Topic Overview:

The development of various drug delivery systems is an area of active research. This research is focused mainly on chemical, biochemical, or physiological aspects. The movement of a delivery system in human body, to the point of the drug action is much less investigated. The drug transport to the targeted site where the drug’s chemical action should take place is essential for its proper function.

Task:

COMSOL-based theme.  Brief literature search on surviving modelling problems in drug delivery, selection of system to be modelled. Creation of a suitable (hydrogel) structure in COMSOL environment, modeling the release, study the effect of relevant paraneters on the release. Writing final report. Depending on applicant’s level and software knowledge the complexitz of the task will be precised after arrival.

Requirements:

Basic knowledge of equations in mathematical physics, capability of using computer software to solve these equations and basic capability of programming.

Outcomes:

Final report on obtained results.

References:

J.Mayr, C.Saldias, D.D.Díaz. Release of small bioactive molecules from physical gels. Chem.Soc.Rev. 47(4), 1484-1515 (2018)

Recommended for:

Students willing to work on computer. Suitable also for student of (applied) mathematics or mathematical modeling. Experience with COMSOL advantageous but not necessary.
Brno University of Technology, Faculty of Chemistry, Materials Research Centre – Laboratory of Biocolloids
Modelling of Transport Phenomena in Controlled Release
University of West Bohemia in Pilsen, Faculty of Applied Sciences, Department of Mathematics
Simulation and Statistical Modeling of Basic Financial Indicators
Topic Overview:

In this project we will investigate basic tools of stochastic calculus and analysis in mathematical finance and their application to different continuous-time market models and to selected financial derivatives such as options. One of the most important indicators in finance is volatility. It is the degree of variation of a trading price series over time, usually measured by the standard deviation of logarithmic returns. Historic volatility measures a time series of past market prices. Implied volatility looks forward in time, being derived from the market price of a market-traded derivative (in particular, an option). We will study appropriate implied volatility surfaces parametrizations as well as fitting to other financial curves or surfaces of interest. Local volatility and stochastic volatility (LVSV) models will be considered.

Task:

General tasks within the project:
a) Studying efficiency of pricing formulas.
b) Simulation of LVSV models.
c) Calibration of LVSV models to real market data.
d) Robustness and sensitivity analysis.
e) Parametrizations of implied volatility surfaces.
Each intern will be responsible probably for one task only. Its detailed specification will be provided on request by the supervisor.

Requirements:

Basic knowledge of financial mathematics is an advantage, but not requisite. Ability to implement financial models on a computer (preferably in MATLAB, alternatively in Python) is highly recommended.

Outcomes:

Implementation of studied models, report on obtained results ideally written as a paper draft.

References:

Papers related to specific tasks will be provided on request by the supervisor. An inspiration can be obtained from recent publications of the supervisor (http://home.zcu.cz/~honik/publications/).
a) Baustian, F., Mrázek, M., Pospíšil, J., and Sobotka, T. (2017), Unifying pricing formula for several stochastic volatility models with jumps. Appl. Stoch. Models Bus. Ind. 33(4), 422–442, ISSN 1524-1904, DOI: 10.1002/asmb.2248, Zbl 1420.91444, MR3690484.
b) Mrázek, M., Pospíšil, J., and Sobotka, T. (2016), On calibration of stochastic and fractional stochastic volatility models. European J. Oper. Res. 254(3), 1036-1046, ISSN 0377-2217, DOI: 10.1016/j.ejor.2016.04.033, Zbl 1346.91238, MR3508893.
c) Mrázek, M. and Pospíšil, J. (2017), Calibration and simulation of Heston model. Open Math. 15(1), 679-704, ISSN 2391-5455, DOI: 10.1515/math-2017-0058, Zbl 1368.60061, MR3657941.
d) Pospíšil, J., Sobotka, T., and Ziegler, P. (2019), Robustness and sensitivity analyses for stochastic volatility models under uncertain data structure. Empir. Econ. 57(6), 1935–1958, ISSN 0377-7332, DOI: 10.1007/s00181-018-1535-3.
University of West Bohemia in Pilsen, Faculty of Applied Sciences, Department of Mathematics
Parameter Estimation in Rough Fractional Stochastic Models
Topic Overview:

In this project we will investigate stochastic models driven by a fractional Brownian motion (fBm) especially in the so called rough regime, i.e. in the case with low regularity. We will primarily focus on parameter estimation and numerical solutions. Models described by stochastic ordinary and partial differential equations (SODEs and SPDEs), or stochastic evolution equations in Hilbert spaces will be of interest. Stochastic models with rough fractional noise have numerous practical applications, but their analysis is theoretically challenging. Nevertheless, parameter estimation and numerical solution are now possible and of high interest. We will study appropriateness of several different parameter estimation techniques known from equations driven by a (standard) Brownian motion, in particular ergodic-type estimators and least squares estimators. To enable Monte Carlo analysis, different numerical solutions of rough fractional SODEs and SPDEs will be developed, implemented and analysed. We plan to introduce novel estimation procedures that integrate these numerical solutions.

Task:

General tasks within the project:
a) Finding effective numerical methods for rough fractional stochastic models, analysis of the numerical solution and comparison of these  methods.
b) Stochastic analysis of drift parameter estimators for linear and semi-linear rough fractional SPDEs.
c) Drift parameter estimation for non-linear equations (ergodic-type and LSE estimators).
Each intern will be responsible probably for one task only. Its detailed specification will be provided on request by the supervisor.

