Basic Information

Corrections

In the lecture on July 9, we discussed the parallelism of the gradient of the committer function in the zero-temperature limit and the potential energy gradient. When I asked the student about this, I led him to conclude that these two gradients should actually be orthogonal to each other because certain terms in the equations are to be zero. My reasoning was incorrect. The following link, which I received from the student, shows a discussion on this topic LINK.

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).

28.5.: select one(!) of the following topics and fill out the checklist according to your actions.
Task 1: Are there any other observables besides chi that satisfy the rate equation given in the lecture? Why are c_1, c_2 (in L chi = -c_1 chi + c_2(1- chi)) positive? When are they the smallest?
Task 2: What are the relations between (K chi_plus, K chi_minus), (s_1,s_2, lambda_2) and (c_1, c_2, mu_2) where we defined (c_1, c_2 as above, K chi = s_1 chi + s_2(1- chi), K f_2 = lambda_2 f_2 and Lf_2 = mu_2 f_2)
Task 3: Formalize the proof for the convergence of the ISOKANN iteration to the chi function.

4.6.: select one(!) of the following topics and fill out the checklist according to your actions.
Task 1: Implement ISOKANN using piecewise linear functions as a basis for chi and test it with the double-well potential.
Task 2: Using ISOKANN.jl, simulate Alanine-Dipeptide and compute the chi function.

11.6.: select one(!) of the following topics and fill out the checklist according to your actions. Send it via eMail to me.
Topic1: Explain the difference between P- and NP-problems. Name one NP-complete or NP-hard problem connected to Boolean algebra/rings. What does the statement "P(Boole) not equal NP(Boole)" proven by Mihai Prunescu mean?.
Topic2: Explain the Buchberger Algorithm to find a Gröbner Basis.
Topic3: Explain the principles of deontic logic. How does it correlate with generating "rules" or "models"?
Topic4: Explain the principles of "subjective probability" theory. A statement: Data analysis is trying to maximize the mutual information between observations and interpretations. Discuss this statement with regard to subjective probability.

18.6.: select one(!) of the following topics and fill out the checklist according to your actions. Send it via eMail to me.
Topic1: Take the "Diagnosis_for_..."-Skript I gave to you in the tutorial/lecture (Julia-code). Try to understand Step 9 in this script by reading the egyptology-article from Sarah Klasse. Invent an own data example (maybe Markov Chain) and test the method with it.
Topic2: Take the "Analysis of Temporal Processes"-paper I gave to you in the tutorial/lecture. Invent an own process with Boolean state vectors (like in Figure 3). Compute a basis of the desired ideal like it is explained in the Examples 2-7.

25.6.: select one(!) of the following topics and fill out the checklist according to your actions. Send it via eMail to me.
Topic1: Create an idea, how you would define a drift-diffusion model for an "innocent agent" who moves in a "more or less infected environment" where V(x) describes the "intensity of infection at point x in space".
Topic2: Imagine a biochemical process consisting of multiple steps (in the end: a ligand activates a receptor). Write down the transition rate matrix "from the perspective of the ligand" and also "from the perspective of the protein".
Topic3: Construct a transition probability matrix which has non-real eigenvalues. Compute a (sorted) Schur decomposition and interpret the leading Schur-Vectors.
Topic4: What is the theory of eigenvalues and eigenvectors of permutation matrices? What is the theory of eigenvalues and eigenvectors of directed graph Laplacians having sinks?

TUTORIALS: The last tutorials (July 8th and July 15th) do not take place in the seminar room. They are open and voluntary for communication with the teaching team in order to prepare for the oral exams. Please, ask (via email) for personal meetings with us.

ORAL EXAMS: I will evaluate your oral presentations based on the assessment scheme (German version) available on the University of Jena's website [LINK]. Your submitted checklists will serve as the foundation for my questions. You may choose to focus the discussion on one of your favorite topics. If you do so, please be prepared to answer more in-depth questions. Oral examinations will last between 20 and 30 minutes. If you have prepared a full-length lecture (90 minutes), it will be assessed using the same scheme, and you should expect questions from the audience.

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