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| {{for|the use in [[geodesy]]|inverse geodetic problem}}
| | If you are going coupon shopping Canada style, you"ll need to find out about the coupon requirements you may use at several retail sites located through the entire Internet t... <br><br>You are trying to find the stingiest methods possible to cut costs through the use of coupons online, when you"re going discount shopping Canada model. There are many coupon using merchants within the Internet group, and you"[http://Www.tumblr.com/tagged/ll+learn ll learn] to form a detailed relationship with every one of them, if you are going coupon shopping Canada type. <br><br>You"ll need to find out about the coupon rules you can use at many retail sites located throughout the Internet to acquire discounts including free shipping or a portion off the cost, if you"re going coupon shopping Canada design. These discounts will magically appear, by entering a promotion code when you are within the phase of your purchase buying process. <br><br>The areas where you can shop are just limited by your imagination. When you can imagine good savings on arts and crafts, automotive needs, or even find good discounts on the settlement and store aisles of many famous suppliers, your savings will become reality when these reductions are used and that are your for a low, low price. <br><br>Some of these stores that you"ll find when going promotion shopping Canada design, will have a favorite item within their store that they need you buy. Many items might be covered by the online discount coupons for their items within their shop, from savings on components, computer and other offers at the Canada retail location. <br><br>If you are imaginative enough, you might even get to purchase some lavish clothier apparel when going promotion shopping Canada design. Canadian suppliers enjoy passing savings to others, and they know the answer can lead them straight back to their retail area, when people start to see the elegant activity wear clothing you are carrying. <br><br>There are great discounts available through these online stores for items that are imperative to everyday life. There are online discount coupons that you need to use to buy [http://search.un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=contacts&Submit=Go contacts] and other cure, beauty and drugs that could save your self your life. When going voucher shopping Canada design, you must realize that these items are not poor, o-r damaged by any means, they"re simply reduced listed items that should go home with you today. <br><br>All the family can take advantage of the coupons you utilize to buy films, musical CDs and also pay for hotel rooms when people continue holiday. When going discount shopping Canada model, the offers are abundant and they will cause you to smile. The free stuff alone will make your heart zing, nevertheless the new dining area table you got at such a discount was undoubtedly a steal. <br><br>Therefore, when going discount shopping Canada model, be ready to cut costs at every stop. This prodound [http://journals.fotki.com/excitingliterature65/Morocco-Layer/ kohls coupons] encyclopedia has assorted majestic suggestions for how to do this activity. From jewelry, to magazines and everything, between the deals get deviously simple most of the time. In a user friendly process, you make your choice, press in a code, and you"ll be on with the savings when contemplating suppliers all over the world, that only Canada companies offer the most of..<br><br>If you adored this post as well as you would want to receive more information with regards to [http://excitedpoker636.shutterfly.com state health insurance] generously stop by our own web-page. |
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| An '''inverse problem''' is a general framework that is used to convert observed measurements into information about a physical object or system. For example, if we have measurements of the Earth's [[gravity field]], then we might ask the question: "given the data that we have available, what can we say about the density distribution of the Earth in that area?" The solution to this problem (i.e. the density distribution that best matches the data) is useful because it generally tells us something about a physical parameter that we cannot directly observe. Thus, inverse problems are some of the most important and well-studied mathematical problems in [[science]] and [[mathematics]]. Inverse problems arise in many branches of [[science]] and [[mathematics]], including [[computer vision]], [[natural language processing]], [[machine learning]], [[statistics]], [[statistical inference]], [[geophysics]], [[medical imaging]] (such as [[computed axial tomography]] and [[electroencephalography|EEG]]/[[event-related potentials|ERP]]), [[remote sensing]], [[ocean acoustic tomography]], [[nondestructive testing]], [[astronomy]], [[physics]] and many other fields.
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| ==History==
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| The field of inverse problems was first discovered and introduced by [[Soviet Union|Soviet]]-[[Armenians|Armenian]] physicist, [[Viktor Ambartsumian]].<ref>http://ambartsumian.ru/en/papers/epilogue-ambartsumian’-s-paper/</ref><ref>http://www.springerlink.com/content/7464423747103122/</ref>
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|
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| While still a student, Ambartsumian thoroughly studied the theory of atomic structure, the formation of energy levels, and the [[Schrödinger equation]] and its properties, and when he mastered the theory of eigenvalues of [[differential equation]]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]]'' 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."
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| 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.
