Inverse problem

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An inverse problem in science is the process of computing a series of observations of the causal factors that produce them: for example, computing images in X-ray computed tomography, reconstructing the source in acoustics, or calculating the Earth's density from the measurement of the gravitational field.

This is called an inverse problem because it starts with the result and then calculates the cause. This is the opposite of the forward problem, which starts with the cause and then calculates the result.

The inverse problem is some of the most important mathematical problems in science and mathematics because they tell us about parameters we can not observe directly. They have wide applications in optics, radar, acoustics, communication theory, signal processing, medical imaging, computer vision, geophysics, oceanography, astronomy, remote sensing, natural language processing, machine learning, nondestructive testing, and many other fields.

Video Inverse problem

Histori

One of the earliest examples of solutions to inverse problems invented by Hermann Weyl and published in 1911, described the asymptotic behavior of the eigenvalues ​​of Laplace-Beltrami carriers. Today known as Weyl's law , perhaps the most easily understood as an answer to the question of whether it is possible to hear the form of a drum. Weyl suspected that the eigenfrequencies of the drums would be related to the area and the perimeter of the drum with a certain equation, which was later corrected by many mathematicians.

Invers problems are then touched by Soviet-Armenian physicist Viktor Ambartsumian.

While still a student, Ambartsumian studied thoroughly the theory of atomic structure, the formation of energy levels, and Schrödinger's equations and properties, and when he mastered the theory of eigenvalues ​​of differential equations, he showed a clear analogy between the discrete energy levels and the eigenvalues ​​of the differential equations. He then asked: given a family of eigenvalues, is it possible to find a form of equation whose eigenvalues ​​they are? Basically Ambartsumian is checking the Sturm-Liouville problem upside down, which deals with determining the vibrating string equation. The paper was published in 1929 in the German physics journal Zeitschrift fÃÆ'¼r Physik and remains unclear for a long time. Describing this situation after decades, Ambartsumian said, "If an astronomer publishes articles with mathematical content in physics journals, then chances are that will happen is forgotten."

Nevertheless, towards the end of World War II, this article, written by the 20-year-old Ambientumian, was discovered by Swedish mathematicians and formed a starting point for the entire research area of ​​the inverse problem, the foundation of the whole discipline.

Maps Inverse problem

Conceptual understanding

The inverted problem can be conceptually formulated as follows:

Data -> Model parameters

The inverse problem is considered "the inverse" of the forward problem that connects the model parameters with the data we observe:

Model parameters -> Data

The transformation from the data to the model parameter (or vice versa) is the result of the physical system interaction with the object we want to infer the property about. In other words, transformation is the physics that connects the physical quantity (ie the model parameters) with the observed data.

The table below shows some examples of physical systems, the physics that govern, the physical quantities we are interested in, and what we actually observe.

Linear algebra is useful in understanding the physical and mathematical constructs of the inverse problem, due to the transformation or "mapping" of data to the model parameters.

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General statement of issue

Tujuan dari masalah invers adalah untuk menemukan parameter model terbaik ${\ displaystyle m}  ÂÂ$ seperti itu (setidaknya kira-kira)

${\ displaystyle \ d = G (m)}  ÂÂ$

di mana ${\ displaystyle G}  ÂÂ$ adalah operator yang menjelaskan hubungan eksplisit antara data yang diamati, ${\ displaystyle d}  ÂÂ$ , dan parameter model. Dalam berbagai konteks, operator ${\ displaystyle G}  ÂÂ$ disebut operator maju , operator pengamatan , atau fungsi pengamatan . Dalam konteks yang paling umum, G mewakili persamaan pengaturan yang menghubungkan parameter model dengan data yang diamati (yaitu, fisika yang mengatur).

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Masalah invers linier

Dalam kasus masalah invers linear diskrit yang menggambarkan sistem linear, ${\ displaystyle d}  ÂÂ$ (data) dan ${\ displaystyle m}  ÂÂ$ (model terbaik) adalah vektor, dan masalahnya dapat ditulis sebagai

${\ displaystyle \ d = Gm}  ÂÂ$

di mana ${\ displaystyle G}  ÂÂ$ adalah matriks (operator), sering disebut matriks observasi .

