# Seminars before September 2013

## Inverse Problems with Experimental Data

Traditionally numerical performance of algorithms for various inverse problems is verified on computationally simulated data. Basically this is because it is not easy to obtain experimental data. Recently our research group at University of North Carolina at Charlotte has built an experimental apparatus for collecting backscattering data of electromagnetic waves propagation. We have learned that there is a huge mismatch between thes data and the theory. In purely mathematical terms this means thousands percent of the noise. Therefore, to make mathematical theory working, it is necessary to pre-process the data. So that the preprocessed data would look somehow similar with computational simulations. Results of performance of the globally convergent inverse algorithm of [1] on these and other real data will be presented.

[1]. L. Beilina and M.V. Klibanov. Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012

## Blood flow modelling in microfluidic devices with biomedical applications

In the treatment of cancer, for exact diagnosis it is crucial to know the amount of circulating tumor cells (CTC) in the peripheral blood of the patients. Due to their rare occurrence, CTC need to be filterred in the blood sample.

We model the flow of blood inside microfluidic channels that will act as filtering devices. The modelling is done on the level of particular blood cells immersed in the blood plasma. We create a functional computational model including fluid-structure interactions, elastic properties of the blood cells and collisions between the immersed objects. The focus of the presentation will be put on the representation of the elastic properties. We raise several questions about the modelling of the cell adhesion to antibody-covered surfaces.

## Approximate Global Convergence and Adaptivity for Hyperbolic Coefficient Inverse Problems

The central question in the numerical solution of a Coefficient Inverse Problem is the question of obtaining a good approximation for the unknown coefficient without any advanced knowledge of any point in a small neighborhood of this coefficient. As soon as this approximation is obtained, Newton-like methods can refine it. This is an ENORMOUSLY challenging question. Especially under the condition that the corresponding numerical method should be numerically efficient: otherwise there is no point to work on.

Because of this challenge, it is inevitable that some reasonable approximations should be made. On the other hand, because of these approximations, it is crucial that the numerical method should be verified computationally. Roughly speaking, the above is what "approximate global convergence" is about.

In the past few years we have developed several versions of such a numerical methods. Their convergence to the exact solution is guaranteed within framework of a certain approximate mathematical model.

These algorithms were verified on both computationally simulated and experimental data. Especially on the most challenging case of blind experimental data. Computational results are accurate.

## Comparative analysis of typical approaches for inverse source problems with boundary and final time measured output data for parabolic equations

This lecture presents a systematic study of typical approaches for inverse source problems for the separated form F(x)H(t) source terms heat conduction (or linear parabolic) equation. 1. Inverse problems of determination the spacewise source function F(x) (ISPF). 2. Inverse problems of determination the time dependent source function H(t) (ISPH). The following approaches, commonly used in literature are analysed for each inverse problem: adjoint problem approach based on weak solution theory for parabolic PDEs; Fourier analysis; Collocation method. The adjoint problem approach permits one to deriive gradients of all corresponding cost functionals via the solutions of appropriate adjoint problems. As a result, a gradient method formally applicable to each inverse problem. However, as computational experiments show, for inverse source problems of type ISPH, it is not possible. Collocation method, applied to all above problems, show that these inverse problems have different condition numbers. Fourier analysis permits to show that ISPF3 has a unique solution.

## Nonlinear inverse problems in magnetic field synthesis

Magnetic field synthesis problems have been discussed at length in scientific literature but they still remain as a topic of research in electrical engineering, physics and medical applications. In these disciplines, there is often a need to design an electromagnetic arrangement which can generate a magnetic field of required distribution. Such an arrangement can also work as an active shield. The aim of an active shield is to generate a specified magnetic field which counteracts the external magnetic fields in a protected region. This presentation will describe the methods of magnetic field synthesis in an axisymmetric region. It is assumed that the region of interest is surrounded by a cylindrical solenoid with an electrical current. Two cases will be considered, namely: magnetic field synthesis on a olenoid's axis and magnetic field synthesis in an axisymmetric three-dimensional finite region. Two independent methods of magnetic fields synthesis, i.e. iteratively regularized Gauss-Newton method and Genetic Algorithm coupled with Bezier curves-based method, will be discussed and compared in this presentation.

