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The department of "Mathematical Modeling and Numerical Analysis" was created In March 2014, by merging the departments: "Mathematical Modeling", "Computational Mathematics" and "Biomathematics".

## Historical Notes on the Department of "Mathematical Modeling"

The Department of Mathematical Modelling of the Institute for Mathematics and Informatics (IMI) of the Bulgarian Academy of Sciences (BAS) is the direct successor of the following units formed in the early 60’s:

- Department of Numerical Methods and Algorithms of the Institute for Mathematics with a Computing Centre of BAS;
- Chair of Computational Mathematics of the Faculty of Mathematics in the Sofia University (successor of the laboratory of Computational Mathematics at the Department for Advanced Analysis)

The Department was formed within the United Centre for Science and Education in Mathematics and Mechanics, following a Decision of the Bulgarian government “for endorsement of a new model in the education in mathematics and mechanics and integration of the activities of the Faculty of Mathematics in the Sofia University and the Institute for Mathematics with a Computing Centre of BAS”. The Department was called sector during the period of functioning of the united centers for science and education - till 1988. The following scientists entered on December 30, 1970 the new Department of Mathematical Modelling:

Acad. Blagovest Sendov, Acad. Borislav Bojanov, Corresponding member of BAS Vassil Popov, Prof. DSc Raycho Lazarov, Prof. DSc Stefka Dimova, Prof. DSc Svetoslav Markov, Prof. DSc Rumen Maleev, Prof. DSc Mihail Kaschiev, Prof. Dr Milko Petkov, Assoc. Prof. Dr Todor Boyanov, Assoc. Prof. Dr Vassil Vesselinov, Assoc. Prof. Dr Andrey Andreev, Assoc. Prof. Dr Nikolay Kyurkchiev, Assoc. Prof. Dr Evgeniya Kolarova-Sendova, Assist. Prof. Dr Mitko Tsvetanov, Assist. Prof. Dr Margarita Nikolcheva, Assist. Prof. Dr Emiliya Mateeva, Assist. Prof. Teodora Kirpikova-Jukanova, Assist. Prof. Mara Apostolova.

As of 1 January 2013 the Department of Mathematical Modelling consists of:

Prof. DSc Kamen Ivanov (head of the Department), Prof. DSc Pencho Petrushev, Prof. Dr Nikolay Kyurkchiev, Prof. Assoc. Prof. Dr Andrey Andreev, Assoc. Prof. Dr Vladimir Hristov, Assoc. Prof. Dr Ognyan Trifonov, Assoc. Prof. Dr Borislav Draganov, Assist. Prof. Dr Irina Georgieva, Daniela Stoyanova.

Two temporary groups existed in the Department of Mathematical Modelling till 1995, namely, a temporary problem group on mathematical modelling of the gravity impact of the Sun and the Moon on the Earth with staff Prof. DSc Ivan Ivanov, Assist. Prof. Petar Dabnishki, and Assist. Prof. Dimitar Dimitrov; Laboratory for Mathematical Chemistry and Chemical Informatics with staff Assoc. Prof. DSc Daniel Bonchev, Assist. Prof. Ivan Bangov.

Members of the Department of Mathematical Modelling were also:

Prof. DSc Ivan Dimov, Prof. DSc Ralitsa Kovacheva, DSc Lili Popova, Assoc. Prof. Dr Georgi Iliev, Assoc. Prof. Dr Spas Tashev, Assoc. Prof. Dr Nataliya Kolkovska, Assoc. Prof. Dr Denka Kutzarova, Assoc. Prof. Dr Tatyana Chernogorova, Assist. Prof. Dr Nikola Naydenov, Assoc. Prof. Dr Panayot Vassilevski, Assist. Prof. Dr Nikola Vladov, Assist. Prof. Dr Milena Moskova, Assist. Prof. Dr Hristo Jijev, Assist. Prof. Angelina Yotova, Assist. Prof. Nikolay Bankov, Assist. Prof. Petar Shopov, Veselin Kyurkchiev, Krasimira Popova, Maksim Chilev, Yanko Chernev.

Several new groups working in closely related mathematical fields originated from Department of Mathematical Modelling. Departments of Numerical methods and Biomathematics of IMI-BAS separated from Department of Mathematical Modelling. Essential parts of the founders of Institute for Parallel Processing of BAS and the Chair of Numerical Methods and Algorithms of the Faculty of Mathematics in the Sofia University were former members of the Department too.

