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Multiscale methods (Introduction to mathematical physics), fall 2011


Paolo Muratore-Ginanneschi




Advanced studies


The course is intended for undergraduate students of mathematics, physics and chemistry
and mathematical biology. Prior courses in advanced calculus and linear algebra are required
(Analyysi I+II, Vektorianalyysi and Lineaarialgebra 1, or Mapu 1-2).

Description and contents

The scope of the course is to give a practical introduction to methods of perturbation theory
of wide use in theoretical and mathematical physics and mathematical biology. In order to
illustrate the use of these techniques the second part of the course will focus on specific
problems discussed in recent research papers.


Weeks 36-42 and 44-50, Tuesday 14-16 in room C123. Two hours of exercise classes per week.

Notice: last two lectures
Friday 02.12 at 14-16 in C123
Friday 09.12 at 14-16 in B321


Exam projects can be based on research articles using multi-scale perturbation theory. Besides those available from Pavliotis's
web-page, research articles of interest may be:

  • M. Vergassola and M. Avellaneda, Scalar Transport in Compressible Flow, Physica D, 106 (1997) 148-166 arXiv:chao-dyn/9612001
  • A. Mazzino, S. Musacchio, A. Vulpiani, Multiple-scale analysis and renormalization for pre-asymptotic scalar transport,
    Phys. Rev. E 71, 011113 (2005), arXiv:nlin/0412004.
  • L. Biferale, A. Crisanti, M. Vergassola, A. Vulpiani, Eddy diffusivities in scalar transport, Phys. Fluids 7, 2725 (1995), cond-mat/9412115
  • Avellaneda, M. & Majda, A. J. An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows Com. Math. Phys., 1991, 138, 339-391

Lecture Notes

The lecture notes cover and sometimes integrate the material expounded in the lections. They also give bibliographic references for the same topics.

Lectures 1-10

Lecture_01-02: Motivating examples

Lecture_02: Functional Analysis Background

Lecture_03: Two-scale convergence

Lecture_04: Elements of Ito calculus


  • Course lecture notes


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Exercise groups







On Fridays

10-12 a.m.


Seyedali Zahabi

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