This course is intended for graduate students (or undergraduate) who need a rapid and uncomplicated introductions to the field of applied mathematics involving computational linear algebra and differential equations. The lecture has two themes-how to understand equations, and how to solve them . This course include numerical linear algebra(QR,SVD, singular system), Newton’s method for minimization, Equilibrium and stiffness matrix, Least squares, Nonlinear problems, Covariances and Recursive Least squares, Differential equations and finite elements, Finite Difference and Fast Poisson, Boundary value problems in Elasticity and Solid mechanics.

The objective of this course is to offer basic knowledge of computational fluid dynamics based on fundamental understanding of fluid mechanics and numerical analysis. This course will include Navier-Stokes equation, Finite ifference method (FDM), Staggered-grid, Projection Method, Numerical simulation of unsteady viscous flows, Visualization, etc. Prerequisites: Fluid mechanics, Numerical Analysis

This course is designed to acquaint students in mathematical and physical sciences and engineering with the fundamental theory of numerical analysis. This course is devoted to nonlinear equations, optimization, approximation theory, numerical quadrature and numerical linear algebra (including linear systems, least squares problems and eigenvalue problems). The course willstress both on analytic and computational aspects of numerical methods. Prerequisites: Advanced Calculus and Linear Algebra (or Engineering Math), Programming skills

This course focuses on the fundamentals of modern numerical techniques for a wide range of linear and nonlinear elliptic, parabolic, and hyperbolic partial differential equations central to a wide variety of applications in science, engineering, and other fields. Topics includes : Mathematical formulations, Finite difference method, Finite element method Prerequisites: Programming skills, Advanced Calculus and Linear Algebra (or Engineering Math)

This is an introductory, graduate-level course on partial differential equations (PDE) for science and engineering. This course focuses on derivation, interpretation, and analysis for model equations including Laplace equation, heat, and wave equations. This course covers maximum principle and uniqueness results, variation principles, Lax-Milgram theorem and applications to boundary value problems. Prerequisites: Advanced Calculus, Linear Algebra (or Engineering Math), Ordinary Differential Equations.

This lecture presents a comprehensive introduction and overview of electromagnetic tissue property imaging techniques using MRI focusing on Magnetic Resonance Electrical Impedance Tomography(MREIT), Electrical Properties Tomography(EPT) and Quantitative Susceptibility Mapping (QSM). The contrast information from these novel imaging modalities is unique since there is currently no other method to reconstruct high-resolution images of the electromagnetic tissue properties including electrical conductivity, permittivity, and magnetic susceptibility. These three imaging modalities are based on Maxwell`s equations and MRI data acquisition techniques. They are expanding MRI`s ability to provide new contrast information on tissue structures and functions. Prerequisites: Vector calculus, Linear algebra, Differential equation, Numerical analysis

Fundamental understanding of fluid phenomena and application to real flow problems. Governing equations. Flow kinematics. Vorticity dynamics, Low-Reynolds number flow, flow instability, turbulence.

Fundamentals for advanced computation technology of turbulence will be treated including spectral method, Simple algorithm, immersed boundary method. Prerequisites: Turbulence theory

This course focuses on the fundamentals of finite element methods for a wide range of linear and nonlinear elliptic, parabolic, and hyperbolic partial differential equations and integral equations central to a wide variety of applications in science, engineering, and other fields. Finite element method is a numerical technique for finding approximate solutions to partial differential equations. It uses variational methods to minimize an error function and produce a stable solution. The theory of finite elements and their applications is such a lively area in science and engineering. This course introduces the finite element methods which features important material for both research and application. Prerequisites: Numerical analysis

The objective of this course is to offer strategies and tools to plan, develop, and deliver scientific presentations. Also students will be naturally exposed to cutting-edge technologies from invited seminars. Through questions and answers (Q & A) sessions, this course encourages the students to have a critical thinking and to enhance their communication skills.

The flow of particles and droplets in fluids has a wide application in thermo-fluid systems: pollution dispersions in the atmosphere, fluidization in combustion process, aerosol deposition, etc. This course introduces graduate students to knowledge of particle-laden flows with an emphasis on the effect of disperse phase particles on flows. This course will include properties of dispersed phase flows, particle-fluid interaction, particle-particle interaction, continuous phase equations, etc. Prerequisites: Fluid mechanics, Viscous fluid Flows

Fundamentals and application of turbulent flows are considered through investigation of theories and analysis on turbulence. Governing equations and various definitions and statistical tools to describe turbulence are introduced. Particularly scales of turbulence are considered for the better understanding of turbulence. Various models for engineering applcations are considered. Prerequisites: Viscous fluid flow

