## Scientific Computing Courses

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Courses for Scientific Computing

#### IAM500 - M.S. Thesis

Credit: 0(0-0); ECTS: 50.0

The Program of research leading to M.S. degree arranged between the student and a faculty member. Students register to this course in all semesters while the research program or write up of thesis is in progress. Student must start registering to this course no later than the second semester of his/her M.S. study.

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Credit: 3(3-0); ECTS: 8.0

The course consists of a detailed description of continuous and discrete dynamical systems. We shall combine the introduction to the general theory with the consideration of bifurcations and chaos, the most important subtopics. The analysis of appropriate mechanical, physical, economic and biological models is an essential part of almost every lecture of the course. To support the course numerical and computational toolbox will be used.

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Credit: 3(3-0); ECTS: 8.0

Mean-Variance (Markowitz) analysis; continuous-time market model in finance; options and exotic options, pricing (valuation) of options; self-financing, optimal strategies, optimal portfolios (problems); martingale method; stochastic control and portfolio optimization.

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Credit: 3(3-0); ECTS: 8.0

Brief introduction to Statistical Learning: Regression versus Classification; Linear Regression: simple and multiple Linear Regression; Classification: Logistic Regression, Discriminant Analysis; Resampling Methods: Cross-Validation, the Bootstrap; Regularization: Subset Selection, Ridge Regression, the Lasso, Principle Components and Partial Least Squares Regression; Nonlinear Models: Polynomial; Splines; Generalized Additive Models; Tree-Based Models: Decision Trees, Random Forest, Boosting; Support Vector Machines; Unsupervised Learning: Principle Component Analysis, Clustering Methods.

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Credit: 3(3-0); ECTS: 8.0

Mathematical modelling of stochastic reaction systems. Deterministic approach: ODE models, Reaction Rate Equations. Stochastic Models: Chemical Master Equation, Chapman-Kolmogorov Equations, Gillespie Algorithms, Explicit Solution Formulas, Hybrid Methods, Tau-Leaping method. Lotka-Volterra Models, Michaelis-Menten Models.

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Credit: 3(3-0); ECTS: 8.0

Computer Arithmetic; Linear Equations: Gauss elimination, LU decomposition; Linear Least Squares: data fitting, normal equations, orthogonal transformations; Eigenvalue Problems; Singular Value Decomposition; Nonlinear Equations: bisection, fixed-point iteration, Newton’s method, optimization; Interpolation: polynomials, piecewise polynomials; Numerical Differentiation and Integration.

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Credit: 3(3-0); ECTS: 8.0

Ordinary Differential Equations: Euler’s method, multistep methods, Runge-Kutta methods, stiff equations, adaptivity; Boundary Value Problems: shooting, collocation, Galerkin; Partial Differential Equations: parabolic, hyperbolic, and elliptic equations; Iterative Methods for Sparse Linear Systems: splitting methods.

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#### IAM563 - Methods of Applied Mathematics

Credit: 3(3-0); ECTS: 8.0

Development of exact, numerical and approximate solution methods towards a qualitative strategy for understanding the behavior of ordinary differential equations. Methods for derivation, solution and computation of partial differential equations models based on several examples taken from classical sciences. Linear PDEs, investigated through the development of various solution techniques: first-order quasilinear equations, elliptic, parabolic and hyperbolic equations, free boundary value problems, quasi-linear equations, eigenfunction expansions, Green's functions and integral transformations, linear and nonlinear wave phenomena.

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Credit: 0(0-4); ECTS: 4.0

Basics of programming, introducing MATLAB, programming with MATLAB, basic algorithms and problem solving in Linear Algebra, Differential Equations, Optimization, and so an. Reporting and presenting problems and their solutions, introducing LATEX and/or Scientific Workplace, Typesetting text and mathematical formulae,graphing, making bibliography and index, packages and defining your own styles.

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Credit: 3(2-2); ECTS: 8.0

This course is intended to all students at the Institute. After a short introduction to Matlab various algorithms, their complexity will be introduced and symbolic, numerical and stochastic algorithms will be followed. Students will be encouraged to carry out several projects in groups. Moreover, students in groups will complete a term project at the end of the semester.

