An inverse problem in science is the process of calculating from a set of observations the
causal
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
factors that produced them: for example, calculating an image in
X-ray computed tomography
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10 nanometers, corresponding to frequencies in the range 30  ...
,
source reconstruction in acoustics, or calculating the density of the Earth from measurements of its
gravity field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
. It is called an inverse problem because it starts with the effects and then calculates the causes. It is the inverse of a forward problem, which starts with the causes and then calculates the effects.
Inverse problems are some of the most important mathematical problems in
science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
and
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
because they tell us about parameters that we cannot directly observe. They have wide application in
system identification
The field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data. System identification also includes the optimal design of experiments for efficiently generating informative data f ...
,
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
,
radar
Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...
,
acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
,
communication theory
Communication theory is a proposed description of communication phenomena, the relationships among them, a storyline describing these relationships, and an argument for these three elements. Communication theory provides a way of talking about a ...
,
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
,
medical imaging
Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to rev ...
,
computer vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the hum ...
,
geophysics
Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' som ...
,
oceanography
Oceanography (), also known as oceanology and ocean science, is the scientific study of the oceans. It is an Earth science, which covers a wide range of topics, including ecosystem dynamics; ocean currents, waves, and geophysical fluid dynamic ...
,
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
,
remote sensing
Remote sensing is the acquisition of information about an object or phenomenon without making physical contact with the object, in contrast to in situ or on-site observation. The term is applied especially to acquiring information about Earth ...
,
natural language processing
Natural language processing (NLP) is an interdisciplinary subfield of linguistics, computer science, and artificial intelligence concerned with the interactions between computers and human language, in particular how to program computers to pro ...
,
machine learning
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence.
Machine ...
,
nondestructive testing
Nondestructive testing (NDT) is any of a wide group of analysis techniques used in science and technology industry to evaluate the properties of a material, component or system without causing damage.
The terms nondestructive examination (NDE), n ...
, slope stability analysis and many other fields.
History
Starting with the effects to discover the causes has concerned physicists for centuries. A historical example is the calculations of
Adams and
Le Verrier
Urbain Jean Joseph Le Verrier FRS (FOR) H FRSE (; 11 March 1811 – 23 September 1877) was a French astronomer and mathematician who specialized in celestial mechanics and is best known for predicting the existence and position of Neptune usin ...
which led to the discovery of
Neptune
Neptune is the eighth planet from the Sun and the farthest known planet in the Solar System. It is the fourth-largest planet in the Solar System by diameter, the third-most-massive planet, and the densest giant planet. It is 17 times ...
from the perturbed trajectory of
Uranus
Uranus is the seventh planet from the Sun. Its name is a reference to the Greek god of the sky, Uranus (mythology), Uranus (Caelus), who, according to Greek mythology, was the great-grandfather of Ares (Mars (mythology), Mars), grandfather ...
. However, a formal study of inverse problems was not initiated until the 20th century.
One of the earliest examples of a solution to an inverse problem was discovered by
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
and published in 1911, describing the asymptotic behavior of eigenvalues of the
Laplace–Beltrami operator
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named af ...
. Today known as ''
Weyl's law In mathematics, especially spectral theory, Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace–Beltrami operator. This description was discovered in 1911 (in the d=2,3 case) by Hermann Weyl for eigenvalues for the Laplace ...
'', it is perhaps most easily understood as an answer to the question of whether it is possible to
hear the shape of a drum. Weyl conjectured that the eigenfrequencies of a drum would be related to the area and perimeter of the drum by a particular equation, a result improved upon by later mathematicians.
The field of inverse problems was later touched on by
Soviet
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, ...
-
Armenian
Armenian may refer to:
* Something of, from, or related to Armenia, a country in the South Caucasus region of Eurasia
* Armenians, the national people of Armenia, or people of Armenian descent
** Armenian Diaspora, Armenian communities across the ...
physicist,
Viktor Ambartsumian
Viktor Amazaspovich Ambartsumian (russian: Виктор Амазаспович Амбарцумян; hy, Վիկտոր Համազասպի Համբարձումյան, ''Viktor Hamazaspi Hambardzumyan''; 12 August 1996) was a Soviet Armenian astr ...
.
While still a student, Ambartsumian thoroughly studied the theory of atomic structure, the formation of energy levels, and the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
and its properties, and when he mastered the theory of
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s, he pointed out the apparent analogy between discrete energy levels and the eigenvalues of differential equations. He then asked: given a family of eigenvalues, is it possible to find the form of the equations whose eigenvalues they are? Essentially Ambartsumian was examining the inverse
Sturm–Liouville problem, which dealt with determining the equations of a vibrating string. This paper was published in 1929 in the German physics journal ''
Zeitschrift für Physik
''Zeitschrift für Physik'' (English: ''Journal for Physics'') is a defunct series of German peer-reviewed physics journals established in 1920 by Springer Berlin Heidelberg. The series stopped publication in 1997, when it merged with other journ ...
'' and remained in obscurity for a rather long time. Describing this situation after many decades, Ambartsumian said, "If an astronomer publishes an article with a mathematical content in a physics journal, then the most likely thing that will happen to it is oblivion."