## Electromagnetic heating of metals and its selected applications

Physical backgrounds of electromagnetic heating of solid and molten metals. Examples of typical industrial and other applications (direct heating, induction heating, induction hardening and levitation melting of solid metals, dosing, pumping and stirring of molten metals). Continuous mathematical models of these processes, coupling of particular physical fields (current or electromagnetic field, temperature field, in specific cases also field of thermoelastic displacements). Possibilities of their numerical solution. Presentation of the fully adaptive higher-order finite element method developed in our group. Illustrative examples and further activities in the domain.

## Application of level set method for groundwater flow problems

The possibilities of level set methods for modelling of moving interfaces and boundaries in groundwater flow and transport problems will be discussed. Particular application of groundwater flow with moving water table will be introduced in details. The main ingredients of level set methods like computations of signed distance function, extrapolation of missing data and solution of advection equation will be described. Several examples of practical relevance will be presented.

## Curve evolution models - numerical solution and applications

n this talk we present 2D and 3D Lagrangean curve evolution models used for online semi-automatic medical image segmentation, simulations of wind-driven forest fire fronts and finding an ideal path of camera in virtual colonoscopy. In all these models, the curve is driven by a properly designed external vector field, the motion is regularized by curvature and numerical computations are stabilized by a suitable asymptotically uniform tangential redistribution in 2D and in 3D. We also present new fast algorithm for treatment of topological changes in Lagrangean approach to curve evolution which is important e.g. in forest fire simulations. Our formulations of 2D and 3D curve evolution models are based on intrinsic advection-diffusion equations with a driving force discretized by the flowing finite volume method allowing large time steps without losing numerical stability. This is a common work with Jozef Urban and Martin Balazovjech.

## A computational multiscale approach for multiphase porous media

We present a multiscale method for multiphase porous media flow based on numerical homogenization. The multiscale algorithm consists of a pore scale phase-field multiphase flow solver coupled to a macroscopic finite volume solver. The coupling between the solvers is done through a macroscopic pressure gradient which enters the pore-scale simulations and averaged microscopic fluxes which are used in the finite volume solver. The method is able to handle arbitrary number of fluid phases and allows to include nonlinear effects such as contact angles and surface tension in a straightforward way. For single phase and simplified two-phase flow problems the approach is consistent with existing homogenization results.

## Thermal Explosion in Polydisperce Fuel Spray (probabilistic approach)

We propose a new, simplified model of the thermal explosion in a combustible gaseous mixture containing vaporizing fuel droplets of  different radii (polydisperse). The polydispersity is modeled using a probability density function (PDF) that corresponds to the initial distribution of fuel droplets.  This approximation of polydisperse spray is more accurate than the traditional ‘parcel’ approximation and permits an analytical treatment of the simplified model. Since the system of the governing equations represents a multi-scale problem, the method of invariant (integral) manifolds is applied. An explicit expression of the critical condition for thermal explosion limit is derived analytically. Numerical simulations demonstrate an essential dependence of these thermal explosion conditions on the PDF type and represent a natural generalization of the thermal explosion conditions of the classical Semenov theory.

## Stochastic approaches in computational electromagnetism

In the field of numerical modelling, the input data (dimensions, material propertiess, external inputs, etc.) are often supposed to be known exactly. The model as well as its outputs are then deterministic. However, the knowledge of input data arises from a set of assumptions. In the real world, input data are stochastic inputs. The dimensions of any device are known within a given uncertainty (tolerance) due to imperfections in the manufacturing process (machining. casting, punching, etc.). The characteristics of materials are also time dependant. The effect of ageing is very difficult to characterize and to model. In addition, uncertainties regarding the composition materials, and its environment (humidity, pressure, temperature, etc.) that influence the characteristics (by means of oxidation for example) are often only partially known. Therefore, the behaviour of materials also obey stochastic laws. In most problems, this stochastic aspect can be neglected. However, if the inputs exert significant influence on model outputs, this assumption cannot be used, particularly if failure analysis is one of the tasks of the model. With advances in computer performance and progresses in Applied Mathematics, numerical models become more and more accurate and the error due to stochastic effects can no longer be neglected in comparison to other errors. This then poses the interest and the need of increasing the deterministic model accuracy by the including stochastic properties of inputs. Stochastic models can be more suitable. There are many models based on the numerical solution of partial derivative equations using numerical methods like the Finite Volume Method, the Finite Element Method, etc. In electromagnetism, the Finite Element Method is widely used to solve Maxwell’s equations. However, in electromagnetism, uncertainties can be encountered either in the behaviour of materials or in electromagnetic sources or in dimensions. The seminar will focus on some stochastic numerical methods which allow broadcasting uncertainties on behaviour laws through a Finite Element Model in static electromagnetism. The presentation will be illustrated by several examples.