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| ==Conceptual understanding==
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| The inverse problem can be conceptually formulated as follows:
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| :Data → Model parameters
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| The inverse problem is considered the "inverse" to the forward problem which relates the model parameters to the data that we observe:
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| :Model parameters → Data
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| The transformation from data to model parameters (or vice versa) is a result of the interaction of a physical system with the object that we wish to infer properties about. In other words, the transformation is the physics that relates the physical quantity (i.e. the model parameters) to the observed data.
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| The table below shows some examples of physical systems, the governing physics, the physical quantity that we are interested, and what we actually observe.
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| {| class="wikitable"
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| |-
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| ! Physical system !! Governing equations !! Physical quantity !! Observed data
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| |-
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| | Earth's gravitational field || [[Newton's law of universal gravitation|Newton's law of gravity]] || Density || [[Gravitational field]]
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| |-
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| | Earth's magnetic field (at the surface) || [[Maxwell's equations]] || [[Magnetic susceptibility]] || [[Magnetic field]]
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| |-
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| | [[Seismic wave]]s (from earthquakes) || [[Wave equation]] || Wave-speed (density) || [[Particle velocity]]
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| |}
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| Linear algebra is useful in understanding the physical and mathematical construction of inverse problems, because of the presence of the transformation or "mapping" of data to the model parameters.
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| ==General statement of the problem==
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| The objective of an inverse problem is to find the best model <math>m</math> such that (at least approximately)
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| :<math>\ d = G(m)</math>
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| where <math>G</math> is an [[Operator (mathematics)|operator]] describing the explicit relationship between the observed data, <math>d</math>, and the model parameters. In various contexts, the operator <math>G</math> is called '''forward operator''', '''observation operator''', or '''observation function'''. In the most general context, G represents the governing equations that relate the model parameters to the observed data (i.e. the governing physics). | |
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| ==Linear inverse problems ==
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| In the case of a discrete linear inverse problem describing a [[linear system]], <math>d</math> (the model parameters) and <math>m</math> (the best model) are vectors, and the problem can be written as
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| :<math>\ d = Gm</math>
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| where <math>G</math> is a [[Matrix (mathematics)|matrix]] (an [[Operator (mathematics)|operator]] ), often called the '''observation matrix'''.
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| ===Examples===
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| ==== Earth's gravitational field ====
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| 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 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:
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| <math> d = a = \frac{K M}{r^2} </math>
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| where <math>a</math> is a measure of the local gravitational acceleration, <math> K </math> is the universal gravitational constant, <math> M</math> is the local mass (density) of the rock in the subsurface and <math> r </math> is the distance from the mass to the observation point.
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| 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 5 measurements on the surface of the Earth. In this case, our data vector, d is a column vector of dimension (5x1). We also know that we only have five unknown masses in the subsurface (unrealistic but used to demonstrate the concept). Thus, we can construct the linear system relating the five unknown masses to the five data points as follows:
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| : <math> d = G m, \, </math>
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| : <math> d =
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| \begin{bmatrix}
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| d_1 \\
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| d_2 \\
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| d_3 \\
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| d_4 \\
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| d_5 \end{bmatrix}, </math>
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| : <math> m =
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| \begin{bmatrix}
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| M_1 \\
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| M_2 \\
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| M_3 \\
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| M_4 \\
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| M_5
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| \end{bmatrix}, </math>
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| : <math> G =
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| \begin{bmatrix}
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| \frac{K}{r_{11}^2} & \frac{K}{r_{12}^2} & \frac{K}{r_{13}^2} & \frac{K}{r_{14}^2} & \frac{K}{r_{15}^2} \\
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| \frac{K}{r_{21}^2} & \frac{K}{r_{22}^2} & \frac{K}{r_{23}^2} & \frac{K}{r_{24}^2} & \frac{K}{r_{25}^2} \\
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| \frac{K}{r_{31}^2} & \frac{K}{r_{32}^2} & \frac{K}{r_{33}^2} & \frac{K}{r_{34}^2} & \frac{K}{r_{35}^2} \\
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| \frac{K}{r_{41}^2} & \frac{K}{r_{42}^2} & \frac{K}{r_{43}^2} & \frac{K}{r_{44}^2} & \frac{K}{r_{45}^2} \\
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| \frac{K}{r_{51}^2} & \frac{K}{r_{52}^2} & \frac{K}{r_{53}^2} & \frac{K}{r_{54}^2} & \frac{K}{r_{55}^2}
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| \end{bmatrix}
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| </math>
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| Now, we can see that the system has five equations, <math> G </math>, with five unknowns, <math> m </math>. To solve for the model parameters that fit our data, we might be able to invert the matrix <math> G </math> to directly convert the measurements into our model parameters. For example:
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| : <math> m = G^{-1} d \, </math>
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| However, not all square matrices are invertible (<math> G </math> is almost never invertible). This is because we are not guaranteed to have enough information to ''uniquely'' determine the solution to the given equations unless we have independent measurements (i.e. each measurement adds unique information to the system). It's important to note that in most physical systems, we do not ever have enough information to uniquely constrain our solutions because the observation matrix does not contain unique equations. From a [[linear algebra]] perspective, the matrix <math> G </math> is rank deficient (i.e. has zero eigenvalues), meaning that is not invertible. Further, if we add additional observations to our matrix (i.e. more equations), then the matrix <math> G </math> is no longer square. Even then, we're not guaranteed to have full-rank in the observation matrix. Therefore, most inverse problems are considered to be underdetermined, meaning that we do not have unique solutions to the inverse problem. If we have a full-rank system, then our solution may be unique. Overdetermined systems (more equations than unknowns) have other issues.
