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4.5. Numerical and Scientific Analysis (NSA)
This area focuses on computational algorithms for obtaining numerical solutions to mathematical problems arising in engineering. It covers error analysis, numerical linear algebra, and methods for solving differential equations by discretization.
4.5.1. NSA/Error Analysis and Floating Point Arithmetic (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Study of how numerical errors originate and propagate in engineering computations, and strategies to control them.
Topics:
Core
- Floating point representation and round-off errors in engineering software
- Conditioning of engineering problems and stability of numerical algorithms
- Truncation error in series approximations and finite difference stencils
- Error propagation through chains of computations in engineering simulations
Learning Outcomes:
Core:
- Explain the sources and consequences of round-off and truncation errors in engineering computations [Familiarity]
- Calculate the condition number of a linear system and interpret its effect on solution accuracy [Assessment]
- Estimate the propagation of measurement uncertainties through an engineering calculation chain [Usage]
- Select numerical methods with appropriate stability properties for engineering simulation requirements [Assessment]
4.5.2. NSA/Numerical Linear Algebra (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Efficient numerical algorithms for solving large linear systems and eigenvalue problems arising in structural, fluid, and electromagnetic engineering.
Topics:
Core
- LU and QR decompositions for solving engineering linear systems
- Iterative solvers: Jacobi, Gauss-Seidel, and Conjugate Gradient for large sparse systems
- Singular Value Decomposition (SVD) for data fitting and model reduction
- Sparse matrix storage formats and efficient solvers for large engineering FEM/FVM systems
Learning Outcomes:
Core:
- Solve engineering linear systems using LU decomposition with partial pivoting [Usage]
- Analyze the convergence rate of iterative solvers applied to engineering stiffness matrices [Assessment]
- Implement a Conjugate Gradient solver for a large symmetric positive-definite engineering system [Usage]
- Apply SVD to compress a structural mode shape dataset and assess the approximation quality [Assessment]
4.5.3. NSA/Numerical Solution of Nonlinear Equations (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Iterative methods for approximating roots of engineering nonlinear equations and analysis of their convergence.
Topics:
Core
- Bracketing methods: bisection and false position
- Newton-Raphson method and the secant method
- Order of convergence and stopping criteria in iterative methods
Learning Outcomes:
Core:
- Explain the basis and convergence guarantees of the bisection and false position methods [Familiarity]
- Apply the Newton-Raphson and secant methods to approximate roots of engineering equations [Usage]
- Compare the convergence order of different iterative methods and select the most appropriate one for a given problem [Assessment]
4.5.4. NSA/Approximation Theory and Interpolation (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Polynomial and spline interpolation, best approximation, Chebyshev polynomials, and rational approximation.
Topics:
Core
- Polynomial interpolation: Lagrange, Newton divided differences, and Runge's phenomenon
- Spline interpolation: cubic splines, B-splines, and piecewise polynomial methods
- Best approximation in normed spaces: Chebyshev (minimax) and least squares
- Chebyshev polynomials: properties, orthogonality, and spectral convergence
- Trigonometric approximation and the fast Fourier transform (FFT)
Learning Outcomes:
Core:
- Explain Runge's phenomenon and justify the choice of Chebyshev nodes to mitigate it [Familiarity]
- Construct cubic spline and Chebyshev interpolants for given data and estimate the interpolation error [Usage]
- Apply the FFT to efficiently compute trigonometric approximations of a sampled function [Assessment]
4.5.5. NSA/Numerical Integration and Quadrature (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Quadrature rules, adaptive integration, Gaussian quadrature, and multi-dimensional integration.
