# Applied Maths Research

Research activity

Imaging, image processing and image analysis deal with wide range of problems in which the relevant data are organized in a spatial way.

Imaging is related to image formation which involves the mathematical solution of physical processes in different media. Examples include Ultra-Sound (US), Computerized Tomography (CT), Magnetic Resonance Imaging (MRI),Sinthetic Aperture Radar (SAR), Microscopic Imaging and more.

Image processing deals with various application of image restoration and enhancement.Examples are de-noising, de-blurring, inpainting, segmentation etc.

Image analysis deals with the task of extracting information from images.Examples include stereo vision, motion analysis in video, classification and image retrieval. There is a large intersection with computer vision, computational learning, classification, categorization and shape analysis.

The applications are numerous: Medical imaging, security, entertainment and product quality are only part of the things that the private and public sectors are interested in. The size of the industry in Israel in which core technologies involve imaging, image processing and image analysis is very large and constantly increasing.

Numerical Analysis deals with developing and analyzing computational methods for calculating the numerical solution of mathematical problems.

A truly effective use of numerical analysis in applications requires both a theoretical knowledge of the subject and computational experience with it. The theoretical knowledge should include an understanding of both the original problem being solved and of numerical methods for its solution, including their derivation, error analysis, and an idea when they will perform well or poorly. Since many problems cannot be solved by a simple application of some standard software, one usually has to devise new methods or to adapt standard methods to the given situation. This requires a good theoretical foundations in numerical analysis.

The Numerical Analysis research in the Applied Mathematics Department at Tel Aviv University includes the following subjects:

- Numerical methods for hyperbolic PDE's,computational solution for the Navier-Stokes equations in aerodynamics, construction of high-order methods for general domains,construction of absorbing boundary conditions.
- Numerical Analysis of Elliptic Problems, Finite Elements.
- Multidimensional approximation and numerical integration.
- Numerical Analysis Education: A program for high school numerical analysis, involving computer lab experiments and an accompanying mathematical analysis.

In many areas of applications, especially in physics, phenomena are described by Partial Differential Equations (PDE).

The development of the theory of such equations often makes use of insights obtained in applications, and in turn sheds light on those areas. Research in PDEs includes both general theory and the in-depth study of particular equations arising in applications.

Aspects of the general theory of PDEs being investigated at TAU include:

- Systems of conservation laws.
- Evolution of singularities in hyperbolic systems.
- Energy methods for diffusion equations.
- Theory of Fluid Dynamics. A major field in the theory of PDE is the theory of Fluid Dynamics.

Here the subjects investigated are:

- The dynamics of an ideal incompressible fluid.
- Weak solutions and asymptotic behavior of 2-dimensional flows.
- Generalized fluid flows

Computations are today of crucial importance to many applications in science and engineering. Typical applications, within the department, include aircraft design, jet engine design, electromagnetic scattering, acoustics, weather forecasting and also medical research and applications involving flow in blood vessels.

The Applied Mathematics Department at Tel Aviv is considered an international center for the design, analysis and application of such numerical schemes.

Some of the topics actively being investigated are:

- Highly accurate computational solution for the Navier-Stokes equations in aerodynamics.
- Fast and efficient methods for solving the finite difference approximations to the steady state Navier-Stokes equations
- Applications to flows around aircraft, which may involve complex phenomena like shock-waves.
- Construction of high-order methods for Maxwell's equations in electrodynamics for general domains.
- Construction of high-order methods for the Helmholtz equations in general domains.
- Construction of absorbing boundary conditions for hyperbolic and elliptic partial differential equations.
- Application of the Finite Elements Method to elasticity.
- Application to fracture mechanics.
- Study of iterative solvers (multigrid) for PDEs.
- Stability and accuracy of numerical schemes.

The activity centers around modeling and analysis of behavior in dynamical systems driven by random processes.These include stochastic differential equations that describe chemical kinetics, the stock market, the physics of nano systems, communication systems, computer networks, and others.

The activity covers such diverse areas as the foundations of quantum mechanics and the theory of measurements, irreversible processes, path integrals, modeling super-conducting devices such as the Josephson junction and the DC-SQUID, activated processes, statistical physics, chemical physics, formation of patterns, nonlinear phenomena in plasma and fluids, kinetic theory of gases and more.

