Scientific Computing Research in the IAM

Scientific Computing Group

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$Multigrid approach to coupled hydraulic fracture equations$

General Information

Mathematical models can be written that describe systems of interest in many fields: engineering, fundamental science, finance, biology and medicine. Analytic investigation of these models can give tremendous insights into the original applications. However, some specific information about the systems often cannot be found using analytical methods. In these cases, the models must be approximated numerically. The numerical computations must be done accurately and, for large-scale problems, efficiently. Numerical approximation is used by many researchers, and so there is significant interest in developing and improving the accuracy and efficiency of these methods. This is the field of Scientific Computing.

There is a strong group of researchers in this field in several departments at UBC, collected in the Institute of Applied Mathematics. Some of these researchers are more interested in the numerical analysis (accuracy, efficiency) of general methods, whereas others have developed improved methods for computations in their application field of interest. Many of the group members are part of the SCAIM (Scientific Computation and Applied & Industrial Mathematics group and of the Computer Science-based Scientific Computing Laboratory.


Radial fracture computation		Computation of a growing set		Moving interface computation

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Core Faculty

The Scientific Computing Group is composed of several core IAM faculty who are actively involved in the IAM activities and supervise IAM students or postdoctoral fellows. Prospective students interested in a research project in Scientific Computing in the IAM are encouraged to contact one or more of the core faculty as potential supervisors and let them know of their interests.

Uri Ascher	Uri is a Professor of Computer Science and a former Director of the IAM (1993-98). The focus of his work is the investigation and promotion of novel, efficient and reliable methods in scientific computation, particularly for approximation problems involving differential equations with constraints. He has contributed to a wide variety of applications and has also written several textbooks on numerical analysis published by SIAM.
Robert Bridson	Robert's work revolves around numerical and geometric algorithms for solving problems in fluid and solid mechanics, including iterative methods for linear systems, discretisation of PDEs in space and time, mesh generation, and collision and contact handling. He often works closely with film studios in applying these algorithms to problems in physics-based animation.
Michael Friedlander	Michael's research is primarily in developing numerical methods for large-scale optimisation. He is especially interested in issues of convergence analysis, robust software implementation, and applications in signal processing and image reconstruction. His recent contributions include software packages for large-scale sparse optimisation (SPGL1), sparse signal reconstruction (Sparco), and a linear operator toolbox (SPOT).
Chen Greif	Chen is interested in numerical linear algebra, and in particular in sparse matrix computations. His main interests include iterative solvers for large and sparse linear systems and preconditioning techniques, especially for saddle point systems. He is also interested in eigenvalue problems related to Markov chains, including the PageRank problem.
Eldad Haber	Eldad is a Professor of Mathematics and of Earth and Ocean Sciences. His main field of interest is the development of computational methods for inverse problems with applications to geophysical and medical imaging. The field is interdisciplinary by nature and includes numerical discretisation of partial differential equations, numerical optimisation and robust statistics. Eldad is an NSERC Industrial Research Chair in Computational Geoscience.
Ian Mitchell	Ian is interested in numerical methods and software for solving ordinary and partial differential equations in the areas of control, robotics and verification. For example, the Toolbox of Level Set Methods is a Matlab software package which can be used for dynamic implicit surfaces in graphics, animation and fluid simulations as well as the Hamilton-Jacobi equation in control and verification.
Carl Ollivier-Gooch	Carl is a Professor in the Department of Mechanical Engineering. His research is in numerical methods using unstructured meshes, with a particular interest in computational aerodynamics. His group also develops algorithms for unstructured mesh generation and studies the interaction of unstructured mesh quality and solution accuracy.
Anthony Peirce	Anthony's principal areas of research expertise are in the application of asymptotic and numerical analysis to industrial problems. Research topics have included optimal control of molecular motion, stability of reactive fronts propagating in layered porous media, analysis of the regularisation effect of microstructure on localisation phenomena in elasto-plastic models, and development of multipole expansion techniques for boundary integral models of large-scale fracture interactions. His most recent research efforts are focussing on the analysis of hydraulic fracture propagation, which is of considerable importance in the oil, gas, and mining industries. Anthony was an IAM Director from 1999 to 2000.
Dominik Schötzau	Dominik's research area is the development and numerical analysis of finite element methods for partial differential equations, with applications in fluid mechanics, solid mechanics and electromagnetics. He has done research on various aspects of discontinuous Galerkin methods, mixed methods and hp-adaptive methods.
Brian Wetton	Brian's major research area is the numerical analysis of various continuum mechanics and materials science problems. He had a long industrial collaboration with Ballard Power Systems, developing simulation tools to aid in the design of hydrogen fuel cells. He has an ongoing interest in electrochemical systems and is pursuing other industrial projects. Brian is currently supervising four graduate students in the IAM.

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Recommended Courses

Students interested in Scientific Computing in the IAM are advised to take the following preliminary, core and optional courses.

Preliminary and Foundational Courses
CPSC 302: Numerical Computation for Algebraic Problems
CPSC 303: Numerical Approximation and Discretization
CPSC 402: Numerical Linear Algebra
CPSC 542G: Scientific Computing
MATH 405/607E: Numerical Methods for Differential Equations

Scientific Computing Courses
CPSC 520: Numerical Methods for Time-Dependent PDEs
MATH 521: Numerical Analysis of PDEs
MECH 510: Computational Methods in Transport Phenomena I

Further Options (Special Topics)
CPSC 517: Sparse Matrix Computations
CPSC 546: Numerical Optimization
MATH 522: Computational Methods in PDE Optimization
MECH 511: Computational Methods in Transport Phenomena II

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Sample Publications

Listed below are some of the recent theses and journal publications by IAM students, IAM faculty, IAM postdoctoral fellows and IAM alumni in the field of Scientific Computing. Click on the item of interest to open a thesis file, an article pdf on the author's web page, or an article abstract on the journal site.

Tyson Brochu and Robert Bridson. Robust Topological Operations for Dynamic Explicit Surfaces. SIAM Journal on Scientific Computation, 31, p. 2472 (2009).

Chen Greif, Dan Li, Dominik Sch�tzau and Xiaoxi Wei. A Mixed Finite Element Method with Exactly Divergence-Free Velocities for Incompressible Magnetohydrodynamics. Computer Methods in Applied Mechanics and Engineering (in press).

Carl Ollivier-Gooch, Amir Nejat and Chris Michalak. On Obtaining and Verifying High-Order Finite-Volume Solutions to the Euler Equations on Unstructured Meshes. The American Institute of Aeronautics and Astronautics Journal, 47(9), pp. 2105-2120 (2009).

Paul Houston, Dominik Sch�tzau and Xiaoxi Wei. A Mixed Discontinuous Galerkin Method for Linearized Incompressible Magnetohydrodynamics. Journal of Scientific Computing, 40, pp. 281-314 (2009).

Lloyd Bridge and Brian Wetton. A Mixture Formulation for Numerical Capturing of a Two-Phase/Vapour Interface in a Porous Medium. Journal of Chemical Physics 225, pp. 2043-2068 (2007).

Ian Mitchell. The Flexible, Extensible and Efficient Toolbox of Level Set Methods. Journal of Scientific Computing, Vol. 35 (2-3), pp 300-329 (2008).

Anthony Peirce and Emmanuel Detournay. An Implicit Level Set Method for Modeling Hydraulically Driven Fractures. Computer Methods in Applied Mechanics and Engineering, 197, pp. 2858�2885 (2008).

Hui Huang. Efficient Reconstruction of 2D Images and 3D Surfaces. Ph.D. Thesis, University of British Columbia (2008).

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Last Update: 29 Nov 2010