Contoh

Medan gravitasi bumi

Only some physical systems are really linear with respect to model parameters. One of the systems of geophysics is from the Earth's gravitational field. Earth's gravity field is determined by the distribution of Earth's density below the surface. Since the lithology of the Earth changes significantly, we can observe minute differences in the Earth's gravitational field on the Earth's surface. From our understanding of gravity (Newton's Law of Gravitation), we know that the mathematical expression for gravity is:

${\ displaystyle d = a = {\ frac {KM} {r ^ {2}}}}$

di mana ${\ displaystyle a}  ÂÂ$ adalah ukuran akselerasi gravitasi lokal, ${\ displaystyle K}  ÂÂ$ adalah konstanta gravitasi universal, ${\ displaystyle M}  ÂÂ$ adalah massa lokal (yang terkait dengan kepadatan) dari batuan di bawah permukaan dan ${\ displaystyle r}  ÂÂ$ adalah jarak dari massa ke titik pengamatan.

Sistem ini memiliki lima persamaan, ${\ displaystyle G}  ÂÂ$ , dengan lima yang tidak diketahui, ${\ displaystyle m}  ÂÂ$ . Untuk memecahkan parameter model yang sesuai dengan data kami, kami mungkin dapat membalikkan matriks ${\ displaystyle G}  ÂÂ$ untuk secara langsung mengonversi pengukuran ke dalam parameter model kami. Sebagai contoh:

${\ displaystyle m = G ^ {- 1} d \,}  ÂÂ$

However, not all square matrices can be reversed ( ${\ displaystyle G}$ is almost never reversible). This is because we are not guaranteed to have enough information to uniquely define solutions for given equations unless we have independent measurements (ie each measurement adds unique information to the system). It is important to note that in most physical systems, we never have enough information to uniquely limit our solution because the observation matrix does not contain any unique equations. From the perspective of linear algebra, matrix ${\ displaystyle G}$ is a ranking deficiency (ie has a zero eigen value), meaning it can not be reversed. Furthermore, if we add additional observations to our matrix (ie more equations), then matrix ${\ displaystyle G}$ is no longer square. Even then, we are not guaranteed to have full rank in the observation matrix. Therefore, most of the inverse issues are considered undetermined, which means that we have no unique solution to the inverse problem. If we have a full ranking system, then our solution may be unique. Overdetermined systems (more equations than unknowns) have other problems.

Since we can not directly reverse the observation matrix, we use a method of optimization to solve the inverse problem. To do so, we define the purpose, also known as the objective function, for the inverse problem. The goal is functional that measures how close the predicted data of the model is recovered according to the observed data. In the case where we have perfect data (ie no sound) and perfect physical understanding (ie we know physics) then the restored model must match the perfectly observed data. Standard Object function, ${\ displaystyle \ phi}$ , usually in the form of:

${\ displaystyle \ phi = || d-Gm || _ {2} ^ {2} \,}$

which represents the L-2 norm of mismatch between observed data and predicted data from the model. We use the L-2 norm here as a general measure of the distance between the predicted data and the observed data, but other norms may be used. The purpose of the objective function is to minimize the difference between the predicted and observed data.

To minimize the purpose of the goal (ie solving the inverse problem) we calculate the objective function gradient using the same reason as we will minimize the function of only one variable. The objective function gradients are:

${\ displaystyle \ nabla _ {m} \ phi = 2 (G ^ {\ mathrm {T}} Gm-G ^ {\ mathrm {T}} d) = 0 \,} Â Â$

di mana G ^T menunjukkan transposisi matriks G . Persamaan ini menyederhanakan:

${\ displaystyle G ^ {\ mathrm {T}} Gm = G ^ {\ mathrm {T}} d \,}  ÂÂ$

Setelah penataan ulang, ini menjadi:

${\ displaystyle m = (G ^ {\ mathrm {T}} G) ^ {- 1} G ^ {T} d \,}  ÂÂ$

Ekspresi ini dikenal sebagai Persamaan Normal dan memberi kita solusi yang mungkin untuk masalah invers. Ini setara dengan kuadrat terkecil biasa:

${\ displaystyle {\ hat {\ beta}} = (X ^ {\ mathrm {T}} X) ^ {- 1} X ^ {\ mathrm {T}} y}  ÂÂ$

In addition, we typically know that our data has random variations caused by random noise, or worse coherent noise. However, errors in the observed data introduce errors in the recovered model parameters by solving the inverse problem. To avoid this error, we may want to limit possible solutions to emphasize certain features that may be in our model. This type of constraint is known as regularization.

