# TEW - Linear Algebra with applications to data analysis and control (2019)

## Speakers and Syllabus

Objectives of the workshop :- With exponential growth of computing power and availability of historical data records in digital form is rapidly transforming industrial production systems. As a consequence, applied mathematics and computations have assumed significant importance in design and operation of engineered systems. Linear Algebra and its applications arguably assume a central position in engineering mathematics. While linear algebra related courses are taught in the first year of engineering, students often fail to grasp importance and relevance of this important tool. Thus, it is important to teach linear algebra with “engineering flavour” to students pursuing under-graduate and post-graduate studies engineering. Providing exposure to different engineering applications when various linear algebra tools are taught can help in assimilating ideas better and significantly enhance the learning experience.This course is aimed at teaching various linear algebra techniques with a strong emphasis on engineering applications. The course material will connect important concepts in linear algebra with real world engineering applications, which can be used to motivate UG and PG students.

Syllabus for the lecture series

 Name of the speakers with affiliation Topic Detailed syllabus Harish Pillai EE Department, IIT Bombay Solutions of Systems of linear equations Linear systems and their solutions -- when does a solution exist, how accurate is the solution, how to solve the equations while minimising the error, how to estimate whether the solution obtained is close to the "real" solution. In the process, I would touch on condition number of a matrix, Givens rotations, Householder transforms, LU decomposition. Sudhir R. GhorpadeMathematics Department, IIT Bombay Applications of linear algebra to differential equations We will begin with a quick review of basics of systems of linear equations and the notions of eigenvalues, eigenvectors and diagonalization of matrices. Then we consider systems of the first order linear differential equations and outline how the methods of linear algebra can be applied to the problem of determining their solutions. We shall also see how this relates to solving homogeneous linear differential equations of nth order. Applications to electrical networks may also be indicated. J. K. Verma Mathematics Department, IIT Bombay Singular Value decomposition of rectangular matrices The singular value decomposition is closely associated with the eigenvalue-eigenvector decomposition of symmetric matrices. It applies to any rectangular matrices. It has numerous applications. We shall discuss applications to the least squares problem and polar decomposition of matrices. Vikram GadreElectrical Engineering Department, IIT Bombay Transforms in image and signal processing Introducing Transforms: Fourier Transform, Wavelet Transform, Radon Transform and generalising to multiple dimensions. Connecting transforms to linear algebra principles and illustration with examples. Mathematical principles of filtering: examples of signal and image filters. S. A. Soman Electrical Engineering Department, IIT Bombay Least squares approximations and power systems Solving Large and Spare Linear Least Squares Problem with Applications to Power System State Estimation. We will explore the following. Overview of Overdetermined Full Rank Linear Least Squares problem. Solving it using normal equations approach and QR decomposition. Sparsity issues and reduction of fill ins during LDU & QR factorisation. Formulation of power system state estimation problem as LSE problem. Sachin Patwardhan, Chemical Engineering Department, IIT Bombay Applications of singular and eigenvalues Module 1 (Applications of Singular Value Decomposition for Big-Data Analysis) Fault (abnormal behaviour) detection and diagnosis using Principle component analysis (PCA), Fault diagnosis using Fisher Discriminant Analysis, Soft sensor development using Partial Least Squares (PLS), time series modelling using Dynamic PCA. Module 2 (Applications of Eigenvalue based Analysis for Design and Control) Analysis of local stability of engineered systems using local linearisation and eigenvalue based analysis, linear feedback controller synthesis using eigenvalue assignment, convergence of numerical schemes for solving linear algebraic equations using eigenvalue based analysis

## Time Table

 Monday 18 Nov Tuesday 19 Nov Wednesday 20 Nov Thursday 21 Nov Friday 22 Nov Saturday 23 Nov 9.30-11.00 Pillai Pillai Verma Verma Soman Soman 11.00-11.15 Tea 11.15-12.45 Ghorpade Ghorpade Gadre Gadre Patwardhan Patwardhan 1.00-2.30 Lunch 2.30-4.30 Lab session Pillai Belur Sudarshan Aditya Ghorpade Gopi Rekha Ruma Verma Belur Sudarshan Aditya Gadre Venkitesh Kriti Hiranya Patwardhan Venkitesh Kriti Hiranya Soman Gopi Rekha Ruma 4.30-5.00 Tea 5.00-6.00 Bapat* Belur Karamchandani Borkar Netrapalli Deshpande * To be confirmed

Lab Instructors

 Name Affiliation Topics Dates ofthe Lab Prof. Madhu Belur P. Sudrshan Aditya Nadkarni EE Department, IIT Bombay 1. Linear equations 2. Singular Value decomposition 18 Nov20 Nov Venkitesh Iyer Kriti Goel Hiranya Kishore Dey Mathematics Department, IIT Bombay 1. Transforms in image and signal processing 2. Applications of singular and eigenvalues 21Nov 22Nov Gopikrishnan C. R. Rekha Khot Ruma Rani Maity Mathematics Department, IIT Bombay 1. ODEs and eigenvalues 2. Least squares approximations 19Nov 23Nov

Public Lectures at 5 p.m.

 Speaker Affiliation Preneeth Netrapalli Microsoft Research Amit Deshpande Microsoft Research Nikhil Karamchandani EE, IIT Bombay Madhu Belur EE, IIT Bombay Ravindra Bapat ISI, Delhi Vivek Borkar EE, IIT Bombay
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