorithms and/or different sets of experiments perform in identifying them? Second: given the forward problem with a smooth continuous pore size distribution (or, with the number of pore sizes greater than <em>N</em>), how should an optimal representation by <em>N</em> effective pore sizes be defined, regardless of the method necessary to find them?</p>

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