Fredholm integral

Salah satu contoh sentral dari masalah invers linear diberikan oleh persamaan integral Fredholm dari jenis pertama:

${\ displaystyle d (x) = \ int _ {a} ^ {b} g (x, y) \, m (y) \, dy}  ÂÂ$

To pretty smooth ${\ displaystyle g}$ The above defined operator is compact in Banach's reasonable space such as < space. Even if the mapping is injected, the opposite will not continue. (However, with the inverted theorem, if the mapping is bijective, then the reverse will be limited (ie continuous). So small errors in data ${\ displaystyle d}$ is strongly reinforced in the solution ${\ displaystyle m}$ . In this sense, the inverse problem concludes ${\ displaystyle m}$ from measurement ${\ displaystyle d}$ is incorrect.

To obtain a numerical solution, the integrals must be approximated by using quadrature, and the data sampled at discrete points. The resulting system of linear equations will be bad.

Computed tomography

Another example is the invasion of Radon transformation, essential for the reconstruction of tomography for computed tomography X-rays. Here the function (originally of two variables) is inferred from its integral along all possible lines. Though from a theoretical point of view many linear inverse problems are well understood, problems involving radon transformation and its generalizations still present many theoretical challenges with still unresolved data adequacy questions. These problems include incomplete data for x-ray transformation in three dimensions and problems involving generalizing x-ray transformations into the tensor field. The solutions explored include Algebra Reconstruction Techniques, filtered reverse projections, and, as computing power has increased, recurrent reconstruction.

Riemann Hypothesis

The final example associated with the Riemann hypothesis is given by Wu and Sprung, the idea is that in the old semiclassical quantum theory the opposite of the potential in the Hamiltonian is proportional to the half-derivative of the eigen (energy) value of the calculating function n (x).

Matching Permeability in Shale-gas Reservoir

To accurately reproduce permeability, a new method based on a combination of Metropolis-Hastings and a genetic algorithm. The new method learns from the previously realized shale realizations and produces models that match the existing permeability data.

Deconvolution

The classic example of an inverse problem is the deblurring image (or signal), ie, the deconvolution problem in the plane. In such a case, the forward problem is the convolution with the kernel of the graduation convention. Considering the integral equation (from Freholm type 1):

$Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âg()\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂxÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â)Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\mathrm{?}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂK()\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âx,Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂyÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â)Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âf()\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂyÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â)Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂdÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Ây,Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\text{Ã,}Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂxÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â?Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â{R}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}^{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â2}}$, Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ? Â Â Â Â Â Â Â Â ? Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â R Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 2 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â , Â Â Â Â Â Â Â Â Â Â (X) = \ int_ {\ Omega} K (x, y) f (y) dy, \ x \ in \ mathbb {R } ^ {2,} \ Omega \ subset \ mathbb {R} ^ {2},} Â Â

di mana ${\ displaystyle K}  ÂÂ$ adalah kernel, ${\ displaystyle x = (x_ {1}, x_ {2})}  ÂÂ$ dan ${\ displaystyle y = (y_ {1}, y_ {2})}  ÂÂ$ . Masalah invers adalah merekonstruksi gambar asli ${\ displaystyle f (x)}  ÂÂ$ berdasarkan gambar yang berisik dan buram ${\ displaystyle g (x)}  ÂÂ$ .

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Masalah inversi non-linear

A more difficult family of inverse problems is collectively referred to as a non-linear inverse problem.

Masalah invers non-linear memiliki hubungan yang lebih kompleks antara data dan model, yang ditunjukkan oleh persamaan:

${\ displaystyle \ d = G (m).}  ÂÂ$

Di sini ${\ displaystyle G}  ÂÂ$ adalah operator non-linear dan tidak dapat dipisahkan untuk mewakili pemetaan linier dari parameter model yang membentuk ${\ displaystyle m}  ÂÂ$ ke dalam data. Dalam riset semacam itu, prioritas pertama adalah memahami struktur masalah dan memberikan jawaban teoritis terhadap tiga pertanyaan Hadamard (sehingga masalah diselesaikan dari sudut pandang teoritis). Hanya kemudian dalam sebuah studi yang regularisasi dan interpretasi solusi (atau solusi, tergantung pada kondisi keunikan) solusi pada parameter dan data/pengukuran (probabilistik atau yang lain) dapat dilakukan. Oleh karena itu bagian berikut tidak benar-benar berlaku untuk masalah ini. Sedangkan masalah invers linier benar-benar dipecahkan dari sudut pandang teoritis pada akhir abad kesembilan belas, hanya satu kelas masalah invers nonlinear begitu sebelum tahun 1970, yaitu inverse spectral dan (satu dimensi ruang) inverse hamburan masalah, setelah seminalis karya sekolah matematika Rusia (Kerin, Gelfand, Levitan, Marchenko). Sebuah tinjauan besar hasil telah diberikan oleh Chadan dan Sabatier dalam buku mereka "Masalah Terbalik dari Teori Hamburan Kuantum" (dua edisi dalam bahasa Inggris, satu dalam bahasa Rusia).