Topics:
Core
- Newton-Cotes rules: trapezoidal, Simpson's, and composite rules; error analysis
- Adaptive quadrature and automatic error control
- Gaussian quadrature: optimal nodes and weights, orthogonal polynomial connection
- Numerical treatment of improper and singular integrals
- Monte Carlo integration and quasi-Monte Carlo methods for high-dimensional integrals
Learning Outcomes:
Core:
- Derive the error formula for composite Simpson's rule and identify its order of accuracy [Familiarity]
- Select and apply an appropriate quadrature rule (Gaussian, adaptive) based on integrand regularity [Usage]
- Apply Monte Carlo integration to estimate high-dimensional integrals and quantify the statistical error [Assessment]
4.5.6. NSA/Numerical Methods for ODEs and PDEs (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Solving differential equations that govern engineering systems through time-stepping and spatial discretization methods.
Topics:
Core
- Runge-Kutta methods and adaptive time-stepping for engineering ODE problems
- Stability analysis and stiffness in engineering ODE integrators
- Finite difference discretization of engineering PDEs (heat equation, Laplace equation)
Non Core
- Finite element and shooting methods for engineering boundary value problems
Learning Outcomes:
Core:
- Implement a fourth-order Runge-Kutta integrator for a mechanical vibration problem [Usage]
- Determine the region of absolute stability for an explicit time-stepping scheme [Assessment]
- Discretize the 2D heat equation using finite differences and set up the resulting linear system [Usage]
NonCore:
- Compare finite difference and finite element approaches for solving an engineering PDE [Assessment]
4.5.7. NSA/Finite Element Method (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Variational formulation, Galerkin methods, finite element spaces, error estimates, and applications to elliptic PDEs.
Topics:
Core
- Weak (variational) formulation of boundary value problems and Sobolev spaces
- Galerkin method: finite-dimensional approximation spaces and the stiffness matrix
- Triangular and quadrilateral finite element spaces: Lagrange elements and conformity
- A priori error estimates: Céa's lemma and interpolation error bounds
Non Core
- A posteriori error estimates and adaptive mesh refinement
Learning Outcomes:
Core:
- Derive the weak formulation of an elliptic BVP and show its equivalence to the strong form [Familiarity]
- Assemble the global stiffness matrix and load vector for a linear finite element discretization [Usage]
- Estimate the \(H^1\) error of a finite element solution using Céa's lemma and interpolation theory [Assessment]
4.5.8. NSA/Optimization Algorithms (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Gradient methods, Newton and quasi-Newton methods, constrained optimization, and convex optimization algorithms.
Topics:
Core
- Gradient descent and line search methods: Armijo-Wolfe conditions and convergence rates
- Newton's method and quasi-Newton methods (BFGS, L-BFGS)
- Constrained optimization: KKT conditions, penalty methods, and sequential quadratic programming
- Convex optimization algorithms: interior-point methods and the alternating direction method of multipliers (ADMM)
- Stochastic gradient descent (SGD), variance reduction, and Adam optimizer
Learning Outcomes:
Core:
- Explain the convergence guarantees of gradient descent for smooth convex functions [Familiarity]
- Apply Newton and quasi-Newton methods to unconstrained optimization problems [Usage]
- Formulate a constrained optimization problem as a KKT system and apply an interior-point method [Assessment]
4.5.9. NSA/Parallel and High-Performance Computing (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Parallel architectures, performance models, parallel algorithms for linear algebra, and GPU computing.
Topics:
Core
- Parallel architectures: shared memory (OpenMP), distributed memory (MPI), and GPU (CUDA)
- Performance models: Amdahl's law, roofline model, and communication complexity
- Parallel algorithms for dense and sparse linear algebra (ScaLAPACK, PETSc)
- Domain decomposition and parallel PDE solvers
Non Core
- Automatic differentiation (forward and reverse mode) and its role in scientific ML
Learning Outcomes:
Core:
- Explain Amdahl's law and identify the bottlenecks limiting parallel speedup in a given algorithm [Familiarity]
- Implement a parallel linear algebra routine using MPI or OpenMP and measure parallel efficiency [Usage]
NonCore:
- Apply automatic differentiation to compute gradients of complex scientific functions for optimization [Assessment]