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| Because we cannot directly invert the observation matrix, we use methods from optimization to solve the inverse problem. To do so, we define a goal, also known as an objective function, for the inverse problem. The goal is a [[Functional (mathematics)|functional]] that measures how close the predicted data from the recovered model fits the observed data. In the case where we have perfect data (i.e. no noise) and perfect physical understanding (i.e. we know the physics) then the recovered model should fit the observed data perfectly. The standard objective function, <math> \phi </math>, is usually of the form:
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| : <math> \phi = || d - G m ||_2^2 \, </math>
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| which represents the [[L-2 norm]] of the misfit between the observed data and the predicted data from the model. We use the L-2 norm here as a generic measurement of the distance between the predicted data and the observed data, but other norms are possible for use. The goal of the objective function is to minimize the difference between the predicted and observed data.
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| To minimize the objective function (i.e. solve the inverse problem) we compute the gradient of the objective function using the same rationale as we would to minimize a function of only one variable. The gradient of the objective function is:
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| : <math> \nabla_G \phi = G^\mathrm{T} G m - G^\mathrm{T} d = 0 \, </math>
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| where ''G''<sup>T</sup> denotes the [[matrix transpose]] of ''G''. This equation simplifies to:
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| : <math> G^\mathrm{T} G m = G^\mathrm{T} d \, </math>
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| After rearrangement, this becomes:
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| : <math> m = (G^\mathrm{T} G)^{-1} G^T d \, </math>
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| This expression is known as the Normal Equation and gives us a possible solution to the inverse problem. It is equivalent to [[Ordinary Least Squares]]
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| : <math>\hat\beta = (X^\mathrm{T}X)^{-1} X^\mathrm{T}y</math>
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| Additionally, we usually know that our data has random variations caused by random noise, or worse yet coherent noise. In any case, errors in the observed data introduces errors in the recovered model parameters that we obtain by solving the inverse problem. To avoid these errors, we may want to constrain possible solutions to emphasize certain possible features in our models. This type of constraint is known as [[Regularization (mathematics)|regularization]].
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| ==== Mathematical ==== | |
| One central example of a linear inverse problem is provided by a [[Fredholm integral equation|Fredholm]] first kind [[integral equation]].
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| : <math> d(x) = \int_a^b g(x,y)\,m(y)\,dy </math>
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| For sufficiently smooth <math>g</math> the operator defined above is [[Compact operator|compact]] on reasonable [[Banach space]]s such as [[Lp space|''L''<sup>''p''</sup> space]]s. Even if the mapping is [[Injective function|injective]] its [[Inverse function|inverse]] will not be continuous. (However, by the bounded inverse theorem, if the mapping is bijective, then the inverse will be bounded (i.e. continuous).) Thus small errors in the data <math>d</math> are greatly amplified in the solution <math>m</math>. In this sense the inverse problem of inferring <math>m</math> from measured <math>d</math> is [[Ill-posed problem|ill-posed]].
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| To obtain a numerical solution, the integral must be approximated using [[Numerical integration|quadrature]], and the data sampled at discrete points. The resulting system of linear equations will be [[Condition number|ill-conditioned]].
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| Another example is the inversion of the [[Radon transform]]. Here a function (for example of two variables) is deduced from its integrals along all possible lines. This is precisely the problem solved in image reconstruction for [[X-ray]] [[Computed axial tomography|computerized tomography]]. 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.