Consecutive heads of Department of Mathematical Modelling are Acad. Blagovest Sendov, Prof. DSc Racho Denchev, Prof. DSc Pencho Petrushev, Assoc. Prof. Dr Vladimir Hristov, Prof. DSc Kamen Ivanov and Prof. Dr Nikolay Kyurkchiev.

The main field of research is the development of mathematical models in astronomy, biology, economics, medicine, fusion synthesis, chemistry. The Bulgarian school in approximation theory was founded and developed as a part of Department of Mathematical Modelling under the leadership of Acad. Bl. Sendov. Other fields of research are numerical analysis, numerical methods for solving differential and integral equations, functional analysis and analytic number theory.

The members of Depatment of Mathematical Modelling are also active as lecturers in a number of domestic and foreign universities. More than 120 PhD dissertations were prepared and defended under the supervision of staff of the Deparment. The Department was the main organizer of the international conferences Constructive Function Theory held in 1971, 1977, 1981, 1984, 1987, 1991, 2002, 2005, 20010, and 2013.

The international recognition of the mathematical results of the school in approximation theory resulted in the only prize for young scientists working in the field - the Vassil A. Popov prize. The prize is awarded by an international committee every third year. It was won by 5 different mathematicians from France, Belgium and the USA. The state Dimitrov Prize was awarded to academician Blagovest Sendov and corresponding member of BAS Vassil Popov, as the later is also a winner of the Nikola Obrechkoff Prize for remarkable achievements in the field of mathematics.

*The old site of the department:* here

## Historical Notes on the Department of "Computational Mathematics"

The former Laboratory of Numerical Methods (presently Department of Computational Mathematics) was created in 1987 by some members of the Department of Mathematical Modeling. The aim was to further strengthen the direction of numerical methods for PDEs as the most important component in mathematical modeling as a research tool and means for industrial applications. Prof. Raytcho Lazarov was appointed for its Head and the first members were: Michail Kaschiev, Stefka Dimova, Ivan Dimov, Natalia Kolkovska, Oleg Iliev, Panayot Vassilevski, Tatyana Chernogorova and Angelina Yotova.

Further heads were Prof. Michail Kaschiev (1991-2004) and Assoc. Prof. Natalia Kolkovska (2004-2012). The scientific staff grew by Petar Shopov, Tanya Kostova-Vassilevska, Atanas Pehlivanov, Ivan Bazhlekov, Ludmil Zikatanov, Milena Dimova, Daniela Vasileva, Mariana Nikolova, Pavlin Entchev, Galin Dimitrov, Ivan Georgiev, Stanislava Stoilova.

Main fields of research are in the area of efficient numerical methods and algorithms for solving linear and nonlinear partial differential and integral equations and systems. Important results are obtained about the construction, stability and convergence analysis for finite element, finite difference and finite volume approximations; the construction and analysis of iterative methods, domain decomposition, a posteriori error control and adaptive grid refinement. The numerical methods and algorithms are applied in mathematical modeling and computer simulation of physical, engineering, environmental and other problems.

A number of applied projects were developed in collaboration with Institute of Metallurgy and Metal Sciences, Institute of Microelectronics, Sofia Technical University, Joint Institute for Nuclear Research (Dubna, Russia), Institute of Mathematical Modeling of Russian Academy of Sciences, Texas A&M University, Darmstadt University of Technology (Germany), Institute of Science and Technology of University of Manchester, the Engineering department of Queen Mary College (University of London), Fraunhofer Institute for Industrial Mathematics (Kaiserslautern, Germany).

The department was involved in the organization of six international conferences on Numerical Methods and Applications in Sofia and Borovets. Members of the department participate in the projects:

- Numerical and Analytical Tools for Localized Solutions of Generalized Wave Equations in Multidimension, Bulgarian Science Fund, 2011-2014.
- Mathematical modeling of nonlinear systems on multiprocessor clusters, Bulgarian State Agency for Atomic Regulation, 2011-2013.
- Efficient mu1tilevel methods and algorithms for problems with heterogeneous coefficients, Bulgarian Science Fund, 2012-2014.

*The old site of the department:* here

## Historical Notes on the Department of "Biomathematics"

Section "Biomathematics" of IMI-BAS is successor of the former Research group on "Mathematical Modelling in Biology" within the Center of Biology at BAS (1977-1989) and the Institute of Biophysics at BAS (1990-1996). In 1996 the group was included in the IMI at BAS as a department on Biomathematics.