The activity centers around modeling, analysis, and simulations of two subjects. The first is in molecular biophysics and In particular in the theory of ionic permeation in protein channels embedded in the cell membrane. The research is directed at predicting the function of a protein channel from its structure.

The ultimate goal is to construct a computational model of a protein that can be used much the same way that mathematical modeling of aircraft isused in design and performance analysis. The second subject is the study of brain tissue’s geometry via diffusion-MRI.

Modern Design and manufacture is nowadays based on CAGD (Computer Aided Geometric Design) software tools. This field and its underlying mathematics has been developed along with the computer revolution in the last two decades. The CAGD group in Tel-Aviv University is involved in the the development and the analysis of new tools for Computer-Aided Geometric Design and Geometric Modeling.

The recent developments in multiresolution analysis, in subdivision algorithms and in triangulation techniques, open new possibilities for their applications in Computer Aided Geometric Design and in Computer Graphics. The new methods and their understanding is now in such an advanced state that calls for practical applications.

The current trend is to produce a new generation of software tools that will replace the existing ones in Geometric Modeling, with many possible applications in 3D Graphics, Animation, Robotics and Geometric Design.

The Tel-Aviv CAGD group has contributed to all aspects of the research and applications in these new directions, and is now planning new

research projects which are important to the development in this area.

Mathematical modeling lies at the interface between mathematics and applications. In the study of complex systems, such as those that arise in medicine and in economy, finding the `right' mathematical model is sometimes the most challenging part of the research effort.

The Applied Mathematics group is involved in modeling in various fields (in many cases, in collaboration with researchers from other departments) such as medicine, economy, computer science and laser-tissue interactions.

Control theory deals with various processes depending on input parameters called controls. The real-time information on the process is provided by various measurements called "outputs". The control task is to adjust the control inputs in time, so as to provide for needed process features.

Control theory is a highly applicable field. Practically all types of human activity involve control problems' solving. Engineers, mathematicians, specialists from the corresponding application area and software developers have to cooperate in solving typical practical control problems. No air-condition, washing machine, automobile engine, airplane, missile, robot, satellite, medical equipment, etc. can operate without numerous control blocks.

Mathematical control theory deals with mathematically formulated control problems. In particular, the classical control theory studies systems of linear differential equations linearly depending on controls. The classical problem is to define the controls as a function of the system state (so-called "feedback function"), so as to provide for the asymptotic system stabilization. Another classical problem is to make a certain function (so-called "output") of the state variables to track a signal measured in real time.

The standard approach is to build a mathematical model of the process and to develop a corresponding feedback control. Unfortunately, it often happens in practice that no exact model is available. Moreover some parts of the system like sensors and special units (so-called "actuators"), which transmit the controls to the system, are often not modeled at all. Such incomplete models are called uncertain. The question arises whether it is possible to treat a system as a “black box”, avoiding direct dependence on the exact model.

Discontinuous control has proved to be the main control theory tool to deal with grave system uncertainties. The first results were obtained in 1950s and were based on so-called sliding modes. Sliding modes are based on high-frequency-switching of discontinuous controls. The idea is to keep convenient artificial connections between the system coordinates, killing the system dimensions one-by-one. Such a connection is described by the equality of a certain function (sliding variable) to zero. The main drawback of the classical sliding mode is the high-frequency system vibration (the chattering effect) caused by control switching. Another problem is that the control should explicitly appear already in the first time derivative of the sliding variable.

The mentioned drawbacks of the classical sliding modes are removed by the high-order sliding-mode control theory matured at the department. Standard controllers have been developed solving the general tracking problem. Practically the only information which is needed, is the order of the sliding-variable derivative directly depending on the control. The developed controllers feature exactness and finite-time convergence, and, the control can be made smooth in time. The controllers are robust with respect to bounded output noises, discrete sampling and small delays. The approach has been recently proved to be also robust with respect to small system disturbances and unaccounted-for fast stable dynamics of sensors and actuators. The dangerous chattering effect caused by high-frequency switching can be excluded.

As an application, an exact arbitrary-order robust differentiator is obtained, featuring finite-time convergence and the optimal asymptotic accuracy in the presence of input noises. It has already found multiple theoretical and practical applications. The research performed at the department has leaded to successful practical applications in flight control. Recently the approach has been found effective in the most difficult case, when the very nature of the model is uncertain. In that case the system is treated as a “black box”, while only input-output relation features are axiomatically introduced. The approach

was applied to experimental automatic insulin diabetes treatment.