A BME Matematika Intézet Analízis és Differenciálegyenletek Tanszékének közös Alkalmazott Analízis Szemináriuma 2016. őszén indult Faragó István (Differenciálegyenletek Tanszék) kezdeményezésére az MTA-ELTE Numerikus Analízis és Nagy Hálózatok Kutatócsoporttal együttműködésben. A szeminárium célja, hogy elősegítse egy alkalmazott analízissel (funkcionálanalízis, differenciálegyenletek, numerikus módszerek) foglalkozó kutatói kör kialakítását az intézeten belül. A szemináriummal fórumot szeretnénk biztosítani az alkalmazott analízissel foglalkozó matematikusok és az analízist alkalmazó kutatók számára az együtt gondolkodásra. További cél az érdeklődő hallgatók (MSc, PhD) bevonása a kutatói munkába.
Szemináriumunk 2017-től felvette a Farkas Miklós Alkalmazott Analízis Szeminárium nevet. Ezzel szeretnénk emléket állítani egyetemünk egykori tanszékvezető matematikaprofesszorának, aki elindította egyetemünkön a matematikus-mérnök képzést, és a stabilitáselmélet valamint a biomatematika terén elért jelentős tudományos eredményeivel ill. könyveivel nagyban hozzájárult az alkalmazott matematika erősödéséhez.
A covid járvány miatt három évig szünetelt a szeminárium. 2023. szeptemberétől indult újra a korábban említett szervezetek utódjainak (BME Analízis és Operációkutatás Tanszék, HUN-REN-ELTE Numerikus Analízis és Nagy Hálózatok Kutatócsoport) szervezésében.
The talks are in English on a regular basis. / Az előadásokat angol nyelven tartjuk.
Scheduled talks in the spring semester of 2024: 22 February: Tamás Fülöp (BME), 7 March: Ferenc Izsák (ELTE), 21 March: Máté Vizi (BME), 11 April: tba, 25 April: tba, 9 May: Research reports of PhD students.
The talk starts with an overview of the so-called symplectic methods used for numerical simulations of reversible, Hamiltonian, dynamical systems, of their advantages, and also of two procedures called Backward Error Analysis and Shadowing. Then the need for the extension of these techniques is demonstrated on irreversible physical systems formulated in a consistent thermodynamical framework, yielding ordinary differential equations and boundary-value problems, the numerical solution of which is often problematic yet necessary. Finally, the talk presents successful implementations of these ideas, in which the thermodynamical structure is shown to be advantageous.
7 December 2023
Research reports of PhD students
PhD students working in the field of numerical matematics present their topics and results.
- 10:15 Menghis Teweldebrhan Bahlibi: Discrete maximum principles for the Courant finite element solution of some nonlinear elliptic problems - presentation
- 10:35 Sebastian Josue Castillo: Newton-Krylov methods for non-linear systems - presentation
- 10:55 Ábel Göde: Supremum and infimum of positive operators - presentation
- 11:15 Noémi Gyúró: On the role of the basic reproduction number in continuous and discrete systems modelling disease propagation - presentation
- 11:35 Adrián Szemenyei: Semi-implicit discretizations for a class of epidemiological models - presentation
23 November 2023
This interactive seminar intends to give you an insight into „hardcore fluid mechanics” in industries. I am going to encourage you to utilize your background of basic physics and mathematics – including mental arithmetic – in quick and relevant approximations for complex fluids engineering solutions. For this purpose, all what you need are as follows: a piece of paper, a pen or pencil, some time, your sharp mind, and being sufficiently brave to make hand-made sketches and to elaborate descriptive formulae being as simple as possible. You can extensively have a practical benefit from this toolkit because it is mostly at your hand – even under extreme circumstances, such as in a desert island, where you can make drawings into the beach sand using a wooden stick… Keeping things simple, such old-school method is an exciting game as well as is of outstanding engineering value, as you will realize via my three case studies, demonstrating an evolutionary approach in the applied mathematics tools: A) Order-of-magnitude estimation for confirmation / rebuttal of feasibility of operational concepts; B) Geometrical optimization of constructions; C) Preliminary design of aerodynamic shapes. These examples will also strengthen your social life given that they A) will give you hints how to make new friends on New Years’ Day using fluid mechanics; B) will aid your mobile phone communication; and C) will contribute to the morning cornflakes of your family! How? Come and see! Let’s cooperate!
9 November 2023
Rita Kiss (BME, Department of Mechatronics, Optics and Mechanical Engineering Informatics)
Biomechanics: corporation between the doctor and the engineer
The modern formulation of the scientific field of biomechanics is linked to the name of the German physician and physicist Hermann Ludwig Ferdinand von Helmholtz (1821-1894): "Biomechanics is the science dealing with the mechanical properties of living beings and their physiological role." The field of biomechanics plays an important role in the development and validation of diagnostic tools, in the modeling of processes in the human body, and in the development of various medical aids.
