Inverse problem

inverse problemsinverse theoryinversionModel inversioninverse methodlinear inverse problemInverse modelingforward problemill-posed inverse probleminverse
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field.wikipedia
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Medical imaging

imagingdiagnostic imagingdiagnostic radiology
They have wide application in system identification, optics, radar, acoustics, communication theory, signal processing, medical imaging, computer vision, geophysics, oceanography, astronomy, remote sensing, natural language processing, machine learning, nondestructive testing, and many other fields.
In this restricted sense, medical imaging can be seen as the solution of mathematical inverse problems.

Remote sensing

remote-sensingremotely sensedremote sensor
They have wide application in system identification, optics, radar, acoustics, communication theory, signal processing, medical imaging, computer vision, geophysics, oceanography, astronomy, remote sensing, natural language processing, machine learning, nondestructive testing, and many other fields.
Generally speaking, remote sensing works on the principle of the inverse problem: while the object or phenomenon of interest (the state) may not be directly measured, there exists some other variable that can be detected and measured (the observation) which may be related to the object of interest through a calculation.

Regularization (mathematics)

regularizationregularizedregularize
In these cases, regularization may be used to introduce mild assumptions on the solution and prevent overfitting.
In mathematics, statistics, and computer science, particularly in machine learning and inverse problems, regularization is the process of adding information in order to solve an ill-posed problem or to prevent overfitting.

Tomographic reconstruction

Reconstruction algorithmreconstructedtomography
In X-ray computed tomography the lines on which the parameter is integrated are straight lines : the tomographic reconstruction of the parameter distribution is based on the inversion of the Radon transform.
Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections.

SAMV (algorithm)

SAMViterative Sparse Asymptotic Minimum Variance
Solutions explored include Algebraic Reconstruction Technique, filtered backprojection, and as computing power has increased, iterative reconstruction methods such as iterative Sparse Asymptotic Minimum Variance.
SAMV (iterative sparse asymptotic minimum variance ) is a parameter-free superresolution algorithm for the linear inverse problem in spectral estimation, direction-of-arrival (DOA) estimation and tomographic reconstruction with applications in signal processing, medical imaging and remote sensing.

Well-posed problem

ill-posedill-posed problemwell-posed
Inverse problems are typically ill posed, as opposed to the well-posed problems usually met in mathematical modeling.
Inverse problems are often ill-posed.

Inverse scattering problem

inverse scattering
Whereas linear inverse problems were completely solved from the theoretical point of view at the end of the nineteenth century, only one class of nonlinear inverse problems was so before 1970, that of inverse spectral and (one space dimension) inverse scattering problems, after the seminal work of the Russian mathematical school (Krein, Gelfand, Levitan, Marchenko).
It is the inverse problem to the direct scattering problem, which is to determine how radiation or particles are scattered based on the properties of the scatterer.

Iterative reconstruction

image reconstructionIterative2-D imaging
Solutions explored include Algebraic Reconstruction Technique, filtered backprojection, and as computing power has increased, iterative reconstruction methods such as iterative Sparse Asymptotic Minimum Variance.
The reconstruction of an image from the acquired data is an inverse problem.

Radon transform

filtered back projectionfiltered backprojectionRadon transformation
In X-ray computed tomography the lines on which the parameter is integrated are straight lines : the tomographic reconstruction of the parameter distribution is based on the inversion of the Radon transform. Solutions explored include Algebraic Reconstruction Technique, filtered backprojection, and as computing power has increased, iterative reconstruction methods such as iterative Sparse Asymptotic Minimum Variance.
Reconstruction is an inverse problem.

Unisolvent functions

unisolvence
Some inverse problems have a very simple solution for instance when one has a set of unisolvent functions, meaning a set of functions such that evaluating them at distinct points yields a set of linearly independent vectors.
* Inverse problem

Mathematical geophysics

mathematical geophysicist
Geophysical inverse theory is concerned with analyzing geophysical data to get model parameters.

Tikhonov regularization

ridge regressionregularizeda squared regularizing function
When the forward map is compact, the classical Tikhonov regularization will work if we use it for integrating prior information stating that the L^2 norm of the solution should be as small as possible : this will make the inverse problem well-posed.
The probabilistic formulation of an inverse problem introduces (when all uncertainties are Gaussian) a covariance matrix C_M representing the a priori uncertainties on the model parameters, and a covariance matrix C_D representing the uncertainties on the observed parameters.

Inverse Problems

It combines theoretical, experimental and mathematical papers on inverse problems with numerical and practical approaches to their solution.

Optimal estimation

In applied statistics, optimal estimation is a regularized matrix inverse method based on Bayes' theorem.

Backus–Gilbert method

Backus-Gilbert inverse
It is a regularization method for obtaining meaningful solutions to ill-posed inverse problems.

Atmospheric sounding

soundingsoundingsAtmospheric sounder
It is usually easy to go from the state space to the measurement space—for instance with Beer's law or radiative transfer—but not the other way around, therefore we need some method of inverting \vec f or of finding the inverse model, \vec f^{-1}.

Grey box model

Grey box completion and validationgrey-box
A third method is model inversion, which converts the non-linear m(f,p,Ac) into an approximate linear form in the elements of A, that can be examined using efficient term selection and evaluation of the linear regression.

Data assimilation

assimilationassimilating3D-Var data assimilation scheme
This increasing non-linearity in the models and observation operators poses a new problem in the data assimilation.

Causality

causalcause and effectcausation
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field.

Science

scientificsciencesscientific knowledge
Inverse problems are some of the most important mathematical problems in science and mathematics because they tell us about parameters that we cannot directly observe.

Mathematics

mathematicalmathmathematician
Inverse problems are some of the most important mathematical problems in science and mathematics because they tell us about parameters that we cannot directly observe.

System identification

identificationmodel order reductionnonlinear system identification
They have wide application in system identification, optics, radar, acoustics, communication theory, signal processing, medical imaging, computer vision, geophysics, oceanography, astronomy, remote sensing, natural language processing, machine learning, nondestructive testing, and many other fields.