In this kind of problem, the data is a property of the spectrum of a linear operator that represents scattering. The spectrum is made up of eigenvalues ​​and eigen functions, forming together a "discrete spectrum", and generalization, called the continuous spectrum. The overwhelming physical point is that scattering experiments provide information only on the continuous spectrum, and that knowing the full spectrum is important and sufficient in recovering the scattering operator. Therefore we have invisible parameters, much more interesting than zero space that has similar properties in linear inverse problem. In addition, there is a physical movement in which the operator's spectrum is preserved as a consequence of the movement. This phenomenon is governed by a special nonlinear partial differential evolutionary differential equation, for example the Korteweg-de Vries equation. If the operator spectrum is reduced to a single eigenvalue, the same motion is the movement of a bulge that spreads at a constant speed and without deformation, a solitary wave called a "soliton".

The perfect signal and its generalizations for the Korteweg-de Vries equation or other integral nonlinear differential equations are very interesting, 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 sciences (acoustics, mechanics, quantum mechanics, electromagnetic scattering - especially radar, seismic sounds, and almost all imaging modalities).

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Apps

Reversed problem theory is widely used in weather prediction, oceanography, hydrology, and petroleum engineering.

The inverse problem is also found in the field of heat transfer, where surface heat flux is estimated to be out of temperature measured data in a rigid body.

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Mathematical considerations

Inversion problems are usually not generated, as opposed to well-posed problems are more common when modeling a physical situation in which model parameters or material properties are known. Of the three conditions for the problem posed by Jacques Hadamard (the existence, uniqueness, and stability of solutions or solutions), the conditions of stability are most often violated. In terms of functional analysis, the inverse problem is represented by the mapping between the metric spaces. While inverse problems are often formulated in unlimited dimensional spaces, limited number of measurements, and practical considerations for recovering only a number of unknown parameters, can cause recurring problems in discrete form. In this case the inverse problem will usually be pain . In this case, regularization can be used to introduce light assumptions on solutions and prevent overfitting. Many examples of regulated inversion problems can be interpreted as special cases of Bayesian inference.

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See also

• Atmosphere sound
• Backus-Gilbert Method
• Computed tomography
  • Algebra Reconstruction Technique
  • Filtered projection projections
  • Recurrent reconstruction
• Data assimilation
• Technical optimization
• Gray box model
• Math geophysics
• Optimal assessment
• Seismic inversion
• Tikhonov's regularization

Academic journals

Four major academic journals include inverted issues in general:

• Flipping Issues
• Journal of Inverse and Ill-posed Issues
• The Backslash in Science and Engineering
• Troubleshooting and Imaging

Many journals on medical imaging, geophysics, non-destructive testing, etc. Dominated by inverse problems in these areas.

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References

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References

• Chadan, Khosrow & amp; Sabatier, Pierre CÃÆ' © lestin (1977). Inverted Problems in Quantum Scatter Theory . Springer-Verlag. ISBNÃ, 0-387-08092-9
• Aster, Richard; Borchers, Brian, and Thurber, Clifford (2012). Parameter Estimates and Fixed Problems , Second Edition, Elsevier. ISBN: 0123850487, ISBN 978-0123850485
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 19.4.Inside Problems and Use of Prior Information". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.

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Further reading

• C. W. Groetsch (1999). Inverted Issue: Activity for Young Undergraduate . Cambridge University Press. ISBN: 978-0-88385-716-8.

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External links

• International Association Inversation Issues
• Eurasian Association on Reversal Problems
• Finnish Public Problems
• Network Issues Reversed
• [http://www.ipgp.jussieu.fr/~tarantola/Albert Tarantola's website, including
• Inverse Problems page at University of Alabama uding free PDF version of his Inverse Problem Theory, and some online articles on Reversed Problems]
• Inversion and Geosta Problems

  Source of the article : Wikipedia