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Credit: 3(3-0); ECTS: 8.0

Unconstrained Optimization: steepest descent, line search methods, trust-region methods, conjugate gradient methods, Newton and quasi-Newton methods, large-scale unconstrained optimization, least-square problems; Theory of Constrained Optimization; Linear Programming: simplex method, interior point method; Quadratic Programming; Active Set Methods; Interior Point Methods; Penalty, Barrier and Augmented Lagrangian Methods; Sequential Quadratic Programming.

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Credit: 3(3-0); ECTS: 8.0

Models and cases studies from biology, engineering and mechanics, in form of ordinary and partial differential equations. Geometric and discrete models. Elements of inverse problems (image and data processing). Stochastic models in finance. Coursework and computer lab with MATLAB.

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Credit: 3(3-0); ECTS: 8.0

Introduction to transport phenomena, heat transfer, mass transfer and momentum transfer. Conservation laws, macroscopic balances with and without generation.

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Credit: 3(2-2); ECTS: 8.0

Wavelets, multiresolution analysis, applications of wavelets and multiresolution techniques. General constructions. Some important wavelets. Compactly supported wavelets. Multivariable wavelets. Estimators and Laplacian Pyramid decomposition. Adaptive de-noising and lossy compression. Parameter estimation. Uses in inverse problems. Uses in modelling and attractor reconstruction.

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Credit: 3(3-0); ECTS: 8.0

Classification of dynamical systems according to resolution, basic qualitative features of dynamical systems, introduction of hybrid systems and their examples, hybrid systems according to different concerns, linear and piecewise linear systems, qualitative features of hybrid dynamical systems , hybrid models of complex non-linear dynamic systems Hybrid models of dynamical systems with delay, inference problem, intervention problem, hybrid control systems.

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Credit: 3(3-0); ECTS: 8.0

Mathematical Fundamentals of Differential Quadrature method based on Polynomials and Fourier expansion, solution techniques for resulting equations, computation of weighting coefficients. Applications to Burger’s, Helmholtz, wave and Navier-Stokes equations. Applications to beams, thin plates, heat transfer, chemical reactor and Lubrication problems. Computer implementations.

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Credit: 3(3-0); ECTS: 8.0

Abstract Finite Element Analysis: weak derivatives, Sobolev spaces, Lax-Milgram lemma; Piecewise Polynomials Approximations 1D and 2D: interpolation, projection; Finite Element Method 1D and 2D: weak formulation, derivation of linear system of equations, a priori estimates; Time Dependent Problems: finite differences for systems of ODE, stability estimates; Semi-elliptic equations; a posteriori Error Analysis: estimator, mesh Refinement

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Credit: 3(3-0); ECTS: 8.0

Estimation theory, various estimation problems, modeling of deterministic systems, modeling of stochastic processes, linear estimators, nonlinear estimation, system identification, maximum likelihood and least squares estimation, denoising, impulse analysis of systems, density estimation.

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Credit: 3(3-0); ECTS: 8.0

Contents vary from year to year according to interest of students and instructor in charge.

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Credit: 2(2-0); ECTS: 6.0

LaTeX and Matlab; Basic Commands and Syntax of LaTeX and Matlab; Working within a Research Group via Subversion; Arrays and Matrices; Scripts and Function in Matlab; Commands and Environments in LaTeX; More on Matlab Functions; Toolboxes of Matlab; Packages in LaTeX; Graphics in Matlab; Handling Graphics and Plotting in LaTeX; Advanced Techniques in Matlab: memory allocation, vectoristaion, object orientation, scoping, structures, strings, file streams.

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Credit: 2(2-0); ECTS: 6.0

Review of Programming and Toolboxes, Packages, Modules; Iterative Linear Algebra Problems; Root Finding Programs; Recursive Functions and Algorithms; Optimisation Algorithms; Data Fitting and Interpolation; Extrapolation; Numerical Integration; Numerical Solutions of Differential Equations: IVPs and BVPs; Selected Topics (algorithms and coding in different fields).

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Credit: 0(0-0); ECTS: 130.0

Program of research leading to Ph.D. degree arranged between the student and a faculty member. Students register to this course in all semesters starting from the beginning of their second semester while the research program or write up of thesis is in progress.

