Advanced Numerical Analysis

Lecture 1: Introduction and Overview

Lecture -2 Fundamentals of Vector Spaces

Lecture 3 : Basic Dimension and Sub-space of a Vector Space

Lecture 4 : Introduction to Normed Vector Spaces

Lecture 5 : Examples of Norms,Cauchy Sequence and Convergence, Introduction to Banach Spaces

Lecture 6 : Introduction to Inner Product Spaces

Lecture 7 : Cauchy Schwaz Inequality and Orthogonal Sets

Lecture 8 : Gram-Schmidt Process and Generation of Orthogonal Sets

Lecture 9 : Problem Discretization Using Appropriation Theory

Lecture 10 : Weierstrass Theorem and Polynomial Approximation

Lecture 11 : Taylor Series Approximation and Newton’s Method

Lecture 12 : Solving ODE – BVPs Using Firute Difference Method

Lecture 13 :Solving ODE – BVPs and PDEs Using Finite Difference Method

Lecture 14 : Finite Difference Method (contd.) and Polynomial Interpolations

Lecture 15 : Polynomial and Function Interpolations,Orthogonal Collocations Method for Solving ODE -BVPs

Lecture 16 : Orthogonal Collocations Method for Solving ODE – BVPs and PDEs

Lecture 17 :Least Square Approximations, Necessary and Sufficient Conditions for Unconstrained Optimization

Lecture 18 : Least Square Approximations :Necessary and Sufficient Conditions for Unconstrained Optimization Least Square Approximations ( contd..)

Lecture 19 :Linear Least Square Estimation and Geometric Interpretation of the Least Square Solution

Lecture 20 : Geometric Interpretation of the Least Square Solution (Contd.) and Projection Theorem in a Hilbert Spaces

Lecture 21 : Projection Theorem in a Hilbert Spaces (Contd.) and Approximation Using Orthogonal Basis

Lecture 22 :Discretization of ODE-BVP using Least Square Approximation

Lecture 23 : Discretization of ODE-BVP using Least Square Approximation and Gelarkin Method

Lecture 24 : Model Parameter Estimation using Gauss-Newton Method

Lecture 25 : Solving Linear Algebraic Equations and Methods of Sparse Linear Systems

Lecture 26 : Methods of Sparse Linear Systems (Contd.) and Iterative Methods for Solving Linear Algebraic Equations

Lecture 27 : Iterative Methods for Solving Linear Algebraic Equations

Lecture 28 : Iterative Methods for Solving Linear Algebraic Equations: Convergence Analysis using Eigenvalues

Lecture 29 :Iterative Methods for Solving Linear Algebraic Equations: Convergence Analysis using Matrix Norms

Lecture 30 : Iterative Methods for Solving Linear Algebraic Equations: Convergence Analysis using Matrix Norms (Contd.)

Lecture 31 : Iterative Methods for Solving Linear Algebraic Equations: Convergence Analysis (Contd.)

Lecture 32 :Optimization Based Methods for Solving Linear Algebraic Equations: Gradient Method

Lecture 33 : Conjugate Gradient Method, Matrix Conditioning and Solutions of Linear Algebraic Equations

Lecture 34 : Matrix Conditioning and Solutions and Linear Algebraic Equations (Contd.)

Lecture 35 : Matrix Conditioning (Contd.) and Solving Nonlinear Algebraic Equations

Lecture 36 : Solving Nonlinear Algebraic Equations: Wegstein Method and Variants of Newton’s Method

Lecture 37 : Solving Nonlinear Algebraic Equations: Optimization Based Methods

Lecture 38 : Solving Nonlinear Algebraic Equations: Introduction to Convergence analysis of Iterative Solution Techniques

Lecture 39 : Solving Nonlinear Algebraic Equations: Introduction to Convergence analysis (Contd.) and Solving ODE-IVPs

Lecture 40 :Solving Ordinary Differential Equations – Initial Value Problems (ODE-IVPs) : Basic Concepts

Lecture 41 :Solving Ordinary Differential Equations – Initial Value Problems (ODE-IVPs) : Runge Kutta Methods

Lecture 42 :Solving ODE-IVPs : Runge Kutta Methods (contd.) and Multi-step Methods

Lecture 43 :Solving ODE-IVPs : Generalized Formulation of Multi-step Methods

Lecture 44 : Solving ODE-IVPs : Multi-step Methods (contd.) and Orthogonal Collocations Method

Lecture 45 : Solving ODE-IVPs: Selection of Integration Interval and Convergence Analysis of Solution Schemes

Lecture 46 : Solving ODE-IVPs: Convergence Analysis of Solution Schemes (contd.)

Lecture 47 :Solving ODE-IVPs: Convergence Analysis of Solution Schemes (contd.) and Solving ODE-BVP using Single Shooting Method

Lecture 48 : Methods for Solving System of Differential Algebraic Equations

Lecture 49 : Methods for Solving System of Differential Algebraic Equations (contd.) and Concluding Remarks