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2022/2023

Applied Stochastic Processes

Code: 42253 ECTS Credits: 6
Degree Type Year Semester
4313136 Modelling for Science and Engineering OT 0 1

Contact

Name:
Daniel Campos Moreno
Email:
daniel.campos@uab.cat

Use of Languages

Principal working language:
english (eng)

Teachers

Aureli Alabert Romero
VicenÁ Mendez Lopez
Alvaro Corral Cano
Daniel Campos Moreno

Prerequisites

Calculus of several variables. Ordinary and partial differential equations. Introduction to probability theory

Objectives and Contextualisation

The main goal of this course is to provide powerfull tools to deal with the analysis and numerical simulations of stochastic processes both for systems affected by external noise of by internal noise. Applications to ecological and biological systems will be discussed in detail.

Competences

  • Apply logical/mathematical thinking: the analytic process that involves moving from general principles to particular cases, and the synthetic process that derives a general rule from different examples.
  • Apply specific methodologies, techniques and resources to conduct research and produce innovative results in the area of specialisation.
  • Apply techniques for solving mathematical models and their real implementation problems.
  • Conceive and design efficient solutions, applying computational techniques in order to solve mathematical models of complex systems.
  • Formulate, analyse and validate mathematical models of practical problems in different fields.
  • Isolate the main difficulty in a complex problem from other, less important issues.

Learning Outcomes

  1. Apply logical/mathematical thinking: the analytic process that involves moving from general principles to particular cases, and the synthetic process that derives a general rule from different examples.
  2. Apply specific methodologies, techniques and resources to conduct research and produce innovative results in the area of specialisation.
  3. Apply stochastic process techniques to predict the behaviour of certain phenomena.
  4. Apply stochastic process techniques to study models associated with practical problems.
  5. Identify real phenomena as models of stochastic processes and extract new information from this to interpret reality.
  6. Implement the proposed solutions reliably and efficiently.
  7. Isolate the main difficulty in a complex problem from other, less important issues.
  8. Use specific software to model stochastic processes and, depending on the situation, estimate the corresponding parameters.

Content

First Part:

  1. Elementary probability
  2. Stochastic processes. Noise and Markov processes
  3. Microscopic description: Stochastic differential equations and their integration. Applications to population dynamics

Second Part:

  1. Mesoscopic description: Master equation. One-step processes. Diffusion approach. Biological and physical examples.
  2. Random Walks. CTRW. Anomalous diffsuion, Lévy flights and First passage-time problems. Ecological and social applications

Third Part:

  1. Simulation of stochastic processes. Gillespie algorithm. Tau-leaping method. Reaction-diffusion methods. Next reaction method.

Methodology

 The methodology of the course will combine traditional lectures in the classroom with autonomous activities by the student to practice the concepts seen in the course.

As a novelty, this year the first and the second part of the course (theory and practical applications) will be done intertwined to facilitate the assimilation of the contents.

 

Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

Activities

Title Hours ECTS Learning Outcomes
Type: Directed      
Classroom sessions 38 1.52 1, 2, 4, 3
Type: Supervised      
Mentoring 9 0.36 5, 6
Type: Autonomous      
Personal work 30 1.2 2, 3, 5
Preparation of assignments 20 0.8 1, 2, 4, 6, 8

Assessment

The avaluation will be divided into the three parts of the course:

 

First part. (35%)

There will be a problems assignment (10% of the final mark) and a general exam of this part (25%)

Second part. (35%)

There will be a short project to work with the concepts and techinques seen in the classroom (15% of the final mark) and a short exam of this part (20%).

Third part (30%)

There will be several assignments to work with the numerical and somulation techniques seen in the classroom. (30%)

 

Assessment Activities

Title Weighting Hours ECTS Learning Outcomes
Exam of the practical part 20% 1 0.04 1, 4, 3, 8
Exam of the theoretical part 25% 2 0.08 1, 2, 4, 3, 5, 6
Simulations and practical works 55% 50 2 1, 2, 4, 3, 7, 5, 6, 8

Bibliography

Basic:

  • V. Méndez, D. Campos, F. Bartumeus. Stochastic Foundations in Movement Ecology, Springer-Verlag, 2014
  • C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer. Berlin. 1990
  • L.J.S. Allen, An Introduction to Stochastic Processes with Applications to Biology. Chapman & Hall/CRC, Boca Ratón. 2011
  • R. Toral, P. Colet. Stochastic Numerical Methods. Wiley-VCH, 2014

Complementary:

  • N. van Kampen, Stochastic Processes in Physics and Chemistry, Third Edition (North-Holland Personal Library) 2007
  • J. Rudnick and G. Gaspari. Elements of the Random Walk. Cambridge Univ. Press, 2004
  • N.C. Petroni. Probability and Stochastic Processes for Physicists. Springer-Verlag, 2020
  • N. Lanchier. Stochastic Modelling. Springer-Verlag, 2017

Software

The practical activities of the course will be carried out using Python language