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1.1.1 Motivation
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applications 
methods 
model problems 
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Numerical methods are a part of the problem solving skills that are
expected to be mastered by most of the university graduates working
in a quantitative field.
The same fundamental concepts of Brownian motion, convection, diffusion,
dispersion and nonlinearity are used to simulate applications in
telecommunication (collisions of datapackets in a network, solitons in
optical fibers), economics (stock options), biology (transport in cellular
tissues), engineering (heat transfer, pollution) and social sciences
(behavior of people in a crowd).
Quantitative answers for the real world can generally be obtained only
from computations.
The goal of this course is to cover a wide range of numerical methods with
simple examples to learn the building blocs that are often used in complex
simulation codes. Indeed, a broader knowledge is often decisive to choose
the method that is best suited when developing a new code: in this course,
we will attribute an equal importance to finitedifferences, finiteelements,
fast Fourier transformation, MonteCarlo simulations and Lagrangian schemes.
In each case, the algorithm will be illustrated with a variety of prototype
problems including the FokkerPlank (transport), BlackScholes (stock
options), Burger (shock waves), KortewegDeVries (solitons) and the
Schrödinger equations (quantum mechanics).
A strong emphasis is put on a problem based learning, where all the steps
are discussed starting from the derivation, implementation and execution of
a model to the discussion of the merits and limitations that appear when
different methods are compared in the same setting.
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