Numerical Analysis and Mathematical Modelling NaM2


Date Title Abstract Contact
08/03/2016Distributed order fractional diffusion-wave equations with time delay on bounded domains: a numerical approach Ahmed Hendy
Ural Federal University, Yekaterinburg, Russia

Distributed order fractional diffusion equations are used recently in describing physical phenomena such as modeling of waves in a viscoelastic rod of finite length and to describe radial groundwater flow to or from a well.
Time delay occurs frequently in realistic world and it has been considered in numerous mathematical models, e.g., automatic control systems with feedback, population dynamics. In the simulation of dynamical systems, two effects (distribution of parameters in space and delay in time) are often existed. We consider a numerical scheme for a class of non-linear time delay fractional diffusion equation with distributed order in time.
This study covers the unique solvability, convergence and stability of the resulted numerical solution by means of the discrete energy method. The derivation of a linearized difference scheme is the main purpose of this study.
Ahmed Hendy
17/12/2015Filtration properties in porous media Jozef Kacur
Comenius University, Bratislava

Jozef Kacur
10/09/2015Source reconstruction from final data in the heat equation Tomas Johansson
Aston University, Birmingham

The inverse ill-posed problem of reconstructing a heat source in the parabolic heat equation from given knowledge of the solution at a final time is considered. An overview of a uniqueness result for spacewise dependent heat sources will be given together with a discussion about some counter examples showing that the conditions obtained for uniqueness cannot be relaxed. A numerical procedure for the stable determination of the source together with some numerical results will also be presented. The results obtained is joint work with prof. M. Slodicka.
Tomas Johansson
10/03/2015Introduction to fractional calculus Katarina Siskova
Ghent University, Ghent

The introduction to the theory of fractional integration and differentiation will be given. We consider the Riemann-Liouville and the Caputo fractional derivative. Basic properties of the considered fractional derivatives will be introduced, including the use of the Laplace transform, and the numerical approximation of a fractional derivative. The Mittag-Leffler function and its important role will be highlighted. Examples of applications in various branches will be given.
Katarina Siskova
18/12/2014Global convergence for inverse problems and phaseless inverse problems Michael V. Klibanov
University of North Carolina at Charlotte U.S.A.

In this talk three topics will be discussed. Corresponding papers were published in 2008-2014, also see These topic are:
1. A globally convergent numerical method of the first type for coefficient inverse problems with single measurement data. Both the theory and numerical results will be presented. Numerical results will be focused on the most challenging case of blind backscattering experimental data for buried targets.
2. A globally convergent numerical method of the second type type for coefficient inverse problems will be presented. This method is based on the construction of a globally strictly convex cost functional. The key element of this functional is the Carleman Weight Function.
3. The first solution of a long standing problem (since 1977). This is uniqueness of the 3-d coefficient inverse scattering problem in the case when only the modulus of the complex valued scattering wave field is measured, whereas the phase is unknown. In quantum inverse scattering only the differential cross-section is measured, which means the modulus. On the other hand, the entire theory of quantum inverse scattering is constructed for the case when both the modulus and phase of the scattering wave field are measured.
Michael V. Klibanov
18/12/2014Numerical Method for solving an boundary value inverse heat conduction problem Natalia Yaparova
South Ural State University (National Research University) Russia

In this talk two different approaches based on the Laplace and Fourier transforms will be discussed. Corresponding papers were published in 2013-2014.
Application of the Laplace transform makes it possible to obtain an integral equation describing the explicit dependence of the unknown boundary value function on the initial data at the other boundary. Regularization methods are then used to solve this equation. This eliminates the unstable procedure of numerical inversion of the Laplace transform in the computational process. The proposed method was used in a computational experiment to obtain a numerical solution of the inverse problem. Experimental error estimates of the obtained solutions show sufficient stability of these solutions.
The approach based on the projection regularization method for the direct and inverse Fourier transforms with respect to the time variable provides regularized solutions with guaranteed accuracy. The estimation of errors of these solutions are the exact with respect to the order. This property provided the basis for comparative analysis of the solutions obtained by the Laplace and Fourier transform methods.
The proposed methods were employed to carry out a computational experiment. The objectives of this experiment were to test effectiveness of the proposed approaches and to evaluate the errors of the regularized solutions provided by each approach. The computational results confirm the stability of the solutions obtained by these methods.
Natalia Yaparova
22/08/2014Numerical solutions of the inverse problem of pharmacokinetics Voronov Dmitriy Andreevich
Novosibirsk State University, Russia

