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Gaussian beams are asymptotic high frequency solutions to hyperbolic partial differential equations that are concentrated on a single curve through space-time. Their superpositions can be used to model more general high frequency wave propagation. In this talk, I will give a brief review of Gaussian beams and discuss some recently obtained results on the asymptotic convergence rate of Gaussian beam superpositions when the initial data is of the WKB form, $a(x) exp[i k \phi(x)]$. In numerical simulations involving Gaussian beams, one of the main challenges is to represent the initial data in terms of Gaussian beams. I will present a numerical method for decomposing general high frequency initial data into a sum of Gaussian beams. Finally, I will describe some open problems in Gaussian beam methods. | Gaussian beams are asymptotic high frequency solutions to hyperbolic partial differential equations that are concentrated on a single curve through space-time. Their superpositions can be used to model more general high frequency wave propagation. In this talk, I will give a brief review of Gaussian beams and discuss some recently obtained results on the asymptotic convergence rate of Gaussian beam superpositions when the initial data is of the WKB form, $a(x) exp[i k \phi(x)]$. In numerical simulations involving Gaussian beams, one of the main challenges is to represent the initial data in terms of Gaussian beams. I will present a numerical method for decomposing general high frequency initial data into a sum of Gaussian beams. Finally, I will describe some open problems in Gaussian beam methods. | ||
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== Bin Dong, University of California San Diego == | |||
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Revision as of 20:27, 10 November 2010
Gheorghe Craciun, UW-Mathematics
| Mathematical results arising from systems biology |
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We describe new sufficient conditions for global injectivity of general nonlinear functions, necessary and sufficient conditions for global injectivity of polynomial functions, and related criteria for uniqueness of equilibria in nonlinear dynamical systems. Some of these criteria are graph-theoretical, others are checked using symbolic computation. We also mention some applications of these methods in the study of Bezier curves and patches, and other types of manifolds used in geometric modeling. Also, we discuss some criteria for persistence and boundedness of trajectories in polynomial or power-law dynamical systems. All these seemingly unrelated results have been inspired by the study of mathematical models in systems biology. |
Jean-Marc Vanden-Broeck, University College London
| The effects of electrical fields on nonlinear free surface flows |
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Abstract. |
Thierry Goudon, INRIA-Lille, France
| Fluid-particle flows |
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We are interested in flows where a disperse phase (particles) is coupled to a dense phase (fluid). The evolution of the mixture is described by a kinetic equation coupled to a hydrodynamic system (Euler or Navier-Stokes). We will discuss several mathematical questions, with a particular attention paid to asymptotic issues. We will also present relevant numerical schemes specifically adapted to the asymptotic regime. |
Sang Dong Kim, Kyungpook National University, Korea
| A non-standard explicit method for solving stiff initial value problems |
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In this talk, we present a non-standard type of an explicit numerical method for solving stiff initial value problems which not only avoids unnecessary iteration process that may be required in most implicit methods but also has such a good stability as implicit methods possess. The proposed methods use both a Chebyshev collocation technique and an asymptotical linear ordinary differential equation of first-order derived from the difference between the exact solution and the Euler's polygon. These methods with or without usages of the Jacobian are analyzed in terms of convergence and stability. In particular, it is proved that the proposed methods have a convergence order up to 4 regardless of the usage of the Jacobian. Numerical tests are given to support the theoretical analysis as evidences. |
Jean-Luc Thiffeault, UW-Mathematics
| Velocity fluctuations in suspensions of swimming microorganisms |
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Abstract. |
Nick Tanushev, University of Texas
| Gaussian beam methods |
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Gaussian beams are asymptotic high frequency solutions to hyperbolic partial differential equations that are concentrated on a single curve through space-time. Their superpositions can be used to model more general high frequency wave propagation. In this talk, I will give a brief review of Gaussian beams and discuss some recently obtained results on the asymptotic convergence rate of Gaussian beam superpositions when the initial data is of the WKB form, $a(x) exp[i k \phi(x)]$. In numerical simulations involving Gaussian beams, one of the main challenges is to represent the initial data in terms of Gaussian beams. I will present a numerical method for decomposing general high frequency initial data into a sum of Gaussian beams. Finally, I will describe some open problems in Gaussian beam methods. |
Bin Dong, University of California San Diego
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Anne Gelb, Arizona State University
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Abstract. |
Paul Milewski, UW-Mathematics
| The Serre equations of shallow water waves |
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Abstract. |
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