Nonetheless, toward the end of the Second World War, this article, written by the 20-year-old Ambartsumian, was found by Swedish mathematicians and formed the starting point for a whole area of research on inverse problems, becoming the foundation of an entire discipline.
Then important efforts have been devoted to a "direct solution" of the inverse scattering problem especially by
Gelfand and Levitan in the Soviet Union.
They proposed an analytic constructive method for determining the solution. When computers became available, some authors have investigated the possibility of applying their approach to similar problems such as the inverse problem in the 1D wave equation. But it rapidly turned out that the inversion is an unstable process: noise and errors can be tremendously amplified making a direct solution hardly practicable.
Then, around the seventies, the least-squares and probabilistic approaches came in and turned out to be very helpful for the determination of parameters involved in various physical systems. This approach met a lot of success. Nowadays inverse problems are also investigated in fields outside physics, such as chemistry, economics, and computer science. Eventually, as numerical models become prevalent in many parts of society, we may expect an inverse problem associated with each of these numerical models.
Conceptual understanding
Since Newton, scientists have extensively attempted to model the world. In particular, when a
mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
is available (for instance, Newton's gravitational law or Coulomb's equation for electrostatics), we can foresee, given some parameters that describe a physical system (such as a distribution of mass or a distribution of electric charges), the behavior of the system. This approach is known as mathematical modeling and the above-mentioned physical parameters are called the model parameters or simply the model. To be precise, we introduce the notion of state of the physical system: it is the solution of the mathematical model's equation. In
optimal control theory
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
, these equations are referred to as the
state equations. In many situations we are not truly interested in knowing the physical state but just its effects on some objects (for instance, the effects the gravitational field has on a specific planet). Hence we have to introduce another operator, called the observation operator, which converts the state of the physical system (here the predicted gravitational field) into what we want to observe (here the movements of the considered planet). We can now introduce the so-called forward problem, which consists of two steps:
* determination of the state of the system from the physical parameters that describe it
* application of the observation operator to the estimated state of the system so as to predict the behavior of what we want to observe.
This leads to introduce another
operator (''F'' stands for "forward") which maps model parameters
into
, the data that model
predicts that is the result of this two-step procedure. Operator
is called forward operator or forward map.
In this approach we basically attempt at predicting the effects knowing the causes.
The table below shows, the Earth being considered as the physical system and for different physical phenomena, the model parameters that describe the system, the physical quantity that describes the state of the physical system and observations commonly made on the state of the system.
In the inverse problem approach we, roughly speaking, try to know the causes given the effects.
General statement of the inverse problem
The inverse problem is the "inverse" of the forward problem: we want to determine the model parameters that produce the data
that is the observation we have recorded (the subscript obs stands for observed).
So that we look for the model parameters
such that (at least approximately)
where
is the forward map. We denote by
the (possibly infinite) number of model parameters, and by
the number of recorded data.
We introduce some useful concepts and the associated notations that will be used below:
* The space of models denoted by
: the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
spanned by model parameters; it has
dimensions;
* The space of data denoted by
:
if we organize the measured samples in a vector with
components (if our measurements consist of functions,
is a vector space with infinite dimensions);
*
: the response of model
; it consists of the data predicted by model
;
*
: the image of
by the forward map, it is a subset of
(but not a subspace unless
is linear) made of responses of all models;
*
: the data misfits (or residuals) associated with model
: they can be arranged as a vector, an element of
.
The concept of residuals is very important: in the scope of finding a model that matches the data, their analysis reveals if the considered model can be considered as realistic or not. Systematic unrealistic discrepancies between the data and the model responses also reveals that the forward map is inadequate and may give insights about an improved forward map.
When operator
is linear, the inverse problem is linear. Otherwise, that is most often, the inverse problem is nonlinear.
Also, models cannot always be described by a finite number of parameters. It is the case when we look for
distributed parameters (a distribution of wave-speeds for instance): in such cases the goal of the inverse problem is to retrieve one or several functions. Such inverse problems are inverse problems with infinite dimension.
Linear inverse problems
In the case of a linear forward map and when we deal with a finite number of model parameters, the forward map can be written as a
linear system
In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.
Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.
As a mathematical abstraction o ...
where
is the
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
that characterizes the forward map.
An elementary example: Earth's gravitational field
Only a few physical systems are actually linear with respect to the model parameters. One such system from geophysics is that of the
Earth's gravitational field. The Earth's gravitational field is determined by the density distribution of the Earth in the subsurface. Because the
lithology
The lithology of a rock unit is a description of its physical characteristics visible at outcrop, in hand or core samples, or with low magnification microscopy. Physical characteristics include colour, texture, grain size, and composition. Lit ...
of the Earth changes quite significantly, we are able to observe minute differences in the Earth's gravitational field on the surface of the Earth. From our understanding of gravity (Newton's Law of Gravitation), we know that the mathematical expression for gravity is:
here
is a measure of the local gravitational acceleration,
is the
universal gravitational constant
The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in th ...
,
is the local mass (which is related to density) of the rock in the subsurface and
is the distance from the mass to the observation point.