Requirements:

Basic knowledge of stochastic and numerical analysis is an advantage, but not requisite. Ability to implement numerical methods on a computer (preferably in MATLAB, alternatively in Python) is highly recommended.

Outcomes:

Implementation of studied method, report on obtained results ideally written as a paper draft.

References:

a) Lord, G. J., Powell, C. E., and Shardlow, T. (2014), An introduction to computational stochastic PDEs. Camb. Texts Appl. Math., Cambridge University Press, New York, ISBN 978-0-521-72852-2/pbk; 978-0-521- 89990-1/hbk; 978-1-139-01732-9/ebook, DOI: 10.1017/CBO9781139017329, Zbl 1327.60011,
b) MR3308418. b) Øksendal, B. (2003), Stochastic differential equations. An introduction with applications. Universitext, Springer-Verlag, Berlin, 6th edn., ISBN 3-540-04758-1/pbk, DOI: 10.1007/978-3-642-14394-6, Zbl 1025.60026, MR2001996.
Papers related to specific tasks will be provided on request by the supervisor.
University of West Bohemia in Pilsen, Faculty of Applied Sciences, Department of Mathematics
Nonlinear Boundary Value Problems for Differential and Difference Equations
Topic Overview:

Many fundamental physical and natural laws can be formulated in terms of differential or difference equations. Therefore, ODEs are the basic tools of mathematical modeling in natural and technical sciences. Even quite simple ODE can illustrate the instability of various processes, their parameter sensitivity or irreversibility. The modern trend is to use nonlinear models that enable us to describe phenomena which are impossible to catch with linear descriptions.

Task:

Simple analysis of chosen nonlinear problems in cooperation with the supervisor, study of literature, simple numerical experiments.

Requirements:

To be familiar with elementary methods of solving linear initial and boundary value problems for ODEs. Any SW skills are welcome.

Outcomes:

Report summarizing obtained results.
University of West Bohemia in Pilsen, Faculty of Applied Sciences, Department of Mathematics
Advanced Symbolic-Numerical Computations in Geometry and Geometric Modelling
Topic Overview:

The research will be devoted to the study of novel mathematical methods of geometric modelling for a purpose of suitable description, data representation, modification and presentation of (exact or perturbed) curves and surfaces with respect to their further application in technical practice and natural sciences. Practical experience with using suitable geometric and mathematical software will also be developed. Potential impact on the curriculum modification and modernization at the university and high school level may be also studied.

Task:

Work on a chosen topic from the field of geometric modelling (and related disciplines) in cooperation with the supervisor, study of literature, simple experiments in selected CAS (Maple, Mathematica).

Requirements:

Fundamental knowledge of different branches of geometry (projective, differential, algebraic); elementary knowledge of representing shapes in computer-aided design;  knowledge of work in some CAS and 3D modeler  is welcome. In case of insufficient background knowledge, the supervisor will suggest suitable materials for self-study.

Outcomes:

Report on obtained results, or poster.
University of West Bohemia in Pilsen, Faculty of Applied Sciences, Department of Mathematics
Numerical Models, Methods and Algorithms: Design and Analysis
Topic Overview:

The research will be devoted to the numerical modeling and numerical methods for solving partial differential equations. Techniques based on the finite volume method, finite element method or isogeometric analysis will be used to model some phenomena in the field of mechanics or physics (fluid flow, heat conduction, etc.).

Task:

Work on a chosen topic from the field of numerical modeling in cooperation with the supervisor, study of literature, numerical experiments in selected software or programming language (Matlab, Mathematica, Python, C++).

Requirements:

To be familiar with basic numerical methods of solving ODEs, PDEs and systems of linear equations. Any SW skills are welcome.

Outcomes:

Report summarizing obtained results and numerical experiments.
University of West Bohemia in Pilsen, Faculty of Applied Sciences, Department of Mathematics
Structural Graph Theory
Topic Overview:

For many graph properties, there are general results concerning the structure of the graphs satisfying the property. Examples include the Kuratowski Theorem which characterises planar graphs in terms of forbidden minors, or the Tutte theorem which is a characterisation of graphs admitting a perfect matching.

Task:

Research of a problem related to graph structure based on discussions with the supervisor: current status, related results, proof methods, work on a selected special case etc.

Requirements:

Knowledge of basic graph theory as taught in an introductory course (cycles, matchings, spanning trees, distance in graphs, colourings, homomorphisms, planar graphs).

Outcomes:

A presentation at the Graph Theory seminar and a written report on the results of the research.

References:

a) J. A. Bondy, U. S. R. Murty: Graph Theory. Springer, 2008.
b) R. Diestel: Graph Theory. Fifth edition, Graduate Texts in Mathematics, Vol. 173. Springer, 2016.
University of West Bohemia in Pilsen, Faculty of Applied Sciences, Department of Mathematics
Algorithms for Hard Problems in Combinatorial Optimization
Topic Overview:

Combinatorial problems arise naturally in many areas such as operations research, computer science, computational biology and artificial intelligence. Typical examples include scheduling, timetabling, routing, resource allocation and genome sequencing.

Task:

To collaborate in algorithm development for a selected problem (e.g. satisfiability, routing or coloring).

Requirements:

To have a solid background in combinatorial optimization including integer programming methods and approximation algorithms. Programming skills are very welcome.

Outcomes:

Report on obtained results or ideally paper draft.
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