Module IV: Inverse Problems in Empirical Modeling

This module provides a comprehensive introduction to solving a variety of inverse problems: linear/nonlinear, over/under determined, deterministic/statistical, weighted/unweighted, batch/sequential modes. Two main approaches are covered: matrix-based decomposition using direct multiplicative methods (LU, LDLT, Cholesky, QR, SVD algorithms), and standard optimization methods (Gradient, Conjugate gradient, Quasi-Newton algorithms).

Schedule

Date & Time Tuesdays and Thursdays; 3:30 PM to 5:00 PM

Classroom Room B303, EE Department, IISc

Duration 4-5 Lectures

Lectures

Static Linear Inverse Problem : Well posed problems

Static Linear Inverse Problem : ill- posed problems

Geometric view of least sqaures

Deterministic Static Non Linear

Least Square Problem

Example of Static Inverse Problem

Variance Bias Tradeoff

Matrix Decomposition Methods

Minimization Algorithm

Statistical Estimation

Max Likelihood Method

Bayesian Estimation

From Gauss to Kalma Sequential linear Minimum Variance Estimation

Resources

J. M. Lewis, S. Lakshmivarahan and S. K. Dhall (2006) Dynamic Data Assimilation: a least squares approach, Cambridge University Press, Encyclopedia in Mathematics and its Applications, Vol 104

G. Golub and C. F. Van Loan (1989) Matrix Computations, Johns Hopkins Press, Baltimore

S. Lakshmivarahan (2024) Lecture Notes on Bigdata Analytics, Handwritten notes used for a set of forty lectures under the auspices of the Swayam Prabha, IIT-Madras

⁠Blendenpik: Supercharging LAPACK's Least-Squares Solver

⁠Faster least squares approximation

Applications of concentration of measure in signal processing

Concentration of measure: fundamentals and tools

IE 498: Online Learning and Decision Making: Lecture 01 & 02