By discretizing the above expression, we are able to relate the discrete data observations on the surface of the Earth to the discrete model parameters (density) in the subsurface that we wish to know more about. For example, consider the case where we have measurements carried out at 5 locations on the surface of the Earth. In this case, our data vector,
is a column vector of dimension (5×1): its
-th component is associated with the
-th observation location. We also know that we only have five unknown masses
in the subsurface (unrealistic but used to demonstrate the concept) with known location: we denote by
the distance between the
-th observation location and the
-th mass. Thus, we can construct the linear system relating the five unknown masses to the five data points as follows:
To solve for the model parameters that fit our data, we might be able to invert the matrix
to directly convert the measurements into our model parameters. For example:
A system with five equations and five unknowns is a very specific situation: our example was designed to end up with this specificity. In general, the numbers of data and unknowns are different so that matrix
is not square.
However, even a square matrix can have no inverse: matrix
can be
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
deficient (i.e. has zero eigenvalues) and the solution of the system
is not unique. Then the solution of the inverse problem will be undetermined. This is a first difficulty. Over-determined systems (more equations than unknowns) have other issues.
Also noise may corrupt our observations making
possibly outside the space
of possible responses to model parameters so that solution of the system
may not exist. This is another difficulty.
Tools to overcome the first difficulty
The first difficulty reflects a crucial problem: Our observations do not contain enough information and additional data are required. Additional data can come from physical prior information on the parameter values, on their spatial distribution or, more generally, on their mutual dependence. It can also come from other experiments: For instance, we may think of integrating data recorded by gravimeters and seismographs for a better estimation of densities.
The integration of this additional information is basically a problem of
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
. This discipline is the one that can answer the question: How to mix quantities of different nature? We will be more precise in the section "Bayesian approach" below.
Concerning distributed parameters, prior information about their spatial distribution often consists of information about some derivatives of these distributed parameters. Also, it is common practice, although somewhat artificial, to look for the "simplest" model that reasonably matches the data. This is usually achieved by
penalizing the
norm of the gradient (or the
total variation
In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval 'a'' ...
) of the parameters (this approach is also referred to as the maximization of the entropy). One can also make the model simple through a parametrization that introduces freedom degrees only when necessary.
Additional information may also be integrated through inequality constraints on the model parameters or some functions of them. Such constraints are important to avoid unrealistic values for the parameters (negative values for instance). In this case, the space spanned by model parameters will no longer be a vector space but a subset of admissible models denoted by
in the sequel.
Tools to overcome the second difficulty
As mentioned above, noise may be such that our measurements are not the image of any model, so that we cannot look for a model that produces the data but rather look for
the best (or optimal) model: that is, the one that best matches the data. This leads us to minimize an
objective function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...
, namely a
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
that quantifies how big the residuals are or how far the predicted data are from the observed data. Of course, when we have perfect data (i.e. no noise) then the recovered model should fit the observed data perfectly. A standard objective function,
, is of the form:
where
is the Euclidean norm (it will be the
norm when the measurements are functions instead of samples) of the residuals. This approach amounts to making use of
ordinary least squares
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the prin ...
, an approach widely used in statistics. However, the Euclidean norm is known to be very sensitive to outliers: to avoid this difficulty we may think of using other distances, for instance the
norm, in replacement of the
norm.
Bayesian approach
Very similar to the least-squares approach is the probabilistic approach: If we know the statistics of the noise that contaminates the data, we can think of seeking the most likely model m, which is the model that matches the
maximum likelihood criterion. If the noise is
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
, the maximum likelihood criterion appears as a least-squares criterion, the Euclidean scalar product in data space being replaced by a scalar product involving the
co-variance of the noise. Also, should prior information on model parameters be available, we could think of using
Bayesian inference
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, a ...
to formulate the solution of the inverse problem. This approach is described in detail in Tarantola's book.
Numerical solution of our elementary example
Here we make use of the Euclidean norm to quantify the data misfits. As we deal with a linear inverse problem, the objective function is quadratic. For its minimization, it is classical to compute its gradient using the same rationale (as we would to minimize a function of only one variable). At the optimal model
, this gradient vanishes which can be written as:
where ''F''
T denotes the
matrix transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
of ''F''. This equation simplifies to:
This expression is known as th
normal equationand gives us a possible solution to the inverse problem.
In our example matrix
turns out to be generally full rank so that the equation above makes sense and determines uniquely the model parameters: we do not need integrating additional information for ending up with a unique solution.
Mathematical and computational aspects
Inverse problems are typically ill-posed, as opposed to the
well-posed problem
The mathematical term well-posed problem stems from a definition given by 20th-century French mathematician Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that:
# a solution exists,
# the sol ...
s usually met in mathematical modeling. Of the three conditions for a
well-posed problem
The mathematical term well-posed problem stems from a definition given by 20th-century French mathematician Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that:
# a solution exists,
# the sol ...
suggested by
Jacques Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations.
Biography
The son of a teac ...
(existence, uniqueness, and stability of the solution or solutions) the condition of stability is most often violated. In the sense of
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, the inverse problem is represented by a mapping between
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s. While inverse problems are often formulated in infinite dimensional spaces, limitations to a finite number of measurements, and the practical consideration of recovering only a finite number of unknown parameters, may lead to the problems being recast in discrete form. In this case the inverse problem will typically be ''
ill-conditioned
In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input ...