Research area: Biomathematics (mathematical modelling in biology, mathematical biology) with an emphasis on mathematical modelling in ecology, neurophysiology, microbiology, enzyme kinetics, botany, paleology, etc. Developed are numerical methods and tools for mathematical modelling under uncertainties. Related mathematical fields: error analysis, reliable computing, set-valued, convex and interval analysis, numerical methods with verification, controlability of dynamical systems, parameter identification under interval data, computer arithmetic, etc.

*The old site of the department:* here

## Former members:

- Prof., DSc, Pencho Petrushev
- Assoc. Prof., Dr., Daniela Vasileva †

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- Category: Mathematical Modeling and Numerical Analysis

- BIOMATH 2018 (International Conference on Mathematical Methods and Models in Biosciences), June 24-29, 2018, Sofia, Bulgaria
- MMSC2016 (International Workshop “Mathematical Modelling and Scientific Computing), 18-24 September, 2016, Dobrich, Bulgaria
- Workshop on Approximation Theory, CAGD, Numerical Analysis, and Symbolic Computation, September 6-11, 2016, Sofia, Bulgaria
- BIOMATH 2016 (International Conference on Mathematical Methods and Models in Biosciences), June 19-25, 2016, Blagoevgrad, Bulgaria
- International Conference CONSTRUCTIVE THEORY OF FUNCTIONS - 2016, June 11-17, 2016, Sozopol, Bulgaria
- BIOMATH 2015 (International Conference on Mathematical Methods and Models in Biosciences), June 14-19, 2015, Blagoevgrad, Bulgaria
- The 104
^{th}European Study Group with Industry, September 23-27, 2014, Sofia, Bulgaria - Workshop on Approximation Theory, CAGD, Numerical Analysis, and Symbolic Computation, August 25-31, 2014, Sozopol, Bulgaria
- The 8
^{th}International Conference on Numerical Methods and Applications, August 20-24, 2014, Borovets, Bulgaria - BIOMATH 2014 (International Conference on Mathematical Methods and Models in Biosciences), June 22-27, 2014, Sofia, Bulgaria

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- Category: Mathematical Modeling and Numerical Analysis

Mathematical Modeling and Numerical Analysis Seminar

On **12.11.2017** at **11:00** in room 478 of IMI-BAS

Theme of the lecture:

*"**A black-box algorithm for fast matrix assembly in isogeometric analysis"*

**Dr. Clemens Hofreither** , Institute of Computational Mathematics, Johannes Kepler University, Linz, Austria

Mathematical Modeling and Numerical Analysis Seminar

On **15.06.2017** at **14:00** in room 403 of IMI-BAS

Theme of the lecture:

*"*** On best uniform approximation by low-rank matrices**"

**Clemens Hofreither** , Institute of Computational Mathematics, Johannes Kepler University, Linz, Austria

**Abstract. **We study the problem of best approximation, in the elementwise maximum norm,of a given matrix by another matrix of lower rank.

We generalize a recent result by Pinkus that describes the bestapproximation error in a class of low-rank approximation problems and givean elementary proof for it.Based on this result, we describe the best approximation error and theerror matrix in the case of approximation by a matrix of rank one lessthan the original one.

For the case of approximation by matrices with arbitrary rank, we givelower and upper bounds for the best approximation error in terms ofcertain submatrices of maximal volume.

We illustrate our results using 2x2 matrices as examples,for which we also give a simple closed form of the bestapproximation error.

Mathematical Modeling and Numerical Analysis Seminar

On **7.12.2016** at **11:00** in room 403 of IMI-BAS

Theme of the lecture:

*"**Processing of high-resolution CT data: denoising, edge detection, and segmentation**"*

Chief Assist. Prof., Ph.D. *Stanislav Harizanov*, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Mathematical Modeling and Numerical Analysis Seminar

On ** 03.10.2016** at

**1**

**3**

**:**

**3**

**0**in room 503 of IMI-BAS

Theme of the lecture:**"Parallel algorithms for some actual problems***"*

Dr. Alexander Ayriyan, Joint Institute for Nuclear research, Dubna

Mathematical Modeling and Numerical Analysis Seminar

On **27.07.2016** at **1****0****:****0****0** in room 403 of IMI-BAS

Theme of the lecture:**"Mathematical model for the dynamics of the autoimmune disease alopecia areata in cycling hair follicles****"**

*Dr.**Atanaska Dobreva, **Department of Mathematics, Florida State University*

* Abstract*. Alopecia areata is an autoimmune condition which causes distinct patterns of hair loss. The disease development mechanisms are still not completely understood, and treatment is often difficult. Hair follicles are organs that constantly cycle through phases of growth, regression, and rest. A main feature of alopecia areata is that it interrupts and makes the growth stage very short. We first construct a model of ordinary differential equations to describe how the condition develops over time in a small cluster of homogeneous growing follicles. The model incorporates interactions between hair follicles and the immune system in accordance with the immune privilege collapse hypothesis for disease pathogenesis. As a next step, we elaborate the dynamical system by including the underlying hair cycle and provide simulations for states of health, disease, and treatment. In addition, we apply parameter sensitivity analysis to determine how the processes reflected in the model influence the growth phase length.

