Stochastic algorithms (CK0191)

The 2018.1 version of CK0191 momentarily takes over the usual content of the course. In 2018.1, we overview some basic facts in the theory of deterministic and random Markovian functions.
  1. Quick refreshers: Probability and random variable theory, dynamical systems analysis;
  2. Deterministic Markov processes: State-space analysis, Lagrange formula, similarity transformations, Jordan's form
  3. Stochastic Markov processes: General features, characterising functions, time-evolution equations;
  4. Discrete Markov processes: Classification of states, irreducibility, important matrixes and distributions;
  5. Continuous Markov processes: Propagators, time-evolution equations;
If time allows, selected topics related to jump Markov processes will be mentioned (Propagators, time-evolutions).



Instructor ♞: Francesco Corona (FC), francesco döt corona ät ufc döt br

Physical location ♜: Fridays, 14:00-18:00, Bloco 951, Sala 10.
Internet location ♖: Here! Or, here (CK0191) for mambojumbo related to administration.

Evaluation ✍: Approximately half a dozen theoretical and practical problem sets will be assigned as homework: Home assignments are for training but are not mandatory, they can be handed-in but they will not be evaluated. The actual evaluation will be based on three or four partial evaluations (APs) in class (weight 70%) and a final project (weight 30%). If needed a final evaluation (AF) will be arranged.

Test (MAY 06)

Go to:   Lectures and schedule | Problem sets | Supplementary material | As it pops out |

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Lectures and schedule

  1. About this course

    A Preliminaries (FC)

    Intro (FEB 23)

  2. Refreshers

    A Dynamical systems (FC)
    • Input-output and state-space representations, characteristics
    • (Elements of ordinary differential equations, theory of distributions, linear algebra)
    B Probability theory (FC)
    • Laws of probability, random variables, useful theorems, random numbers


  3. Deterministic Markov processes

    A Input-output and state-space representation (FC)
    • Representation and analysis
    • Homogeneus equation and modes

    • Analysis and state transition matrix
    • Sylvester expansion
    B Time evolution (FC)
    • Force-free evolution and types of modes
    • Impulse response and forced evolution
    • Durhamel's integral

    • Lagrange formula, force-free and forced evolution
    • Similarity transformations and diagonalisation
    • Jordan's form, generalised eigenvectors
    • Generalised modal matrix and state transition matrix
    • State transition matrix and modes


  4. Stochastic Markov processes

    A Markov chains (FC)

    (MAY 11, MAY 18, MAY 25 and JUN 01)
    • Definitions, dependence and embeddeability
    • Chapman-Kolmogorov equations
    • Probability distributions
    • Classification of states
    B General processes (FC)
    • State density function, Chapman-Kolmogorov equations
    • Propagators and time-evolution equations
    C Continuous Markov processes (FC)
    • Propagator and time-evolution equations
    • Stable processes and weak noise approximation

  5. Slides/DDMP/Part A (Last update MAY 24)
  6. Slides/DDMP/Part B (Last update JUN 01)
  7. Exercises

  8. Jump Markov processes

    A Continuous states (FC)
    • Propagator and time-evolution equations
    • Self-diffusion and brownian motion
    B Discrete states (FC)
    • Propagator, time-evolution equations
    • Master equations

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Problem sets

As we use problem set questions covered by books, papers and webpages, we expect you not to copy, refer to, or look at the solutions in preparing your answers. We expect you to want to learn and not google for answers: If you do happen to use other material, it must be acknowledged clearly with a citation on the submitted solution.

☯ The purpose of problem sets is to help you think about the material, not just give us the right answers.

Homeworks must be done individually: Each of you must hand in his/her own answers. In addition, each of you must write his/her own code when requested. It is acceptable, however, for you to collaborate in figuring out answers. We are assuming that you take the responsibility to make sure you personally understand the solution to any work arising from collaboration (though, you must indicate on each homework with whom you collaborated).

⚛ To typeset assignments, students are encouraged to use this LaTeX template: Source (PDF).

⚠ Assignments must be returned via SIGAA.

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Material

Course slides will suffice. Slides are mostly based on the following textbook: The material can be complemented using material from the following textbooks (list not exhaustive):
  1. Stochastic processes: From applications to theory, by Pierre Del Moral.
  2. Stochastic processes, by Joseph Doob.
☠ Copies of these books are floating around.

>>>>>> Course material is prone to a typo or two - Please inbox FC to report <<<<<<

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