''. In these cases,
regularization
Regularization may refer to:
* Regularization (linguistics)
* Regularization (mathematics)
* Regularization (physics)
* Regularization (solid modeling)
* Regularization Law, an Israeli law intended to retroactively legalize settlements
See also ...
may be used to introduce mild assumptions on the solution and prevent
overfitting
mathematical modeling, overfitting is "the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit to additional data or predict future observations reliably". An overfitt ...
. Many instances of regularized inverse problems can be interpreted as special cases of
Bayesian inference
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, a ...
.
Numerical solution of the optimization problem
Some inverse problems have a very simple solution, for instance, when one has a set of
unisolvent functions
In mathematics, a set of ''n'' functions ''f''1, ''f''2, ..., ''f'n'' is unisolvent (meaning "uniquely solvable") on a domain Ω if the vectors
: \beginf_1(x_1) \\ f_1(x_2) \\ \vdots \\ f_1(x_n)\end, \beginf_2(x_1) \\ f_2(x_2) \\ \vdots \\ f_ ...
, meaning a set of functions such that evaluating them at distinct points yields a set of
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
vectors. This means that given a linear combination of these functions, the coefficients can be computed by arranging the vectors as the columns of a matrix and then inverting this matrix. The simplest example of unisolvent functions is polynomials constructed, using the
unisolvence theorem
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points of the dataset.
Given a set of data points (x_0,y_0), \ldots, (x_n,y_n), with no ...
, so as to be unisolvent. Concretely, this is done by inverting the
Vandermonde matrix
In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an matrix
:V=\begin
1 & x_1 & x_1^2 & \dots & x_1^\\
1 & x_2 & x_2^2 & \dots & x_2^\\
1 & x_3 ...
. But this a very specific situation.
In general, the solution of an inverse problem requires sophisticated optimization algorithms. When the model is described by a large number of parameters (the number of unknowns involved in some diffraction tomography applications can reach one billion), solving the linear system associated with the normal equations can be cumbersome. The numerical method to be used for solving the optimization problem depends in particular on the cost required for computing the solution
of the forward problem. Once chosen the appropriate algorithm for solving the forward problem (a straightforward matrix-vector multiplication may be not adequate when matrix
is huge), the appropriate algorithm for carrying out the minimization can be found in textbooks dealing with numerical methods for the solution of linear systems and for the minimization of quadratic functions (see for instance Ciarlet or Nocedal).
Also, the user may wish to add physical constraints to the models: In this case, they have to be familiar with
constrained optimization methods, a subject in itself. In all cases, computing the gradient of the objective function often is a key element for the solution of the optimization problem. As mentioned above, information about the spatial distribution of a distributed parameter can be introduced through the parametrization. One can also think of adapting this parametrization during the optimization.
Should the objective function be based on a norm other than the Euclidean norm, we have to leave the area of quadratic optimization. As a result, the optimization problem becomes more difficult. In particular, when the
norm is used for quantifying the data misfit the objective function is no longer differentiable: its gradient does not make sense any longer. Dedicated methods (see for instance Lemaréchal) from non differentiable optimization come in.
Once the optimal model is computed we have to address the question: "Can we trust this model?" The question can be formulated as follows: How large is the set of models that match the data "nearly as well" as this model? In the case of quadratic objective functions, this set is contained in a hyper-ellipsoid, a subset of
(
is the number of unknowns), whose size depends on what we mean with "nearly as well", that is on the noise level. The direction of the largest axis of this ellipsoid (
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
associated with the smallest eigenvalue of matrix
) is the direction of poorly determined components: if we follow this direction, we can bring a strong perturbation to the model without changing significantly the value of the objective function and thus end up with a significantly different quasi-optimal model. We clearly see that the answer to the question "can we trust this model" is governed by the noise level and by the eigenvalues of the
Hessian
A Hessian is an inhabitant of the German state of Hesse.
Hessian may also refer to:
Named from the toponym
*Hessian (soldier), eighteenth-century German regiments in service with the British Empire
**Hessian (boot), a style of boot
**Hessian f ...
of the objective function or equivalently, in the case where no regularization has been integrated, by the
singular value In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self- ...
s of matrix
. Of course, the use of regularization (or other kinds of prior information) reduces the size of the set of almost optimal solutions and, in turn, increases the confidence we can put in the computed solution.
Stability, regularization and model discretization in infinite dimension
We focus here on the recovery of a distributed parameter.
When looking for distributed parameters we have to discretize these unknown functions. Doing so, we reduce the dimension of the problem to something finite. But now, the question is: is there any link between the solution we compute and the one of the initial problem? Then another question: what do we mean with the solution of the initial problem? Since a finite number of data does not allow the determination of an infinity of unknowns, the original data misfit functional has to be regularized to ensure the uniqueness of the solution. Many times, reducing the unknowns to a finite-dimensional space will provide an adequate regularization: the computed solution will look like a discrete version of the solution we were looking for. For example, a naive discretization will often work for solving the
deconvolution
In mathematics, deconvolution is the operation inverse to convolution. Both operations are used in signal processing and image processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deco ...
problem: it will work as long as we do not allow missing frequencies to show up in the numerical solution. But many times, regularization has to be integrated explicitly in the objective function.
