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Wednesday, March 15 • 5:00pm - 5:20pm
Algorithms and Performance: An Efficient and High Order Accurate Solution Technique for Scattering in Variable Media

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Classic numerical partial differential equation techniques face two big
problems when applied to problems with highly oscillatory solutions. First there is the so-called ``pollution'' (dispersion) error, demanding an increasing number of degrees of freedom per wavelength in order to maintain fixed accuracy as wavenumber grows. Second, iterative solvers that are typically used to solve the linear system resulting from discretization are slow to converge due to

While there is much on going work to resolving these problems, this talk presents an alternative solution technique for high frequency scattering problems in variable media. This method does not observe pollution and has an efficient direct solver is called the Hierarchical Poincaré-Steklov (HPS) scheme. The technique uses a classical spectral collocation method on a collection of disjoint leaf
boxes whose union is the domain. On each leaf box approximate solution
operators and Poincaré-Steklov operators such as Dirichlet-to-Neumann operators are
constructed. Then boxes are ``glued'' together in a hierarchical fashion creating
approximate solution and Poincaré-Steklov operators for the union of two boxes.
Once this precomputation is complete, new boundary conditions and source functions
can be processed by applying the solution operators via a collection of small matrix vector multiplies. The resulting method has computational cost that is asymptotically the same as the nested dissection method but has high order accuracy even for problems with highly oscillatory solutions. For example when applied to the Helmholtz problem with a fixed twelve points per wavelength, the HPS method achieves eight digits of accuracy.

In addition to presenting a high level view of the HPS method, we will illustrate
it performance in both the forward and inverse scattering setting. Next, the
adaptive version of the method will be present with numerical results to report
on its performance.


Henri Calandra

Henri Calandra obtained his M.Sc. in mathematics in 1984 and a Ph.D. in mathematics in 1987 from the Universite des Pays de l’Adour in Pau, France. He joined Cray Research France in 1987 and worked on seismic applications. In 1989 he joined the applied mathematics department of the French Atomic Agency. In 1990 he started working for Total SA. After 12 years of work in high performance computing and as project leader for Pre-stack Depth... Read More →
avatar for Ernesto Prudencio

Ernesto Prudencio

Senior Software Engineer, Schlumberger
Ernesto E. Prudencio combines a BSc in electronics engineering (1990, Brazil), a MSc in applied mathematics (domain decomposition methods, 2001, Brazil), and a PhD in computer science / numerical analysis (PDE-constrained optimization, 2005, Boulder, CO), with professional experience in industry (IBM, Integris, Schlumberger), in national laboratories (ANL, SLAC) and academia (UT Austin). He has been working in Schlumberger since September of... Read More →


Wednesday March 15, 2017 5:00pm - 5:20pm
Room 280

Attendees (2)