As with artificial materials, knowledge of the mechanical behavior of bones, muscles, ligaments connecting bones, and tendons connecting bone and muscle is essential not only for assessing the effects of various loads and simulating surgical methods, but also helps in planning the replacement of certain human materials. One of the most common orthopedic knee joint surgeries is the replacement of the anterior cruciate ligament, which can be done with one's own tendon (autograft) or tendon taken from another person (allograft). The verified measurement method can be used to determine which tendon is most suitable for replacing the anterior cruciate ligament. Through own research results, it is presented how the storage and sterilization of the tendon changes the mechanical properties of the tendon.
Movement and balance can be analyzed using measurement methods and characteristics also used in the technical field, so movement tests help establish the diagnosis as accurately as possible, choose the correct treatment strategy, and follow up the healing process. The presentation shows that the authentication based on elaborated geodetic measurements, the modification of the data filtering, and the clarification of the point selection increase the reliability of the measurements. In the case of sports movements, orthopedic changes, and injuries, the effect of various surgical and rehabilitation protocols can be characterized numerically with movement tests performed and evaluated together with doctors, and the treatment can be tailored to the individual. Nowadays, with the increase in life expectancy, the analysis of the ability to balance is one of the defining research areas of movement studies. The last part of the presentation shows through examples that with the sudden change of direction test, changes in balance can be detected at an early stage, and with appropriate treatment strategies and their simulation, the risk of falling in old age can be reduced.
30 October 2023
Frédéric Magoulès (Université Paris-Saclay, France)
Asynchronous Iterative Domain Decomposition Methods
Existing numerical algorithms face their limits when running on a large number of cores. For instance, parallel iterative methods meet serious scalability limitation due to the synchronization procedure occurring between the processors at the end of each iteration. The traditional scheme for parallel iterative algorithms is synchronous iterations. This describes a method where a new iteration is only started when all the data from the previous one has been received. These iterative algorithms have been widely studied and are often simply called parallel iterative algorithms, synchronous being omitted. Another kind of iterative scheme, called asynchronous iterations, can help solve these scalability problems, but lead to several convergence issues.
Domain decomposition methods, through arti?cial subdivisions of the domain, is a means for introducing parallelism. Iterative domain decomposition strategies include in one way or another the following ingredients : (i) a decomposer to split a mesh into subdomains ; (ii) local solvers to find solutions for the subdomains ; (iii) interface conditions enforcing compatibility and equilibrium between subdomains ; (iv) an iterative solution strategy for the interface problem. Synchronization during iterations of the interface problem, leads to intrinsic scalability limitations of iterative domain decomposition methods.
This talk addresses asynchronous parallel iterative domain decomposition methods (in space and in time). The iterative scheme is here modi?ed, and asynchronous iterations are proposed for the solution strategy of the interface problem. After the presentation of the methods and their convergence properties, numerical experiments are performed in parallel on large scale computational fluid dynamics problems to illustrate the robustness and efficiency of the proposed method which outperform their synchronous counterparts. Besides, the programming library developped is proved stable and powerful for the implementation of any asynchronous iterative methods.
 G. Gbikpi-Benissan, F. Magoulès, Asynchronous sub-structuring method with alternating local and global iterations. Journal of Computational and Applied Mathematics, 113531, 2021
 F. Magoulès, G. Gbikpi-Benissan, Asynchronous Parareal time discretization for partial differential equations. SIAM Journal on Scientific Computing, 40 :6, C704-C725, 2018.
 F. Magoulès, D.B. Szyld, and C. Venet. Asynchronous optimized Schwarz methods with and without overlap. Numerische Mathematik, 137(1) :199-227, 2017.
 A. Frommer and D.B. Szyld. On asynchronous iterations. Journal of Computational and Applied Mathematics 123(1-2) :201-216, 2000.
26 October 2023
András A. Sipos (BME, Department of Morphology and Geometric Modeling)
On curves evolving under curvature-driven flows
The evolution of non-intersecting, smooth, closed curves under curvature-driven flows is investigated. Assuming sole generic saddle-node bifurcations, we show that the number of critical points of the curvature function κ is non-increasing. As the number of critical points of the curvature is closely related to the vertices of the evolute, the flow associated with the evolution of the evolute is derived. Finally, results about the evolution of the support function are collected and applied to explain unexpected shapes in nature.
The presented results are based on joint work with Gábor Domokos.
12 October 2023
Stanislav Sysala (Institute of Geonics of the Czech Academy of Sciences, Ostrava)
Selected Newton methods in computational elasto-plasticity
This talk is about Newton-like methods with the applications to elasto-plasticity and computational geotechnics. It is considered an abstract algebraic system of non-linear equations and its properties are specified to be in accordance with discretized elasto-plastic problems defined in terms of displacements. A particular attention is devoted to associated models with and without hardening and their differences are highlighted on the algebraic level. With respect to the expected properties of elasto-plastic algebraic systems, we introduce convenient Newton-like methods, study their local and global convergence and illustrate it on numerical examples. Especially, we build on the semismooth Newton and its quasi-Newton, smoothing, damped and continuation variants. Finally, for the elasto-plastic models without hardening (elastic-perfect plasticity), we introduce a related auxiliary problem on determining the so-called limit load which is important for safety assessment of structures. An appropriate algorithmic solution is suggested and examples from geotechnical practice are illustrated. The presented results are based on joint work with O. Axelsson, J. Haslinger, J. Karátson, M. Béreš and other coworkers.