In order to understand what may happen, we have to keep in mind that solving such a linear inverse problem amount to solving a Fredholm integral equation of the first kind:
where
is the kernel,
and
are vectors of
, and
is a domain in
. This holds for a 2D application. For a 3D application, we consider
. Note that here the model parameters
consist of a function and that the response of a model also consists of a function denoted by
. This equation is an extension to infinite dimension of the matrix equation
given in the case of discrete problems.
For sufficiently smooth
the operator defined above is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
on reasonable
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s such as the
.
F. Riesz theory states that the set of singular values of such an operator contains zero (hence the existence of a null-space), is finite or at most countable, and, in the latter case, they constitute a sequence that goes to zero. In the case of a symmetric kernel, we have an infinity of eigenvalues and the associated eigenvectors constitute a hilbertian basis of
. Thus any solution of this equation is determined up to an additive function in the null-space and, in the case of infinity of singular values, the solution (which involves the reciprocal of arbitrary small eigenvalues) is unstable: two ingredients that make the solution of this integral equation a typical ill-posed problem! However, we can define a solution through the
pseudo-inverse
In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized in ...
of the forward map (again up to an arbitrary additive function). When the forward map is compact, the classical
Tikhonov regularization
Ridge regression is a method of estimating the coefficients of multiple-regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. Also ...
will work if we use it for integrating prior information stating that the
norm of the solution should be as small as possible: this will make the inverse problem well-posed. Yet, as in the finite dimension case, we have to question the confidence we can put in the computed solution. Again, basically, the information lies in the eigenvalues of the Hessian operator. Should subspaces containing eigenvectors associated with small eigenvalues be explored for computing the solution, then the solution can hardly be trusted: some of its components will be poorly determined. The smallest eigenvalue is equal to the weight introduced in Tikhonov regularization.
Irregular kernels may yield a forward map which is not compact and even
unbounded if we naively equip the space of models with the
norm. In such cases, the Hessian is not a bounded operator and the notion of eigenvalue does not make sense any longer. A mathematical analysis is required to make it a
bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector s ...
and design a well-posed problem: an illustration can be found in. Again, we have to question the confidence we can put in the computed solution and we have to generalize the notion of eigenvalue to get the answer.
Analysis of the spectrum of the Hessian operator is thus a key element to determine how reliable the computed solution is. However, such an analysis is usually a very heavy task. This has led several authors to investigate alternative approaches in the case where we are not interested in all the components of the unknown function but only in sub-unknowns that are the images of the unknown function by a linear operator. These approaches are referred to as the " Backus and Gilbert method",
Lions
The lion (''Panthera leo'') is a large cat of the genus ''Panthera'' native to Africa and India. It has a muscular, broad-chested body; short, rounded head; round ears; and a hairy tuft at the end of its tail. It is sexually dimorphic; adult ...
's sentinels approach, and the SOLA method: these approaches turned out to be strongly related with one another as explained in Chavent Finally, the concept of
limited resolution, often invoked by physicists, is nothing but a specific view of the fact that some poorly determined components may corrupt the solution. But, generally speaking, these poorly determined components of the model are not necessarily associated with high frequencies.
Some classical linear inverse problems for the recovery of distributed parameters
The problems mentioned below correspond to different versions of the Fredholm integral: each of these is associated with a specific kernel
.
Deconvolution
The goal of
deconvolution
In mathematics, deconvolution is the operation inverse to convolution. Both operations are used in signal processing and image processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deco ...
is to reconstruct the original image or signal
which appears as noisy and blurred on the data
.
From a mathematical point of view, the kernel
here only depends on the difference between
and
.
Tomographic methods
In these methods we attempt at recovering a distributed parameter, the observation consisting in the measurement of the integrals of this parameter carried out along a family of lines. We denote by
the line in this family associated with measurement point
. The observation at
can thus be written as:
where
is the arc-length along
and
a known weighting function. Comparing this equation with the Fredholm integral above, we notice that the kernel
is kind of a
delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
that peaks on line
. With such a kernel, the forward map is not compact.
= Computed tomography
=
In
X-ray computed tomography
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10 nanometers, corresponding to frequencies in the range 30  ...
the lines on which the parameter is integrated are straight lines: the
tomographic reconstruction
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. The mathematical basis for tomographic imaging was laid down by Johann ...
of the parameter distribution is based on the inversion of the
Radon transform
In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the l ...
. Although from a theoretical point of view many linear inverse problems are well understood, problems involving the Radon transform and its generalisations still present many theoretical challenges with questions of sufficiency of data still unresolved. Such problems include incomplete data for the x-ray transform in three dimensions and problems involving the generalisation of the x-ray transform to tensor fields. Solutions explored include
Algebraic Reconstruction Technique,
filtered backprojection, and as computing power has increased,
iterative reconstruction
Iterative reconstruction refers to iterative algorithms used to reconstruct 2D and 3D images in certain imaging techniques.
For example, in computed tomography an image must be reconstructed from projections of an object. Here, iterative recons ...
methods such as
iterative Sparse Asymptotic Minimum Variance.
