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.

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.

The Fundamental Theorem of Linear Algebra

1993 ◽  
Vol 100 (9) ◽  
pp. 848 ◽  
Author(s):  
Gilbert Strang
Keyword(s):  
Linear Algebra ◽  
Fundamental Theorem

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.

Variable Density Linear Acoustic Inverse Problem

1993 ◽  
Vol 15 (3) ◽  
pp. 255-266 ◽  
Author(s):  
M. Moghaddam ◽  
W.C. Chew
Keyword(s):  
Inverse Problem ◽  
Variable Density ◽  
Diffraction Tomography ◽  
Frequency Diversity ◽  
Low Contrast ◽  
Spatial Frequencies ◽  
Basic Ideas ◽  
Measured Pressure ◽  
Density Linear ◽  
Wave Incidence

The linear acoustic inverse problem is solved simultaneously for density (ρ) and compressibility (κ) using the basic ideas of diffraction tomography (DT). The key to solving this problem is to utilize frequency diversity to obtain the required independent measurements. The receivers are assumed to be in the far field of the object, and plane wave incidence is also assumed. The Born approximation is used to arrive at a relationship between the measured pressure field and two terms related to the spatial Fourier transform of the two unknowns, ρ and κ. The term involving compressibility corresponds to monopole scattering and that for density to dipole scattering. Measurements at several frequencies are used and a least squares problem is solved to reconstruct ρ and κ at the same time. It is observed that the low spatial frequencies in the spectra of ρ and κ produce inaccuracies in the results. Hence, a regularization method is devised to remove this problem. Several results are shown. Low contrast objects for which the above analysis holds are used to show that good reconstructions are obtained for both density and compressibility after regularization is applied.

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.

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>

The SVD-Fundamental Theorem of Linear Algebra

2006 ◽  
Vol 11 (2) ◽  
pp. 123-136 ◽  
Author(s):  
A. G. Akritas ◽  
G. I. Malaschonok ◽  
P. S. Vigklas
Keyword(s):  
Linear System ◽  
Linear Algebra ◽  
Fundamental Theorem ◽  
Gaussian Elimination ◽  
Linear Equations ◽  
Orthogonal Complement ◽  
Schmidt Method ◽  
Orthonormal Bases ◽  
Svd Analysis ◽  
Systems Of Linear Equations

Given an m × n matrix A, with m ≥ n, the four subspaces associated with it are shown in Fig. 1 (see [1]). Fig. 1. The row spaces and the nullspaces of A and AT; a1 through an and h1 through hm are abbreviations of the alignerframe and hangerframe vectors respectively (see [2]). The Fundamental Theorem of Linear Algebra tells us that N(A) is the orthogonal complement of R(AT). These four subspaces tell the whole story of the Linear System Ax = y.  So, for example, the absence of N(AT) indicates that a solution always exists, whereas the absence of N(A) indicates that this solution is unique. Given the importance of these subspaces, computing bases for them is the gist of Linear Algebra. In “Classical” Linear Algebra, bases for these subspaces are computed using Gaussian Elimination; they are orthonormalized with the help of the Gram-Schmidt method. Continuing our previous work [3] and following Uhl’s excellent approach [2] we use SVD analysis to compute orthonormal bases for the four subspaces associated with A, and give a 3D explanation. We then state and prove what we call the “SVD-Fundamental Theorem” of Linear Algebra, and apply it in solving systems of linear equations.

Inverse Analysis of the Residual Stress in Laser–Assisted Milling

2019 ◽  
Author(s):  
Yixuan Feng ◽  
Tsung-Pin Hung ◽  
Yu-Ting Lu ◽  
Yu-Fu Lin ◽  
Fu-Chuan Hsu ◽  
...  
Keyword(s):  
Residual Stress ◽  
Inverse Problem ◽  
Process Parameters ◽  
Inverse Analysis ◽  
Kinematic Hardening ◽  
Cutting Parameters ◽  
Forward Problem ◽  
Depth Of Cut ◽  
Stress Profile ◽  
Analytical Prediction

In laser-assisted milling, higher temperature in shear zone softens the material potentially resulting in a shift of mean residual stress, which significantly affects the damage tolerance and fatigue performance of product. In order to guide the selection of laser and cutting parameters based on the preferred mean residual stress, inverse analysis is conducted by predicting residual stress based on guessed process parameters, which is defined as the forward problem, and applying iterative gradient search to find process parameters for next iteration, which is defined as the inverse problem. An analytical inverse analysis is therefore proposed for the mean residual stress in laser-assisted milling. The forward problem is solved by analytical prediction of mean residual stress after laser-assisted milling. The residual stress profile is predicted through the calculation of thermal stress, by treating laser beam as heat source, and plastic stress by first assuming pure elastic stress in loading process, then obtaining true stress with kinematic hardening followed by the stress relaxation. The variance-based recursive method is applied to solve inverse problem by updating process parameters to match the measured mean residual stress. Three cutting parameters including depth of cut, feed per tooth, and cutting speed, and two laser parameters including laser-tool distance and laser power, are updated with respected to the minimization of resulting residual stress and measurement in each iteration. Experimental measurements are referred on the laser-assisted milling of Ti-6Al-4V grade 5 and ELI. The percentage difference between experiments and predictions is less than 5% for both materials, and the selection is completed within 50 loops.

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