Basic Information

Exercises

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".

14.5.: select one(!) of the following topics and fill out the checklist according to your actions. Send it via eMail to me.
Topic1: Explain the derivation and the meaning of equation (3.1) in this article about rebinding effects.
Topic2: Explain the HMC-method as a special case of MCMC-methods. Why is it important that Hamiltonian Dynamics is symplectic? You can refer to the dissertation of Alexander Fischer (Chapter 4) or of Marcus Weber (Chapter 2).
Topic3: Invent a set S of 10 states. Invent a target distribution according to which you want to sample from S. Generate an MCMC algorithm (acceptance probablity should not be according to Metropoilis min{1,...}) and demonstrate how well your algorithm samples from the targeted distribution.
Topic4: How do Renata Sechi et al. estimate the infinitesimal generator of a Markov chain using an extrapolation method?

21.5.: select one(!) of the following topics and fill out the checklist according to your actions. Send it via eMail to me.
Topic1: Explain how PCCA+ turns into a linear program in Marcus Weber (Chapter 3.4.3).
Topic2: Explain with your own words the Gelman-Rubin criterion and demonstrate its application with simulating Markov Chains.
Topic3: How does Susanna Röblitz estimate the error of eigenvector computation of a Markov chain (e.g. in Theorem 3.1.5) based on the error of the entries of the transition probability matrix?
Topic4: Regarding the generation of matrix T, explain how you would refine the Galerkin basis or extend the horizontal sampling based on "indicators". (Inspiration you can find Chapter 3.4).

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