The main areas of activity of our laboratory are associated with the development of methods to solve inverse problems of synthesis and diagnosis. Mathematically the tasks considered fall into the category of ill-posed problems. The theory of the solution of ill-posed problems, which was developed in the late 20th century, is by far one of the most powerful results in the mathematics of the past century. The most important results in this theory were obtained by the mathematicians of Tikhonov's school, to which I have the honor to belong. The results obtained make it possible to develop methods for the approximate solution of ill-posed problems, assess the optimality of algorithms, estimate the error of the approximate solution, etc. [1,2]
We try not to limit our studies just to the development of algorithms. We took part in the development of synthetic-aperture radars for Earth surface control operating at centimeter-wave frequencies. The underlying idea of synthetic aperture is very simple. An orbiting spacecraft can record the signal of its radar reflected from the Earth surface. If performed over a several kilometers long segment of the trajectory, this procedure allows a virtual antenna several thousand times the size of the actual radar antenna to be synthesized on a computer, resulting in a thousand-fold improvement of the resolution of syntehtic-aperture radars.
Our laboratory took part in the development of x-ray tomographs for the control of microelectronic products. If an object can be studied from all sides, developing algorithms for the solution of inverse problems in x-ray diagnostics is not a particularly challenging task, and hundreds of efficient methods can be proposed in this case. The situation becomes much more challenging if the range of viewing angles is limited. In this case the inverse problem has a non-unique solution and the only approach is to use supplementary information. Such supplementary information may consist, e.g., in the fact that the object has a layered structure. We developed an x-ray tomograph capable of performing reliable quality contol of the layers of multilayer circuit boards with the sizes of up to 50×50 cm at a resolution of about 20 μm.
Inverse problems of the synthesis of flat optical elements hold a special place among the activities of our laboratory. As is well known, the first flat optical elements were proposed by Augustin Fresnel back in the 18th century. Currently, flat computer-generated diffractive elements are used to address a wide range of tasks of radiation shaping.
The most interesting achievements of flat computer-generated optics include flat optical elements used to combat counterfeiting. The synthesis of security elements operating at optical wavelengths consists in computing the microrelief of the optical element and actually shaping this microrelief. For the visible light wavelengths, the accuracy of microrelief shaping should be of about 10 – 20 nm. Such elements are referred to as "nanooptical" and are synthesized via electron-beam lithography. Protection against counterfeit is guaranteed by the highly knowledge-intensive nature of the technology and expensiveness of the equipment used for e-beam lithography. These solutions developed by our laboratory are used to protect documents in the Russian Federation.
The development of efficient methods to solve inverse problems of ultrasound tomography is currently a priority area of work for our laboratory. These studies are highly targeted toward the development of high-resolution ultrasound tomographs for differential diagnosis of breast cancer. It is one of the most pressing problems in the world, because cancer is the primary cause of female mortality. X-ray tomographs cannot be used for regular examinations in this case because of the high radiation exposure involved.
Several companies in the USA and Germany have been working in this direction and have reached the stage of prototype testing. Our solutions differ from those already proposed in that we suggest using substantially lower ultrasound frequencies. Our simulations show that a resolution of about 2 mm can be achieved with 5-mm wavelength even using stationary tomographs with no rotating elements and with few sources. Breakthrough results are made possible by the accurate computation of the gradient of the residual functional, which allows efficient iterative procedures to be developed for solving nonlinear inverse problems of wave tomography.
Wave tomography problems are computationally very expensive. Suffice it to say that the number of unknowns in the 3D nonlinear problem exceeds 10 000 000. Such problems cannot be solved without a supercomputer. We run our simulations on "Lomonosov" supercomputer of Moscow State University. It is obvious that general-purpose supercomputers cannot be used in real tomographs. At the same time, the algorithms developed for solving inverse problems of ultrasound tomography are well suited for GPU computing technology. GPU-based supercomputers have much lower cost for the same processing power, and can be incorporated into medical facilities.
Our laboratory team includes 10 people, half of whom hold academic degrees. All team members are highly skilled experts in the field of mathematical simulation, solution of inverse problems of mathematical physics, computer tomography, optics, electronics, and supercomputer technologies.