Scientific Computing Core

Scientific Computing I and II

This sequence covers fundamental algorithms applicable to modern high-performance computing systems for discrete systems, linear and nonlinear systems, optimization, ordinary and partial differential equations, and other applications. Scientific Computing II can be taken independent of Scientific Computing I provided the student has an adequate background. Doctoral students must take both of these courses while masters students need only take one.

Scientific Computing I (Fall Term)

This course presents fundamental techniques of and the use of software packages in scientific computing. It will cover topics in Monte Carlo simulations, linear and nonlinear systems, optimization, and ordinary differential equations.

PREREQUISITES

• MAPL 460 or MAPL 466, or a knowledge of basic numerical methods (linear and nonlinear equations, integration, interpolation) with the permission of the instructor.
• some knowledge of C or Fortran.

SYLLABUS

• Monte Carlo Simulations

  random variables, distributions, mean, variance, central limit theorem.

  basic Monte-Carlo simulation, Monte-Carlo integration, convergence.

  pseudo-random-numbers, generation of nonuniformly distributed numbers.

  variance reduction: stratified sampling, importance sampling.

• Numerical Linear Algebra

  linear systems: estimation of errors, implementation using Matlab, BLAS and block algorithms in LAPACK.

  brief overview of least squares problems, data fitting, eigenvalue problems, singular value decompositions.

• Nonlinear Systems

  inverse function theorem, Newton and quasi-Newton methods.

  implicit fuction theorem, continuation methods, bifurcations, predictor-corrector algorithms.

• Optimization

  unconstrained optimization, line searches, descent and conjugate gradient methods, Newton and quasi-Newton methods, Davidson-Flectcher-Powell method.

  constrained optimization, SQP.

  simulated annealing, Metropolis algorithms.

• Ordinary Differential Equations

  explicit and implicit methods, basic facts about stability and convergence.

  GEAR packages, stiffness.

  applications: combustion.

Scientific Computing II (Spring Term)

This course presents fundamental techniques of and the use of software packages in scientific computing. It will cover topics in Fourier and wavelet transforms, elliptic partial differential equations, sparse matrices, and time-dependent partial differential equations. It can be taken independent of Scientific Computing I provided the student has an adequate background.

PREREQUISITES

• MAPL 460 or MAPL 466, or a knowledge of basic numerical methods (linear and nonlinear equations, integration, interpolation, ordinary differential equations) with the permission of the instructor.
• some knowledge of C or Fortran.

SYLLABUS

• tranform methods

  continuous and discrete Fourier transforms, Nyquist frequency, sampling theorem, discrete cosine transform, fast Fourier transform (FFT) algorithm.

  applications: discrete convolution and deconvolution, Wiener filtering, approximation using truncated series, trigonometric interpolation, Gibbs phenomenon, analysis of time series, signal compression.

  continuous and discrete wavelet transforms, Haar and Daubechies wavelets, approximation properties, fast wavelet transform (FWT) algorithms.

  applications: filtering, signal compression.

• elliptic partial differential equations

  variational and weak formulations, smooth and nonsmooth solutions.

  finite difference methods, convergence.

  finite element methods, finite element spaces, local stiffness matrices and assembly of stiffness matrix, convergence.

  adaptive refinement.

• sparse matrices

  direct methods: symmetric positive definite case: RCM, min. degree reorderings, nested dissection; description of Matlab's algorithms.

  iterative methods: Jacobi, SOR, conjugate gradient, preconditioning (SSOR, IC, ILU), GMRES, BiCG, QMR.

  multigird methods.

• time-dependent partial differential equations

  well-posedness, dispersion analysis.

  time discretization with explicit and implicit methods, basic facts about stability and convergence.

  method of lines for spatial discretization with finite differences or finite elements.

  applications: diffusive, dispersive, and hyperbolic equations.

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Advanced Scientific Computing I and II

Advanced Scientific Computing I (Fall Term)

This course presents fundamentals of code development on modern high-performance computing systems with parallelism. It requires doctoral students to implement many of these in a year-long project. Masters students will do the same but for a term-long project. The project includes the development of software designed to carry out a computational scientific task, a written and oral proposal within one month, and a written and oral progress or final report at the end of the term.

PREREQUISITES

• Scientific Computing I or a comparable knowledge of scientific computing,
• the permission of the instructor.

SYLLABUS

• Project Design Proposal (6 classes)

  by the end of one month each student must find a faculty advisor and identify a suitable project that includes a deliverable suite of software that is designed to carry out a computational scientific task, propose appropriate algorithms, languages, and platforms for the development of this software, write a short project proposal that also includes a scientific justification, and present the proposal orally.

• Computational Environments (6 classes)

  high-performance computational and graphical platforms, advanced architectures, high-speed networks, network protocols, communication tools.

• Code Development (12 classes)

  compiled and interpreted languages, object-oriented languages, memory management, local and global variables, modularity, portability, debugging and timing tools, interactive programming, snapshot and restart dump files, history dump files, post-processing, scientific visualization, documentation and version management.

• Parallel Computing (12 classes)

  vectorization, distributed computing and clusters, types of parallel computers, parallel programming models, distributed memory, local caches, message passing, communication, synchronization, process spawning, MPI/PVM software, granularity, data parallel programming, parallel numerical linear algebra and ScaLAPACK, High Performance Fortran, parallel algorithms for PDEs.

• Project Progress Report (6 classes)

  each student must give a written and oral report on the state of his or her project, explain how the software has been developed and tested, give his or her current vision of the finished product, and detail how that vision has evolved over the course of the project.

Advanced Scientific Computing II (Spring Term)

This course presents fundamentals of code development and validation on modern high-performance computing systems. It requires students to implement many of these in their year-long project. The project includes a seminar lecture and a written final report.

PREREQUISITES

• Scientific Computing I and II or a comparable knowledge of scientific computing,
• Advanced Scientific Computing I,
• the permission of the instructor.

SYLLABUS

• Code Development and Validation (10 classes)

  continuation of code development topics from the fall, round-off sensitivity, performance analysis, tuning algorithms to platforms, symmetry and stability studies, analytical, experimental, and computational benchmark comparisions, convergence studies, parametric sensitivity studies, investigation of asymptotic regimes.

• Parallel Computing (10 classes)

  continuation of parallel computing topics from the fall, more parallel algorithms for PDEs (for example, adaptive mesh refinement, multi-grid, independent time-steps, domain decomposition, and operator splitting), more parallel algorithms for linear algebra, parallel algorithms for solving systems, parallel algorithms for unconstrained and constrained optimization.

• Computational Science Seminars (10 classes)

  local and visiting computational scientists representing broad selection of scientific disciplines will explain the computational aspects of their work, these aspects should be reflected in the choice of topics covered above, each student is expected to listen to these seminars critically.

• Project Final Report (12 classes)

  each student must give a one hour seminar and a complete written report on his or her project, explain how the software has been developed, tested and validated, document how the software is used, show results of its use, detail how his or her vision of the finished product has evolved over the course of the project.

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