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MRES.A.02 – Scientific Computing and Mathematical Modeling

 

Mathematical Modeling

  • Deterministic and stochastic mathematical models.
  • Mathematical modeling with dynamic systems and differential equations.

Introduction to Scientific Programming (S.P.), Modern S.P. Environments. Computer Errors

  • Solving mathematical problems in scientific programming environments (Matlab, Mathematica, Python, Fortran). Numerical and symbolic calculations on a computer. Double, quadruple and higher precision calculations.
  • Numerical calculation errors on the computer.

Numerical Linear Algebra in S.P. environments

  • Numerical Linear Algebra Methodologies in an S.P. environment. (solving linear systems, factorizations of matrices, calculation of eigenvalues, SVD).

Methodologies of approximation of functions and scientific data in S.P. environments.

  • Interpolation and Approximation of functions and data.
  • Interpolatory Procedures.
  • Least Squares Approximation.
  • Statistical processing and data analysis methodologies.

Optimization Methodologies in S.P. Environments

  • Optimization Methodologies with or without conditions.
  • Finding minimum of cost functions with classical or differential-evolutionary algorithms.
  • Solving equations of non-linear systems.

Differentiation, Integration, Differential Equations

  • Numerical Integration and Differentiation.
  • Numerical Solution of Ordinary Differential Equations
  • Methodologies of solving Partial Differential Equations.

Introduction of parallel computation in modern S.P. Environments

Students who successfully complete this module are expected to be able to:

  • understand basic scientific programming methodologies for solving mathematical problems.
  • implement solutions using the capabilities provided by modern scientific programming environments rather than programming them from scratch.
  • understand the mathematical nature of the problem that they are asked to solve, determine its parameters and develop the solution using tools provided by modern scientific programming environments.
  • Undergraduate courses on Mathematical Analysis
  • A course on Introduction to Linear Algebra
  • A course on programming (Matlab, Python, Julia, R, …)
  • A course on Numerical Analysis (optional).

Student evaluation comes from

  • Class participation and contribution in the discussions held in class and online x 20%
  • Average Grade of Homework Assignments (best 4 out of the total of 5 grades obtained) x 40%
  • Final written exam on computer x 40%
  1. Numerical Analysis, Burden R., Faires J. D, Brooks\Cole.
  2. A First Course in Numerical Analysis, A. Ralston, Ph. Rabinowitz, Mc Graw Hill.
  3. Numerical Methods using Matlab, J. Mathews, K. Fink, Pearson Prentice Hall.
  4. Applied Numerical Analysism C. Gerald, P. O. Wheatley, Addison Wesley.
  5. Applied Numerical Analysis Using Matlab, L. Fausett, Pearson Prentice Hall.
  6. Numerical Methods for Engineers, With Software and Programming Applications Fourth Edition, S.C. Chapra, R.P. Canale , MC Geaw Hill, 2002
  7. Numerical Python, Scientific Programming and Data Science Applications with Numpy, Scipy and Matplotlib, R. Johansson, Apress
  8. Practical Numerical and Scientific Computing with MATLAB and Python”, 1st edition, Eihab B. M. Bashie, CRC Press “
  9. Learning Scientific Programming with Python, Christias Hill

Relevant Scientific Journals:

  1. SIAM Journal on Numerical Analysis
  2. International Journal for Numerical Methods in Engineering
  3. Applied Numerical Mathematics
  4. Journal of Computational and Applied Mathematics
  5. Numerical Algorithms
  6. Numerische Mathematik
  7. Scientific Programming

TOOLS

WEBSITES

Instructor(s) : Ioannis Th. FAMELIS  –  Christos Chorianopoulos – Perikles Papadopoulos

MRES.A.02 – Scientific Computing and Mathematical Modeling