= Diffraction tomography
=
Diffraction tomography is a classical linear inverse problem in exploration seismology: the amplitude recorded at one time for a given source-receiver pair is the sum of contributions arising from points such that the sum of the distances, measured in traveltimes, from the source and the receiver, respectively, is equal to the corresponding recording time. In 3D the parameter is not integrated along lines but over surfaces. Should the propagation velocity be constant, such points are distributed on an ellipsoid. The inverse problems consists in retrieving the distribution of diffracting points from the seismograms recorded along the survey, the velocity distribution being known. A direct solution has been originally proposed b
Beylkinand Lambaré et al.: these works were the starting points of approaches known as amplitude preserved migration (see Beylkin and Bleistein). Should geometrical optics techniques (i.e
rays be used for the solving the wave equation, these methods turn out to be closely related to the so-called least-squares migration methods derived from the least-squares approach (see Lailly,
Tarantola).
= Doppler tomography (astrophysics)
=
If we consider a rotating stellar object, the spectral lines we can observe on a spectral profile will be shifted due to Doppler effect. Doppler tomography aims at converting the information contained in spectral monitoring of the object into a 2D image of the emission (as a function of the radial velocity and of the phase in the periodic rotation movement) of the stellar atmosphere. As explained by
Tom Marsh this linear inverse problem is tomography like: we have to recover a distributed parameter which has been integrated along lines to produce its effects in the recordings.
Inverse heat conduction
Early publications on inverse heat conduction arose from determining surface heat flux during atmospheric re-entry from buried temperature sensors.
Other applications where surface heat flux is needed but surface sensors are not practical include: inside reciprocating engines, inside rocket engines; and, testing of nuclear reactor components. A variety of numerical techniques have been developed to address the ill-posedness and sensitivity to measurement error caused by damping and lagging in the temperature signal.
Non-linear inverse problems
Non-linear inverse problems constitute an inherently more difficult family of inverse problems. Here the forward map
is a non-linear operator. Modeling of physical phenomena often relies on the solution of a partial differential equation (see table above except for gravity law): although these partial differential equations are often linear, the physical parameters that appear in these equations depend in a non-linear way of the state of the system and therefore on the observations we make on it.
Some classical non-linear inverse problems
Inverse scattering problems
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 problem
In mathematics and physics, the inverse scattering problem is the problem of determining characteristics of an object, based on data of how it scatters incoming radiation or particles. It is the inverse problem to the direct scattering problem, wh ...
s, after the seminal work of the Russian mathematical school (
Krein,
Gelfand ''Gelfand'' is a surname meaning "elephant" in the Yiddish language and may refer to:
* People:
** Alan Gelfand, the inventor of the ollie, a skateboarding move
** Alan E. Gelfand, a statistician
** Boris Gelfand, a chess grandmaster
** Israel Gel ...
, Levitan,
Marchenko). A large review of the results has been given by Chadan and Sabatier in their book "Inverse Problems of Quantum Scattering Theory" (two editions in English, one in Russian).
In this kind of problem, data are properties of the spectrum of a linear operator which describe the scattering. The spectrum is made of
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s and
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s, forming together the "discrete spectrum", and generalizations, called the continuous spectrum. The very remarkable physical point is that scattering experiments give information only on the continuous spectrum, and that knowing its full spectrum is both necessary and sufficient in recovering the scattering operator. Hence we have invisible parameters, much more interesting than the null space which has a similar property in linear inverse problems. In addition, there are physical motions in which the spectrum of such an operator is conserved as a consequence of such motion. This phenomenon is governed by special nonlinear partial differential evolution equations, for example the
Korteweg–de Vries equation. If the spectrum of the operator is reduced to one single eigenvalue, its corresponding motion is that of a single bump that propagates at constant velocity and without deformation, a solitary wave called a "
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
".
A perfect signal and its generalizations for the Korteweg–de Vries equation or other integrable nonlinear partial differential equations are of great interest, with many possible applications. This area has been studied as a branch of mathematical physics since the 1970s. Nonlinear inverse problems are also currently studied in many fields of applied science (acoustics, mechanics, quantum mechanics, electromagnetic scattering - in particular radar soundings, seismic soundings, and nearly all imaging modalities).
A final example related to the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
was given by Wu and Sprung, the idea is that in the
semiclassical old quantum theory
The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory ...
the inverse of the potential inside the Hamiltonian is proportional to the
half-derivative
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D
:D f(x) = \frac f(x)\,,
and of the integration ...
of the eigenvalues (energies) counting function ''n''(''x'').
Permeability matching in oil and gas reservoirs
The goal is to recover the diffusion coefficient in the
parabolic partial differential equation
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivati ...
that models single phase fluid flows in porous media. This problem has been the object of many studies since a pioneering work carried out in the early seventies. Concerning two-phase flows an important problem is to estimate the relative permeabilities and the capillary pressures.
Inverse problems in the wave equations
The goal is to recover the wave-speeds (P and S waves) and the density distributions from
seismogram
A seismogram is a graph output by a seismograph. It is a record of the ground motion at a measuring station as a function of time. Seismograms typically record motions in three cartesian axes (x, y, and z), with the z axis perpendicular to the ...
s. Such inverse problems are of prime interest in seismology and
exploration geophysics
Exploration geophysics is an applied branch of geophysics and economic geology, which uses physical methods, such as seismic, gravitational, magnetic, electrical and electromagnetic at the surface of the Earth to measure the physical properties of ...