The objective of this course is to study on numerical analysis for fluid mechanics and thermal/mass transports. To this end, the course will include projection methods on staggered grid for solving Navier-Stokes equation, immersed boundary methods for complex geometry, level-set methods for tracking phase interface, etc. Prerequisites: CSE5002 Basic of Computational Fluid Dynamics

Image processing has become one of the most important components in medical imaging modalities such as MRI, CT, ultrasound and other functional imaging modalities. Image processing techniques such as image restoration and sparse sensing are being used to deal with various imperfections in data acquisition processes of the imaging modalities. Image segmentation, referred to the process of partitioning an image into multiple segments, has numerous applications including tumor detection, quantification of tissue volume, computerguided surgery, study of anatomical structure and so on. In this lecture, we begin with mathematical theories on PDE-based image restoration, sparse sensing, level set methods. We apply their theories to MRI, CT, QSM, and other medical imaging area. Prerequisites: Basic numerical analysis, Partial differential equations

This course is designed to provide the scenario of the medical imaging, its mathematical and physical aspects. This course is devoted to the studies on physics based modeling and simulation in medical imaging. The course will discuss about mathematical and physical aspects of magnetic resonance electrical impedance tomography (MREIT) and electrical impedance tomography (EIT) along with the studies on their simulation and modeling. Prerequisites: Basic Electrical Engineering, Matrix Algebra, Partial Differential Equation, Numerical Analysis, Programming skills in MATLAB or C

The objective of this course is to offer advanced knowledge of turbulence computations. This course will focus on turbulent simulation techniques and turbulent flow physics. Prerequisites: Fluid mechanics, Numerical analysis, Basic CFD, Viscous Flows, Turbulence, Special Topics in Turbulence, etc.

This course is to focus on fundamental aspects of numerical methods for stochastic computations. We consider a class of numerical methods including generalized polynomial chaos (gPC). Designed to simulate complex systems subject to random inputs, these methods are widely used in many areas of computer science and engineering. In this course we introduce polynomial approximation theory and probability theory; describes the basic theory of gPC methods through numerical examples and rigorous development; details the procedure for converting stochastic equations into deterministic ones; using both the Galerkin and collocation approaches; and discusses the distinct differences and challenges arising from high-dimensional problems. The last section is devoted to the application of gPC methods to critical areas such as inverse problems and data assimilation. Prerequisites: Numerical Analysis, PDE for Science and Engineering

This course studies the advanced theories and schemes in numerical analysis for various differential equations and integral equations of applications in science, engineering, and other fields. It focuses on the Sobolev Spaces and Domain Decomposition Methods. Prerequisites: Numerical Analysis, Advanced linear algebra, Finite element method, Numerical partial differential equations

This course is to study various numerical methods for convection dominated diffusion equations. Mathematical models that involve a combination of convective and diffusive processes are among the most widespread in all of science and engineering. Following topics will be discussed: finite difference method, finite element method, finite volume method, stability, streamline diffusion methods, discontinuous Galerkin methods, variational multiscale method.

This course is to study various numerical methods for convection dominated diffusion equations. Mathematical models that involve a combination of convective and diffusive processes are among the most widespread in all of science and engineering. Following topics will be discussed: finite difference method, finite element method, finite volume method, stability, streamline diffusion methods, discontinuous Galerkin methods, variational multiscale method.

This course covers numerical optimization. We will concentrate on convex optimization. For such purpose we will briefly cover the convex theory including convex set, convex function, sublevel set, epigraph, separating hyper-plane theorem, conjugate function, etc. Concerning duality, we will study Lagrange dual function, weak and strong duality, Karush-Khun-Tucker (KKT) conditions. For unconstrained optimization, we study algorithm including descent method, line search method, Newton’s method, for constrained optimization, Newton method will be covered for equality constraint problems and interior point methods including barrier method and primal dual method for inequality constraint problems. After that we will consider applications, approximation and fitting, l1 minimization, for examples. If time permits, other methods may be introduced: penalty methods, augmented Lagrangian methods, etc. Prerequisites: CSE5810 Numerical Analysis, MATLAB Coding skill

The Fourier transform is everywhere in physics and mathematics because it diagonalizes time-invariant convolution operators. It rules over linear time invariant signal processing, the building blocks of which are frequency filtering operators. However, wavelets are well localized and few coefficients are needed to represent local transient structures. As opposed to a Fourier basis, a wavelet basis defines a sparse representation of piecewise regular signals, which may include transients and singularities. In images, large wavelet coefficients are located in the neighborhood of edges and irregular textures. We will start from brief review on Fourier analysis in terms of signal processing. After that we will focus on wavelets, its multi-resolution analysis and sparse representation, and consider applications in signal and image processing. Application of such idea in data analysis may be partially considered. Prerequisites: Numerical Analysis 1, MATLAB coding skill

This course is to study advanced topics in finite element methods. The methods under consideration include standard Galerkin methods, Adaptive FEMs, Nonstandard finite element methods, Mixed FEMs, and discontinuous Galerkin (DG) methods. Applications will include Darcy equations, Stokes equations, and linearized Navier-Stokes known as Oseen equations.