Mathematical Modeling and Numerical Analysis Seminar

On **27.07.2016** at **1****0****:****0****0** in room 403 of IMI-BAS

Theme of the lecture:

* "Low-Rank Approximation and Its Applications*"

*Prof. Ivan Markovsky,* *Vrije Universiteit, Brussel*

**Abstract**. Mathematical engineering continuously addresses new applications and solves new problems. The expansion of existing methods and applications makes it difficult to maintain a common theoretical framework. This talk shows the potential of the structured low-rank approximation setting to unify problems of data modeling from diverse application areas. An example treated in more details in the presentation is identification of a linear time-invariant system from observed trajectories of the system. We present an optimization method based on the variable projection principle. The method can deal with data with exact as well as missing (unknown) values.

Mathematical Modeling and Numerical Analysis Seminar

On **26.05.2016** at **1****4****:****0****0** in room 403 of IMI-BAS

Theme of the lecture:**"On best uniform approximation with low-rank matrices"**

Clemens Hofreither, Institute of Computational Mathematics, Johannes Kepler University, Linz, Austria.

**Abstract. **Given a matrix, we study the best-approximation error by a matrix of lower rank. Here we are interested in a class of elementwise norms, of which the Frobenius and the Chebyshev (entrywise maximum) norms are special cases. Based on a result by A. Pinkus (2012), we derive both lower and upper bounds for the best-approximation error by a rank k matrix in the maximum norm. The quantities used in the estimates are absolute determinants of certain maximal submatrices, and we establish some connections to the theory of pseudoskeleton approximation, where this "maximal volume concept" has been applied previously. We illustrate the results by the simple example of some 2x2 matrices.

Mathematical Modeling and Numerical Analysis Seminar

On** 07.03.2016** at **1****3****:****0****0** in room 403 of IMI-BAS

Theme of the lecture:**"Steady states of polynomial ODEs****"**

*Prof. Carsten Conradi, ** HTW, Berlin*

**Abstract**. Polynomial Ordinary Differential Equations are an important tool in many areas of quantitative biology. Due to large measurement errors few experimental repetitions and a limited number of measurable components, confidence intervals of estimated parameter values are often several orders of magnitude. One therefore has to study families of parametrized polynomial ODEs. In this talk a class of ODEs is discussed that admits a monomial parameterization of the steady state variety. To this class belong, for example, multisite phosphorylation systems. For a special instance of this subclass, one can formulate parameter conditions that guarantee the existence of three steady states.

Mathematical Modeling and Numerical Analysis Seminar

On** 02.11.2015** at

**1**

**4**

**:**

**15**in room 403 of IMI-BAS

Theme of the lecture:

**"Modelling of intrinsic Josephson junctions in high temperature superconductors"**prof. Yury Madzhnunovich Shukrinov, Joint Institute for Nuclear research, Dubna

Mathematical Modeling and Numerical Analysis Seminar

On** 02.11.2015** at

**1**

**5**

**:**

**15**in room 403 of IMI-BAS

Theme of the lecture:

**"Cavity and background oscillations inintrinsic Josephson junctions"**

Dr. Ivan Hristov, FMI, Sofia University

Mathematical Modeling and Numerical Analysis Seminar

On** 29 .05.2015** at

**1**

**4**

**:00**in room 403 of IMI-BAS

Theme of the lecture:**"All-at-once multigrid methods for optimal control problems with inequality constraints"**

Dr. DI. Stefan Takacs от Technische Universität Chemnitz

**Abstract.** In this talk we will discuss the solution of all-at-once multigrid methods for optimal control problems of tracking type with box constraints for the control variable.

One possibility to solve such problems using multigrid methods is to use a (semi-smooth) Newton solver.

Here, a standard multigrid method would be used as an solver for linear sub-problems that occur within this iteration.

Recently, it was possible to construct and analyze multigrid solvers that can be directly applied to elliptic problems with box constraints (like obstacle problems).

Those methods have shown good convergence behavior in practice. In the present talk we will discuss how the ideas of such methods can be extended to control problems.