.
We can basically consider two mathematical models:
* The
acoustic wave equation
In physics, the acoustic wave equation governs the propagation of acoustic waves through a material medium resp. a standing wavefield. The form of the equation is a second order partial differential equation. The equation describes the evolut ...
(in which S waves are ignored when the space dimensions are 2 or 3)
* The
elastodynamics equation in which the P and S wave velocities can be derived from the
Lamé parameters
In continuum mechanics, Lamé parameters (also called the Lamé coefficients, Lamé constants or Lamé moduli) are two material-dependent quantities denoted by λ and μ that arise in strain-stress relationships. In general, λ and μ are indi ...
and the density.
These basic
hyperbolic equations can be upgraded by incorporating
attenuation
In physics, attenuation (in some contexts, extinction) is the gradual loss of flux intensity through a medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, and water and air attenuate both light and sound at variable att ...
,
anisotropy
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
, ...
The solution of the inverse problem in the 1D wave equation has been the object of many studies. It is one of the very few non-linear inverse problems for which we can prove the uniqueness of the solution.
The analysis of the stability of the solution was another challenge.
Practical applications, using the least-squares approach, were developed.
Extension to 2D or 3D problems and to the elastodynamics equations was attempted since the 80's but turned out to be very difficult ! This problem often referred to as Full Waveform Inversion (FWI), is not yet completely solved: among the main difficulties are the existence of non-Gaussian noise into the seismograms, cycle-skipping issues (also known as phase ambiguity), and the chaotic behavior of the data misfit function. Some authors have investigated the possibility of reformulating the inverse problem so as to make the objective function less chaotic than the data misfit function.
Travel-time tomography
Realizing how difficult is the inverse problem in the wave equation, seismologists investigated a simplified approach making use of geometrical optics. In particular they aimed at inverting for the propagation velocity distribution, knowing the arrival times of wave-fronts observed on seismograms. These wave-fronts can be associated with direct arrivals or with reflections associated with reflectors whose geometry is to be determined, jointly with the velocity distribution.
The arrival time distribution
(
is a point in physical space) of a wave-front issued from a point source, satisfies the
Eikonal equation
An eikonal equation (from Greek εἰκών, image) is a non-linear first-order partial differential equation that is encountered in problems of wave propagation.
The classical eikonal equation in geometric optics is a differential equation of ...
:
where
denotes the
slowness (reciprocal of the velocity) distribution. The presence of
makes this equation nonlinear. It is classically solved by shooting
rays
Ray may refer to:
Fish
* Ray (fish), any cartilaginous fish of the superorder Batoidea
* Ray (fish fin anatomy), a bony or horny spine on a fin
Science and mathematics
* Ray (geometry), half of a line proceeding from an initial point
* Ray (gra ...
(trajectories about which the arrival time is stationary) from the point source.
This problem is tomography like: the measured arrival times are the integral along the ray-path of the slowness. But this tomography like problem is nonlinear, mainly because the unknown ray-path geometry depends upon the velocity (or slowness) distribution. In spite of its nonlinear character, travel-time tomography turned out to be very effective for determining the propagation velocity in the Earth or in the subsurface, the latter aspect being a key element for seismic imaging, in particular using methods mentioned in Section "Diffraction tomography".
Mathematical aspects: Hadamard's questions
The questions concern well-posedness: Does the least-squares problem have a unique solution which depends continuously on the data (stability problem)? It is the first question, but it is also a difficult one because of the non-linearity of
.
In order to see where the difficulties arise from, Chavent
proposed to conceptually split the minimization of the data misfit function into two consecutive steps (
is the subset of admissible models):
* projection step: given
find a projection on
(nearest point on
according to the distance involved in the definition of the objective function)
* given this projection find one pre-image that is a model whose image by operator
is this projection.
Difficulties can - and usually will - arise in both steps:
# operator
is not likely to be one-to-one, therefore there can be more than one pre-image,
# even when
is one-to-one, its inverse may not be continuous over
,
# the projection on
may not exist, should this set be not closed,
# the projection on
can be non-unique and not continuous as this can be non-convex due to the non-linearity of
.
We refer to Chavent
[ for a mathematical analysis of these points.
]
Computational aspects
A non-convex data misfit function
The forward map being nonlinear, the data misfit function is likely to be non-convex, making local minimization techniques inefficient. Several approaches have been investigated to overcome this difficulty:
* use of global optimization techniques such as sampling of the posterior density function and Metropolis algorithm
A metropolis () is a large city or conurbation which is a significant economic, political, and cultural center for a country or region, and an important hub for regional or international connections, commerce, and communications.
A big ci ...
in the inverse problem probabilistic framework, genetic algorithms (alone or in combination with Metropolis algorithm: see for an application to the determination of permeabilities that match the existing permeability data), neural networks, regularization techniques including multi scale analysis;
* reformulation of the least-squares objective function so as to make it smoother (see for the inverse problem in the wave equations.)
