MathMods :: Joint MSc

In the class “Numerical Methods for Nonlinear Equations,” we discuss the treatment of nonlinear problems in general Banach spaces(finite and infinite dimensional). We examine convergence properties of general iterative methods and apply these principles to specific iterative methods, such as Newton's method.
The course includes a thorough convergence analysis of Newton's method and explores various Newton-type methods. Typically, the iterative methods covered provide a single solution to a given problem. However, the deflation method can be employed to obtain multiple solutions.
Additionally, for challenging but smooth nonlinear problems, continuation methods (homotopy methods) can be used to find solutions. These methods are also part of the lecture content.

This course is designed to provide an introduction to mathematical foundations, concepts, and constructs for artificial intelligence and machine learning algorithm design. A subset from the below exhaustive list of topics would be covered.
Linear Algebra and Matrix Analysis:
Systems of Linear Equations, Vector Spaces, Linear Independence, Basis & Rank, Orthonormal Basis, Orthogonal Complement, Orthogonal Projections, Rotations, Eigenvalue Decomposition & Diagonalization, Singular Value Decomposition, Matrix Approximation.
Vector Calculus and Continuous Optimization:
Gradients of Vector-Valued Functions, Gradients of Matrices, Automatic Differentiation, Higher-Order Derivatives, Linearization and Multivariate Taylor Series, Unconstrained Optimization, Constrained Optimization & Lagrange Multipliers.
Probability and Distributions:
Probability Space, Discrete and Continuous Probabilities, Sum Rule, Product Rule, & Bayes’ Theorem, Summary Statistics and Independence, Gaussian Distribution, Conjugacy and the Exponential Family, Change of Variables/Inverse Transform.
Models and Data:
Models Learning & Selection, Empirical Risk Minimization, Parameter Estimation, Probabilistic Modeling & Inference, Directed Graphical Models, Bayesian Linear Regression, Dimensionality Reduction with Principal Component Analysis, Density Estimation with Gaussian Mixture Models, Classification with Support Vector Machines.