Numerical Analysis
Prerequisite(s):
Fundamental Algebra and Analysis II (or Calculus A II for non-math majors)
Course Description:
This course provides a rigorous exploration of the mathematical foundations and computational techniques used to solve problems that are analytically intractable. Moving beyond traditional "idealized" mathematics, the curriculum is designed to bridge the gap between theoretical modeling and practical prediction.
Unlike traditional introductory courses, this program focuses deeply on the transition from static approximation to dynamic and stochastic systems. A significant portion of the course is dedicated to the numerical treatment of Ordinary Differential Equations (ODEs)—including stability for stiff equations and geometric integration—and Stochastic Simulation, which addresses the inherent uncertainty in modern physical, biological, and financial models. Students will learn to evaluate numerical methods based on the three pillars: Accuracy, Stability, and Efficiency.
Learning Objectives:Course Syllabus & Detailed Topics
By the end of this course, students will be able to:
Analyze Errors and Computer Arithmetic: Understand the impact of floating-point representation, machine epsilon, and catastrophic cancellation on numerical reliability.
Construct Optimal Approximations: Master polynomial approximation in both infinity norms (Minimax via the Oscillation Theorem) and 2-norms (via Orthogonal Polynomials).
Implement High-Precision Interpolation: Apply Lagrange, Newton, and Hermite forms while mitigating the Runge phenomenon through spline interpolation and Chebyshev nodes.
Solve Complex ODEs: Develop and analyze one-step (Euler, Runge-Kutta) and linear multistep methods (Adams-Bashforth, BDFs), specifically addressing challenges like stiffness and stability.
Preserve Physical Structures: Apply geometric integrators to preserve conserved quantities like energy and symplecticity in Hamiltonian systems.
Execute Stochastic Simulations: Utilize Monte Carlo principles and Markov Chain Monte Carlo (MCMC) to sample from complex distributions and solve high-dimensional problems.
Approximate SDEs: Understand the Ito calculus framework and implement numerical schemes for Stochastic Differential Equations (SDEs) such as the Euler-Maruyama and Milstein methods.
Course Syllabus & Detailed Topics
I. Preliminary Foundations
Modeling Cycle: State variables, modeling laws, and the "Analytical Wall."
Error Analysis: Truncation vs. Round-off error; stability and conditioning; backward error analysis.
Solving Nonlinear Equations: Iterative solvers (Bisection, Newton-Raphson) and modern acceleration techniques (Anderson Acceleration).
Minimax and Orthogonality: Chebyshev polynomials, best approximation in 2-norm, and Legendre polynomials.
Modern Paradigms: Beyond polynomials—Radial Basis Functions (RBFs) for scattered data and the Universal Approximation Theorem for Neural Networks.
Trigonometry Interpolation: The Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT).
Numerical Integration: Newton-Cotes, Gaussian Quadrature, and Romberg integration.
One-Step and Multistep Methods: Runge-Kutta families (Butcher Tableaus) and Adams-Moulton/Bashforth schemes.
Stiffness and Absolute Stability: A-stability, L-stability, and Backward Differentiation Formulas (BDFs).
Geometric Integration: Symplectic integrators (Symplectic Euler, Stormer-Verlet) and Hamiltonian flows.
Boundary Value Problems (BVPs): Shooting methods, Finite Difference methods, and Spectral Collocation.
Monte Carlo Principles: Probability review, Law of Large Numbers, and variance reduction (Control Variates, Antithetic Variates).
Sampling Algorithms: Acceptance-Rejection method, Box-Muller, and Metropolis-Hastings(MCMC).
Stochastic Dynamics: Chemical Master Equation (SSA/Gillespie), Brownian motion, and Ito's Formula.
Numerical SDEs: Strong vs. Weak convergence; Euler-Maruyama and Milstein schemes.