This course is to focus on fundamental aspects of numerical methods for problems in sciencen and engineering. We will consider a various of numerical methods including adaptive finite element methods, stochastic computations, and generalized polynomial chaos (gPC).

The purpose of this course is twofold; sparse and redundant representation in signal and image processing and compressed sensing or compressive sampling in imaging. Actually those two are closely connected each other. Their motivation and history will be briefly introduced and basic concepts, definitions, analysis and algorithms will be explained. At the end of the semester students have to present topics which are related to their research area for an immediate application. Basic knowledge in numerical analysis including linear algebra, least squares, iterative method and optimization will be assumed. Prerequisites: CSE5810 Numerical Analysis 1, MATLAB Coding skill

This is very advanced CSE graduate course aiming to train Ph.D. students to bridge this gap, by providing insights into the various interfaces among mathematical theories, scientific computation and visualization of real world problems. Prerequisites for students with mathematical background are partial differential equation (graduate level) and numerical analysis (FEM). You could be graded on the quality of your accomplishment (level of your Toolbox) and your report covering all the process (modeling-analysis-numerical simulation-visualization-experiment-software, minimum 50 pages). The report should contain the following: 1. Mathematical Model: A. Understand underline physical phenomena and the constraints imposed on the problem. Understand PDE models which usually are the processes of information loss. B. Understand what are measurable using available engineering techniques. Practical limitations associated with the measurement noise, interface between target object and instrument, data acquisition time and so on must be properly understood and analyzed. C. Formulate problems in such a way that we can deal with them systematically and quantitatively. 2. Develop computer programs and properly address critical issues of numerical analysis. 3. Validate results by simulations and experiments. 4. Make software for visualization (Matlab Toolkit) We will deal with the following subjects: - Computational Electromagnetism (Maxwell`s equations) - Computational Elasticity - Computational Fluid Dynamics (Navier-Stokes equations) - Image processing (Level set methods, Denoising, Sparse sensing, Regularization) Programming Language: We will mostly use MATLAB with parallel computing toolbox. However, students are free to use language/environment of their own choices: (e.g Fortran, C/C++, OpenMP, MPI, CUDA, OpenCL, etc). Prerequisites: Partial differential equations, Numerical analysis

This is very advanced CSE graduate course combined with "Mathematical Modeling and Simulation and Visualization for Science I" aiming to train Ph.D. students to bridge this gap, by providing insights into the various interfaces among mathematical theories, scientific computation and visualization of real world problems. Prerequisites for students with mathematical background are partial differential equation (graduate level) and numerical analysis (FEM). Prerequisites: Partial differential equations, Numerical analysis, Numerical partial differential equations, Numerical linear algebra, Programming skills

Imaging techniques in science, engineering, and medicine have evolved to expand our ability to visualize internal information of an object such as the human body. Examples may include X-ray computed tomography (CT), magnetic resonance imaging (MRI), and positron emission tomography (PET). They provide cross-sectional images of the human body, which are solutions of corresponding linear inverse problems. Information embedded in such an image depends on the underlying physical principle which is described in its forward problem. Since each imaging modality has a limited viewing capability in terms of the kind of image contrast they provide, there have been numerous research efforts to develop a new technique producing contrast information not available from existing methods. To facilitate technical progress and also avoid reinventing the wheel by newcomers, we would like to offer a course of study on nonlinear inverse problems associated with these imaging modalities. Research outcomes during last three decades have accumulated enough knowledge and experience that we can deal with these topics in graduate programs of applied mathematics and engineering. This lecture covers nonlinear inverse problems associated with lately developed new imaging modalities. It focuses on methods rather than applications. The methods mainly comprise mathematical and numerical tools to solve the problems. Prerequisites: Vector calculus, Linear algebra, Differential equation, Numerical analysis

This course covers the main aspects of mathematical modeling, analysis, computation and control in biology employing epidemiology models. Various kinds of models and mathmatical tools that will be useful analyzing and implementing models will be presented. Prerequisites: Calculus, Linear algebra, Differential equtions, Programming skills, Basic knowlege in epidemic models

Mathematical models that involve a combination of convective and diffusive processes are among the most widespread in all of science and engineering. This course is to study advanced topics of finite element methods such as adaptivity, saddle point problems, stability. The methods under consideration include standard Galerkin, mixed finite element, and discontinuous Galerkin. Applications will include Darcy equations, Stokes equations, and linearized Navier-Stokes known as Oseen equations.