## Higher order Sobolev estimates and nonlinear stability of stationary solutions for the mean the curvature flow with triple junction

We are interested in the motion of a network of three planar curves with a speed proportional to the curvature of the arcs, having perpendicular intersections with the outer boundary and a common intersection at a triple junction. We derive higher order energy estimates yielding a priori estimates for the H2-norm of the curvature of moving arcs. As a consequence of these estimates we will be able to prove exponential decay of the H2-norm of the curvature. As a consequence, we will show that a linear stability criterion due to Ikota and Yanagida is also sufficient for nonlinear stability of stationary solutions for curvature flow with triple junction. This is a joint work with Harald Garcke and Yoshihito Kohsaka.

## Nonlinear and Multiscale Problems in Low-frequency Electromagnetism

This talk is devoted to the study of two distinctive topics in low-frequency electromagnetism. In the first part, a nonlinear eddy current problem is studied. The second part studies the properties of a material composed from particles of different electromagnetic characteristics. To do so, two different approaches are used. Homogenization, particularly the theory of two-scale convergence to obtain a rigorous analytical model and Heterogeneous Multiscale Method which provides us with a numerical approximation of the homogenized properties of composite materials.

## Applications of Galerkin methods in solid mechanics

Spatially-discontinuous Galerkin methods constitute a generalization of weak formulations, which allow for discontinuities of the problem unknowns in its domain interior. This is particularly appealing for problems involving high-order derivatives, since discontinuous Galerkin methods can also be seen as a means of enforcing higher-order continuity requirement as for non-linear Kirchhoff-Love shell theories. The resulting new one-field formulation takes advantage of the weak enforcement in such a way that the displacements are the only discrete unknowns, while the C1 continuity is enforced weakly.

## On numerical methods for direct and inverse problems in electromagnetism

This talk is devoted to the study of processes in the propagation of electromagnetic fields. We deal with direct problems as well as with inverse ones, low-frequency electromagnetism is discussed and eventually the wave propagation problem in high-frequency domain is investigated. First we present a time dependent eddy current model in electromagnetism with all necessary background for this type of problem. Second, we focus on the linearization and the full discretization of a slightly modified problem. In this case the non-linear relation between the magnetic and the electric field on the boundary is supposed to be Lipschitz continuous. The problem is formulated in the high-frequency domain and includes the study of electromagnetic waves and propagation of energy through matter. The third part is devoted to inverse problems in low-frequency electromagnetism.

## Singular Perturbed Vector Fields and Model Reduction

The problem of modelling of complex systems arising in combustion and chemical kinetics requires more and more sophisticated methods of qualitative study and/or numerical simulations. During the last decades the concept of invariant slow manifolds has proven to be an efficient tool for such models. In particular, it allows to decompose the original, complicated model to a number of low-dimensional submodels (models decomposition). In order to evaluate such a decomposition, the concept of Singular Perturbed Vector Fields (SPVF) has been suggested a few years ago. Roughly speaking, it is a coordinate-free version of singularly perturbed systems that is adopted to evaluation of the "hidden" slow-fast structure which is typical for real combustion and chemical kinetic models. A numerical algorithm based on the Singular Perturbed Vector Fields concept is presented. Some model examples and applications to combustion models will be discussed.

## Assessment and analysis of highly irregular data

Often data suffer from various imperfections, involving censoring, high correlation, small samples or missing data. In our talk we illustrate possible remedies for such a situations, and we will address the exact scale and homogeneity testing for complete generalized gamma samples, censored exponential samples, samples with missing data or information, and correlated data.

## An alternating procedure for boundary data identification for the Laplace equation in semi-infinite regions

We consider a Cauchy problem for the Laplace equation in a two dimensional semi-infinite region containing a bounded inclusion. The Cauchy data are given on the unbounded part of the boundary of the region and the aim is to find the solution on the boundary of the inclusion. To reconstruct the solution on the inclusion we employ the alternating method proposed by V. A. Kozlov and V. G. Maz'ya in 1989 for general strongly elliptic and formally self-adjoint systems. In each iteration step well-posed mixed boundary value problems for the Laplace equation are solved in the unbounded domain. For the numerical implementation of this procedure an efficient boundary integral equation method is outlined based on the indirect variant of the boundary integral equation approach. The mixed problems are each reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are discussed showing the feasibility of the proposed method. The results presented are a joint work together with R. Chapko from the Ivan Franko National University of L'viv in the Ukraine.