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| A final example related to the [[Riemann Hypothesis]] was given by Wu and Sprung, the idea is that in the Semiclassical (old) Quantum theory the inverse of the potential inside the Hamiltonian is proportional to the [[half-derivative]] of the eigenvalues (energies) counting function n(x)
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| ==Non-linear inverse problems==
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| An inherently more difficult family of inverse problems are collectively referred to as non-linear inverse problems.
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| Non-linear inverse problems have a more complex relationship between data and model, represented by the equation:
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| :<math>\ d = G(m).</math>
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| Here <math>G</math> is a non-linear operator and cannot be separated to represent a linear mapping of the model parameters that form <math>m</math> into the data. In such research, the first priority is to understand the structure of the problem and to give a theoretical answer to the three Hadamard questions (so that the problem is solved from the theoretical point of view). It is only later in a study that regularization and interpretation of the solution's (or solutions', depending upon conditions of uniqueness) dependence upon parameters and data/measurements (probabilistic ones or others) can be done. Hence the corresponding following sections do not really apply to these 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]] problems, after the seminal work of the Russian mathematical school ([[Mark Grigoryevich Krein|Krein]], [[Israel Gelfand|Gelfand]], Levitan, [[Vladimir Marchenko|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).
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| 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]]s and [[eigenfunction]]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]]".
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| 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).
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| ==Applications ==
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| Inverse problem theory is used extensively in weather predictions, and oceanography. Another important application is constructing computational models of oil reservoirs. <ref> http://www.sciencedirect.com/science/article/pii/S0920410513003227</ref>
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| == Mathematical considerations ==
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| Inverse problems are typically [[ill-posed problem|ill posed]], as opposed to the [[well-posed problem]]s more typical when modeling physical situations where the model parameters or material properties are known. Of the three conditions for a [[well-posed problem]] suggested by [[Jacques Hadamard]] (existence, uniqueness, stability of the solution or solutions) the condition of stability is most often violated. In the sense of [[functional analysis]], the inverse problem is represented by a mapping between [[metric space]]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 ''[[condition number|ill-conditioned]]''. In these cases, [[regularization (mathematics)|regularization]] may be used to introduce mild assumptions on the solution and prevent [[overfitting]]. Many instances of regularized inverse problems can be interpreted as special cases of [[Bayesian inference]].
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| ==Inverse problems societies==
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| *[http://www.inverse-problems.net/ Inverse Problems International Association]
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| *[http://venda.uef.fi/research/FIPS/ Finnish Inverse Problems Society]
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| ==See also==
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| * [[Atmospheric sounding]]
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| * [[Data assimilation]]
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| * [[Mathematical geophysics]]
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| * [[Backus–Gilbert method]]
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| * [[Optimal estimation]]
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| * [[Tikhonov regularization]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * Chadan, Khosrow & Sabatier, Pierre Célestin (1977). ''Inverse Problems in Quantum Scattering Theory''. Springer-Verlag. ISBN 0-387-08092-9
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| * Aster, Richard; Borchers, Brian, and Thurber, Clifford (2012). ''Parameter Estimation and Inverse Problems'', Second Edition, Elsevier. ISBN 0123850487, ISBN 978-0123850485
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| *{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 19.4. Inverse Problems and the Use of A Priori Information | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=1001}}
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| ==External links==
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| *[http://www.mth.msu.edu/ipnet/ Inverse Problems Network]
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| *[http://www.me.ua.edu/inverse/ Inverse Problems page at the University of Alabama]
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| *[http://www.ipgp.jussieu.fr/~tarantola/ Albert Tarantola's website, including a free PDF version of his Inverse Problem Theory book, and some on-line articles on Inverse Problems]
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| *[http://staff.washington.edu/aganse/invresources/index.html Andy Ganse's Geophysical Inverse Theory Resources Page]
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| *[http://math.tkk.fi/inverse-coe Finnish Centre of Excellence in Inverse Problems Research]
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| *[http://www.intelnics.com/opennn OpenNN: Open Neural Networks Library]
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| ===Academic journals===
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| There are four main academic journals covering inverse problems in general.
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| * [http://www.iop.org/EJ/journal/IP Inverse Problems]
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| * [http://www.reference-global.com/loi/jiip Journal of Inverse and Ill-posed Problems]
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| * [http://www.tandf.co.uk/journals/titles/17415977.asp Inverse Problems in Science and Engineering]
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| * [http://aimsciences.org/journals/ipi/ipi_online.jsp Inverse Problems and Imaging]
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| In addition there are many journals on medical imaging, geophysics, non-destructive testing etc. that are dominated by inverse problems in those areas.
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| {{DEFAULTSORT:Inverse Problem}}
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| [[Category:Inverse problems| ]]
| |
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