Inverse problem
They can be found in system identification, optics, radar, acoustics, communication theory, signal processing, medical imaging, computer vision,[1][2] geophysics, oceanography, astronomy, remote sensing, natural language processing, machine learning,[3] nondestructive testing, slope stability analysis[4] and many other fields.
A historical example is the calculations of Adams and Le Verrier which led to the discovery of Neptune from the perturbed trajectory of Uranus.
One of the earliest examples of a solution to an inverse problem was discovered by Hermann Weyl and published in 1911, describing the asymptotic behavior of eigenvalues of the Laplace–Beltrami operator.
[5] Today known as Weyl's law, it is perhaps most easily understood as an answer to the question of whether it is possible to hear the shape of a drum.
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."
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.
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.
Additional data can come from physical prior information on the parameter values, on their spatial distribution or, more generally, on their mutual dependence.
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.
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 of the objective function or equivalently, in the case where no regularization has been integrated, by the singular values of matrix
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.
When the forward map is compact, the classical Tikhonov regularization will work if we use it for integrating prior information stating that the
A mathematical analysis is required to make it a bounded operator and design a well-posed problem: an illustration can be found in.
[23] 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.
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[31] 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.
[34] 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.
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).
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).
The goal is to recover the diffusion coefficient in the parabolic partial differential equation that models single phase fluid flows in porous media.
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.
[43][44] 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.
But this tomography like problem is nonlinear, mainly because the unknown ray-path geometry depends upon the velocity (or slowness) distribution.
In order to see where the difficulties arise from, Chavent[45] proposed to conceptually split the minimization of the data misfit function into two consecutive steps (
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: Inverse problems, especially in infinite dimension, may be large size, thus requiring important computing time.
