Numerical analysis
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Numerical Analysis — scope drawn from open graduate numerical analysis texts and typical US graduate numerical analysis syllabi.
Approximation and stability theory
- Best approximation in normed spaces (introduction) — Best approximation in normed spaces (introduction)
- Polynomial — Polynomial
- Spline approximation theory — Spline approximation theory
- Stability and consistency — Stability and consistency
- Convergence for numerical methods — Convergence for numerical methods
- A-stability and stiff ODEs — A-stability and stiff ODEs
- Conditioning of linear — Conditioning of linear
- Nonlinear problems — Nonlinear problems
Linear and nonlinear systems
- Direct methods — Direct methods
- QR factorizations — QR factorizations
- Iterative methods — Iterative methods
- GMRES (introduction) — GMRES (introduction)
- Preconditioning strategies — Preconditioning strategies
- Newton–Kantorovich convergence analysis (introduction) — Newton–Kantorovich convergence analysis (introduction)
- Eigenvalue algorithms — Eigenvalue algorithms
- Lanczos method — Lanczos method
Numerical PDEs (introduction)
- Finite difference methods for elliptic and parabolic PDEs — Finite difference methods for elliptic and parabolic PDEs
- Hyperbolic PDEs — Hyperbolic PDEs
- Consistency and stability — Consistency and stability
- Convergence (Lax equivalence overview) — Convergence (Lax equivalence overview)
- Finite element method: Galerkin formulation (introduction) — Finite element method: Galerkin formulation (introduction)
- Multigrid methods (conceptual overview) — Multigrid methods (conceptual overview)
- Adaptive mesh refinement (introduction) — Adaptive mesh refinement (introduction)
High-performance and applied computation
- Implementation on modern architectures (vectorization, parallelism overview) — Implementation on modern architectures (vectorization, parallelism overview)
- Large-scale sparse linear solvers in practice — Large-scale sparse linear solvers in practice
- Uncertainty quantification — Uncertainty quantification
- Monte Carlo methods — Monte Carlo methods
- Inverse problems — Inverse problems
- Regularization (Tikhonov) — Regularization (Tikhonov)
- Validation against analytical — Validation against analytical
- Experimental benchmarks — Experimental benchmarks
Domain applications
- Computational fluid dynamics discretizations (introduction) — Computational fluid dynamics discretizations (introduction)
- Numerical optimization in engineering design loops — Numerical optimization in engineering design loops
- Image processing — Image processing
- Numerical linear algebra pipelines — Numerical linear algebra pipelines
- Time-stepping strategies for multiphysics simulation — Time-stepping strategies for multiphysics simulation
- Software engineering for scientific computing projects — Software engineering for scientific computing projects
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Approximation and stability theory
Best approximation in normed spaces (introduction)
Coming soon - Linear and nonlinear systems
Direct methods
Coming soon - Numerical PDEs (introduction)
Finite difference methods for elliptic and parabolic PDEs
Coming soon - High-performance and applied computation
Implementation on modern architectures (vectorization, parallelism overview)
Coming soon - Domain applications
Computational fluid dynamics discretizations (introduction)
Coming soon
Notes
Graduate Numerical Analysis — scope drawn from open graduate numerical analysis texts and typical US graduate numerical analysis syllabi. Topic outline: `content/topics/graduate/numerical_analysis.json`.