Computation of the gradient of the objective function
Inverse problems, especially in infinite dimension, may be large size, thus requiring important computing time. When the forward map is nonlinear, the computational difficulties increase and minimizing the objective function can be difficult. Contrary to the linear situation, an explicit use of the Hessian matrix for solving the normal equations does not make sense here: the Hessian matrix varies with models. Much more effective is the evaluation of the gradient of the objective function for some models. Important computational effort can be saved when we can avoid the very heavy computation of the Jacobian (often called "Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued fu ...
s"): the adjoint state method, proposed by Chavent and Lions, is aimed to avoid this very heavy computation. It is now very widely used.
Applications
Inverse problem theory is used extensively in weather predictions, oceanography, hydrology, and petroleum engineering.
Inverse problems are also found in the field of heat transfer, where a surface heat flux is estimated outgoing from temperature data measured inside a rigid body; and, in understanding the controls on plant-matter decay. The linear inverse problem is also the fundamental of spectral estimation
In statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density (also known as the power spectral density) of a signal from a sequence of time samples of the signa ...
and direction-of-arrival (DOA) estimation in signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
.
Inverse lithography
In semiconductor device fabrication, the inverse lithography technology (ILT) is an approach to photomask design. This is basically an approach to solve an inverse imaging problem: to calculate the shapes of the openings in a photomask ("source") ...
is used in photomask
A photomask is an opaque plate with holes or transparencies that allow light to shine through in a defined pattern. They are commonly used in photolithography and the production of integrated circuits (ICs or "chips") in particular. Masks are used ...
design for semiconductor device fabrication
Semiconductor device fabrication is the process used to manufacture semiconductor devices, typically integrated circuit (IC) chips such as modern computer processors, microcontrollers, and memory chips such as NAND flash and DRAM that are pres ...
.
See also
*Atmospheric sounding Atmospheric sounding or atmospheric profiling is a measurement of vertical distribution of physical properties of the atmospheric column such as pressure, temperature, wind speed and wind direction (thus deriving wind shear), liquid water content, ...
* Backus–Gilbert method
* Computed tomography
** Algebraic reconstruction technique
** Filtered backprojection
**Iterative reconstruction
Iterative reconstruction refers to iterative algorithms used to reconstruct 2D and 3D images in certain imaging techniques.
For example, in computed tomography an image must be reconstructed from projections of an object. Here, iterative recons ...
*Data assimilation
Data assimilation is a mathematical discipline that seeks to optimally combine theory (usually in the form of a numerical model) with observations. There may be a number of different goals sought – for example, to determine the optimal state es ...
*Engineering optimization Engineering optimization
is the subject which uses optimization techniques to achieve design goals in engineering. It is sometimes referred to as design optimization.
Topics
* structural design (including pressure vessel design and welded beam ...
*Grey box model
In mathematics, statistics, and computational modelling, a grey box modelKroll, Andreas (2000). Grey-box models: Concepts and application. In: New Frontiers in Computational Intelligence and its Applications, vol.57 of Frontiers in artificial int ...
*Mathematical geophysics Geomathematics (also: mathematical geosciences, mathematical geology, mathematical geophysics) is the application of mathematical methods to solve problems in geosciences, including geology and geophysics, and particularly geodynamics and seismology ...
*Optimal estimation In applied statistics, optimal estimation is a regularized matrix inverse method based on Bayes' theorem.
It is used very commonly in the geosciences, particularly for atmospheric sounding.
A matrix inverse problem looks like this:
:
\mathbf \vec ...
*Seismic inversion In geophysics (primarily in oil-and-gas exploration/development), seismic inversion is the process of transforming seismic reflection data into a quantitative rock-property description of a reservoir. Seismic inversion may be pre- or post- stack, d ...
*Tikhonov regularization
Ridge regression is a method of estimating the coefficients of multiple-regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. Also ...
*Compressed sensing
Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a Signal (electronics), signal, by finding solutions to Underdetermined ...
Academic journals
Four main academic journals cover inverse problems in general:
*''Inverse Problems
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 ...
''
*''Journal of Inverse and Ill-posed Problems''
*''Inverse Problems in Science and Engineering''
*''Inverse Problems and Imaging''
Many journals on medical imaging, geophysics, non-destructive testing, etc. are dominated by inverse problems in those areas.
References
References
*Chadan, Khosrow & Sabatier, Pierre Célestin (1977). ''Inverse Problems in Quantum Scattering Theory''. Springer-Verlag.
*Aster, Richard; Borchers, Brian, and Thurber, Clifford (2018). ''Parameter Estimation and Inverse Problems'', Third Edition, Elsevier. ,
*
Further reading
*
External links
Inverse Problems International Association
Eurasian Association on Inverse Problems
Finnish Inverse Problems Society
Inverse Problems Network
Albert Tarantola's website
includes a free PDF version of his Inverse Problem Theory book, and some online articles on Inverse Problems
Inverse Problems page at the University of Alabama
Inverse Problems and Geostatistics Project
Niels Bohr Institute, University of Copenhagen
Andy Ganse's Geophysical Inverse Theory Resources Page
Finnish Centre of Excellence in Inverse Problems Research
{{DEFAULTSORT:Inverse Problem