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Credit: 3(3-0); ECTS: 8.0

Numerical Methods for Discrete Time Models: binomial method for options; discrete time optimal control problems. Reminders on Continuous Models: Ito process and its applications in stock market, Black-Scholes equation and its solution; Hedging, Volatility smile. Monte Carlo Method for Options: generating random numbers, transformation of random variables and generating normal variates; Monte Carlo integration; pricing by Monte Carlo integration; variance reduction techniques, quasi-random numbers and quasi-Monte Carlo method. Finite Difference Methods for Options: explicit and implicit finite difference schemes, Crank-Nicolson method; Free-Boundary Problems for American options. Finite Difference Methods for Control Problems: Markov Chain approximation method, elliptic Hamiltion-Jacobi-Bellman equations, computational methods.

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Credit: 3(3-0); ECTS: 8.0

Classification of inverse problems, linear regression, discretizing continuous inverse problems, rank-deficiency, Tikhonov regularization, iterative methods, other regularization techniques, Fourier techniques, nonlinear inverse problems, Bayesian methods. Computer applications and MATLAB exercises are important elements of the course.

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Credit: 3(3-0); ECTS: 8.0

Globalization techniques, semidefinite and conic optimization, derivative free optimization, semi-infinite optimization methods, Newton Krylov methods, nonlinear parameter estimation and advanced spline regression, multi-objective optimization, nonsmooth optimization, optimization in support vector machines.

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Credit: 3(3-0); ECTS: 8.0

Generating Random Numbers; Basic Principles of Monte Carlo; Numerical Schemes for Stochastic Differential Equations; Simulating Financial Models; Jump-Diffusion and Levy Type Models; Simulating Actuarial Models; Markov Chain Monte Carlo Methods.

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Credit: 3(3-0); ECTS: 8.0

Reduced Order Modeling: proper orthogonal decomposition (POD), evolution problems; Active Subspaces: parametrized models in physics and engineering, discover the active subspaces, exploit the active subspaces, active subspaces in action; Dynamic Mode Decomposition: introduction, Koopman analysis; PDE-constrained optimization: elliptic and parabolic linear optimal control problems; equality and inequality constraints; numerical algorithms for PDE-constrained optimization; reduced order modeling.

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Credit: 3(3-0); ECTS: 8.0

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge.

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Credit: 3(3-0); ECTS: 8.0

Adaptive Finite Elements and Optimal Control.

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Credit: 3(3-0); ECTS: 8.0

Description of fluids and flows. Derivation of Navier-Stokes equations (conservation of mass, conservation of momentum and conservation of energy). Numerical treatment of the Navier-Stokes equations by finite difference method. Applications: lid driven cavity problem, flow over a backward-facing step, free boundary problems, heat flow, natural convection with heated lateral walls, chemical transport. Extension to three-dimensions with examples from environmental sciences, architecture and engineering. Finite element method and applications.

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Credit: 3(3-0); ECTS: 8.0

Variational formulation of differential equations, ﬁnite element spaces and ﬁnite elements discretization, a posteriori error estimators for elliptic problems, residual error estimators, parabolic problems, spacetime adaptivity, implementation and numerical examples.

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Credit: 3(3-0); ECTS: 8.0

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge.

See the course in IAM Catalogue or METU Catalogue

Credit: 3(3-0); ECTS: 8.0

Iterative Solvers for linear and nonlinear systems of equations, precondtioning technics, analysis and implementation of the algorithms

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Credit: 3(3-0); ECTS: 8.0

Probability, Random Processes, and Statistics; Markov Chains; Sampling and Monte Carlo Methods; Parameter Estimation; Uncertainty Propagation in Models; Stochastic Spectral Methods; Surrogate Models and Advanced Topics.

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Credit: 3(3-0); ECTS: 8.0

Introduction into dynamical systems concepts and numerical methods for solving reaction-diffusion equations occurring in biology, chemistry and physics.

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Credit: 3(3-0); ECTS: 8.0

DG in One Spatial Dimension: linear system, implementation in MATLAB; Higher Dimensional Elliptic Problems: interior penalty methods, variational formulation, a priori error estimates, implementation in MATLAB, local discontinuous Galerkin method; DG for Convection Diffusion Problems: upwind scheme; Construction of Finite Element Spaces: Lagrange, Hermite, etc.; A Posteriori Error Analysis: residual-based, goal-oriented, hierarchical, equilibrated error estimators; Hybrid Discontinuous Galerkin Methods.

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Credit: 3(3-0); ECTS: 8.0

Convexity; Gradient Descent; Stochastic Gradient Methods; Noise Reduction Methods; Second-Order Methods; Adaptive Methods; Methods for Regularized Models; Introduction to Machine Learning: support machine vector, neural network.

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