| Website in VV: | Lecture, Exercise (please register for both lecture and exercise!) |
| Lecture dates (2 SWS): | Wednesday 14:15 - 15:45, Seminar room 119, Arnimallee 3. |
| Exercise dates (2 SWS): | Tuesday 12:15 - 13:45, Seminar room 140, Arnimallee 7. |
| Lecture period: | 16.4.2025 - 16.07.2025 |
| Exercise period: | 22.4.2025 - 15.07.2025 |
| Final exam: | An oral exam at the end of the semester |
| Other prerequisite: | Active and regular participation in the exercises |
| Active participation: | You prove this by completing and submitting checklists |
| Regular participation: | You prove this by mandatory attendance in the exercise sessions |
| Date | Markov Processes... | Mathematics | Level | Material* |
| 16.4. | Model-Based | Discrete | Theory | Intro Frobenius Theorem Eigenvalues |
| 23.4. | Model-Based | Discrete | Application | Intro Generator Matrix Spectral Clustering SqRA lag-time |
| 30.4. | Model-Based | Continuous | Theory | Intro 1 Intro 2 Operators (L. Donati) Fokker-Planck |
| 7.5. | Model-Based | Continuous | Application | Notebook (L. Donati) Metzner PhD, TPT MAZE |
| 14.5. | Data-Based | Discrete | Theory | (Ulam, "Count"-Galerkin...) |
| 21.5. | Data-Based | Discrete | Application | (PCCA+ ...) |
| 28.5. | Data-Based | Continuous | Theory | (ISOKANN) |
| 4.6. | Data-Based | Continuous | Application | (ISOKANN) |
| 11.6. | Expert-Based | Logic | Theory | (Boole...) |
| 18.6. | Expert-Based | Logic | Application | (Gröbner basis ...) |
| 25.6. | Expert-Based | Coordination | Theory | (What is relevance? Ring homomorphism) |
| 2.7. | Expert-Based | Coordination | Application | (Coordination) |
| 9.7. | Special topic | Importance Sampling | Theo/App | (Optimal Control?) |
| 16.7. | Special topic | Reaction pathways | Theo/App | (Explainable A.I.?) |
16.4.: select one(!) of the following topics and fill out the checklist according to your actions. Send it via eMail to me.
Topic1: Perron-Frobenius-Theorem. What does it say? How is it proven?
Topic2: Find a process which is not Markovian. (In your example the process itself has to be non-Markovian. Do not try to "hide away Markovianity" by an "inappropriate" modelling or by a clustering of states)
Topic3: If a transition probability matrix P is perturbed into a transition probability matrix P+E, what do we know about the changes of its eigenvalues?
Topic4: A doubly-stochastic matrix can be represented by a convex combination of permutation matrices. How does the Birkhoff-von Neumann method find this decomposition?
23.4.: select one(!) of the following topics and fill out the checklist according to your actions. Send it via eMail to me.
Topic1: Invent matrices Q and compute P(tau)=expm(Q*tau). Learn properties of expm and logm. Invent matrices P(tau) and compute Q=1/tau*logm(P(tau)).
Topic2: Solve minCut problem for a given undirected graph using tutorial about Spectral Clustering and by computing a graph Laplacian.
Topic3: Implement an algorithm to generate a Markov Chain given a transition probability matrix or given an infinitesimal generator.
Topic4: Invent a graph + stationary distribution of molecular states. Use SqRA to find the Q-matrix (Lemma 1 in the article). Compute for different lag-times the matrix P(tau)=expm(tau*Q). Compute the mean first passage time matrix given P(tau) (Theorem 1.17 and Corollary 1.16 in the printed papers).
30.4.: select one(!) of the following topics and fill out the checklist according to your actions. Send it via eMail to me.
Topic1: Explain the main steps: How the Fokker-Planck equation is derived based on the Chapman-Kolmogorov property? You can use this script from L.Donati.
Topic2: What is the "Kolmogorov Forward Equation" and the "Kolmogorov Backward Equation"? Explain their relation with Fokker-Planck equation and with the Master-Equation.
Topic3: Explain the Feynman-Kac formula. How can it be used?
7.5.: select one(!) of the following topics and fill out the checklist according to your actions. Send it via eMail to me.
Topic1: Explain and visualize the relations between the eigenvalues and eigenfunctions of the infinitesimal generator, the Fokker-Planck operator, the Koopman operator, and the propagator (in case of reversible Markov processes). Use, e.g., the linked scripts of L. Donti.
Topic2: Generate a maze with different exits (dicretized into small "squares", some of the squares are "exits"). Generate a transition rate matrix between all squares. Compute the committor function with regard to one exit square. Visualize the result. Start somewhere in the maze and follow the "gradient".
Topic3: Generate a maze with one exit (dicretized into small "squares", one of the squares is an "exit square"). Generate a transition rate matrix between all squares. Compute the mean holding time with regard to the exit square. Visualize the result. Start somewhere in the maze and follow the "gradient".
Topic4: Generate a maze without exit (dicretized into small "squares"). Generate a transition rate matrix between all squares. Compute an eigenvector of the matrix (an eigenvector with an eigenvalue close to zero but not exactly zero). Visualize the result. Start somewhere in the maze and follow the "gradient".