Pharmacokinetics deals with kinetics of absorption, distribution, metabolism and excretion of drugs and their corresponding pharmacologic, therapeutic or toxic responses in man and animals.
What happens to the drug in the body can be visualized by considering the body as being made up of a large number of compartments, each of which has a volume where the drug is well mixed. Drug is then transferred between these compartments, either transported by the blood from one to another, or by passing an interior membrane in some body organ. We can visualize this whole process as a dynamic system described by a system of ordinary differential equations.
Parameter identifiability analysis for dynamic system ODE models addresses the question of which unknown parameters can be quantifies from given input-output data. The linear compartment models that we focus on in this report are never identifiable, except in the trivial case of a model with only one compartment. This forces us to look for identifiable reparametrization of our model. In this report we consider scaling reparametrization that is obtained by replacing an unobserved variable by a scaling version of itself, and updating coefficients accordingly. In real life we determine a series of time points at which blood samples are taken and plasma concentrations are measured. Here inverse problem arises: it is required to find rate constants (entries of matrix) knowing concentration of a drug at the given moments of a time in one compartment. The quality of those data depends on our choice of time points. An inappropriate choice may make up miss the peak concentration or we may not have sampled long enough to obtain a good estimate of the rate constants. It is demonstrated that the resolving ability of the inverse problem can be improved by varying of the location of measurement data points. The Frechet derivative matrix was constructed. Different types of three-compartment models with central elimination and two-compartment model with extravascular drug administration with absorption are covered in this report. Also an algorithm for solving inverse problem in case of n-compartment is covered in this report.
Inverse problem is solved by different algorithms: Landweber iterations method, Newton-Kantorovich method and Singular Value Decomposition. The question of choosing initial approximations is covered in this report. It is shown that physical properties of initial approximations strongly affect on obtained solutions. The results of numerical experiments are presented.
Voronov Dmitriy Andreevich
21/02/2014 IUAP workshop "Mathematical modelling of electrical networks and devices" Prof. Dr. E. Jan W. ter Maten
TU Eindhoven and Bergische Univ. Wuppertal
1) Advanced techniques in time-domain circuit simulation
2) Sampling, Failure Probabilities and Uncertainty Quantification

Prof. Dr. Marc Timme
+ Head, Network Dynamics
+ Adjunct Professor, University of Goettingen
Max Planck Institute for Dynamics and Self-Organization
1) Synchronization and Collective Dynamics in Oscillator Networks
2) Modern Power Grids: (In-)Stability, Braess' Paradox and Power Outage
E. Jan W. ter Maten and Marc Timme
7/10/2013 Sparse 3D reconstructions in Electrical Impedance Tomography using real data We present a 3D reconstruction algorithm with sparsity constraints for Electrical Impedance Tomography (EIT). EIT is the inverse problem problem of determining the distribution of conductivity in the interior of an object from simultaneous measurements of currents and voltages on its boundary. The feasibility of the sparsity reconstruction approach is tested with real data obtained from a new planar EIT device developed at the Institut of Physics, Johannes Gutenberg University, Mainz, Germany. The complete electrode model is adapted for the given device to handle incomplete measurements and the inhomogeneities of the conductivity are a priori assumed to be sparse with respect to a certain basis. This prior information is incorporated into a Tikhonov-type functional by including a sparsity-promoting l1-regularization term. The functional is minimized with an iterative soft shrinkage-type algorithm. Cristiana Sebu
17/12/2012 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 Mikhail Klibanov
12/01/2012Comparative analysis of typical approaches for inverse source problems with boundary and final time measured output data for parabolic equationsThis 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. Hovewer, 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. Alemdar Hasanoglu (Hasanov)
08/12/2011Application of level set method for groundwater flow problemsThe 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.Peter Frolkovic
08/12/2011Curve evolution models — numerical solution and applicationsIn 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.Karol Mikula
01/12/2011A computational multiscale approach for multiphase porous mediaWe 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.Lubomir Banas
29/11/2011Blood flow modelling in microfluidic devices with biomedical appIn 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 occurance, 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.Ivan Cimrak
24/05/2011Stochastic approaches in computational electromagnetismIn 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.S. Clenet
4/11/2009Higher order Sobolev estimates and nonlinear stability of stationary solutions for the mean the curvature flow with triple junctionWe 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.D. Sevcovic
3/11/2009Nonlinear and Multiscale Problems in Low-frequency ElectromagnetismThis 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.J. Busa
7/10/2009Applications of Galerkin methods in solid mechanicsSpatially-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.L. Noels
21/09/2009On numerical methods for direct and inverse problems in electromagnetismThis 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.V. Zemanova
13/08/2009Singular Perturbed Vector Fields and Model ReductionThe 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.V. Goldshtein
26/11/2008Assessment and analysis of highly irregular dataOften 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.M. Stehlik
14/11/2008An alternating procedure for boundary data identification for the Laplace equation in semi-infinite regionsWe 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.B.T. Johansson