Numerical analysis
Graduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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Topics typically covered
Standard syllabus
Approximation and stability theory
- Best approximation in normed spaces (introduction)
- Polynomial and spline approximation theory
- Stability, consistency, and convergence for numerical methods
- A-stability and stiff ODEs
- Conditioning of linear and nonlinear problems
Linear and nonlinear systems
- Direct methods: LU, Cholesky, and QR factorizations
- Iterative methods: Krylov subspaces and GMRES (introduction)
- Preconditioning strategies
- Newton–Kantorovich convergence analysis (introduction)
- Eigenvalue algorithms: QR iteration and Lanczos method
Numerical PDEs (introduction)
- Finite difference methods for elliptic, parabolic, and hyperbolic PDEs
- Consistency, stability, and convergence (Lax equivalence overview)
- Finite element method: Galerkin formulation (introduction)
- Multigrid methods (conceptual overview)
- Adaptive mesh refinement (introduction)
STEM / applied
High-performance and applied computation
- Implementation on modern architectures (vectorization, parallelism overview)
- Large-scale sparse linear solvers in practice
- Uncertainty quantification and Monte Carlo methods
- Inverse problems and regularization (Tikhonov)
- Validation against analytical and experimental benchmarks
Domain applications
- Computational fluid dynamics discretizations (introduction)
- Numerical optimization in engineering design loops
- Image processing and numerical linear algebra pipelines
- Time-stepping strategies for multiphysics simulation
- Software engineering for scientific computing projects
Notes
Topics reflect common graduate numerical analysis syllabi at US universities. Applied sections emphasize large-scale computation; theoretical sections mirror qualifying-exam numerical analysis cores.