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- TRIANGULAR MESHES, JIANLIANG QIAN, YONG-TAO ZHANG, HONG-KAI ZHAO
- 2007

The original fast sweeping method, which is an efficient iterative method for stationary Hamilton–Jacobi equations, relies on natural ordering provided by a rectangular mesh. We propose novel ordering strategies so that the fast sweeping method can be extended efficiently and easily to any unstructured mesh. To that end we introduce multiple reference… (More)

- Yong-Tao Zhang, Hongkai Zhao, Jianliang Qian
- J. Sci. Comput.
- 2006

We construct high order fast sweeping numerical methods for computing viscosity solutions of static Hamilton-Jacobi equations on rectangular grids. These methods combine high order weighted essentially non-oscillatory (WENO) approximation to derivatives, monotone numerical Hamiltonians and Gauss Seidel iterations with alternating-direction sweepings. Based… (More)

We design an Eulerian Gaussian beam summation method for solving Helmholtz equations in the high-frequency regime. The traditional Gaussian beam summation method is based on Lagrangian ray tracing and local ray-centered coordinates. We propose a new Eulerian formulation of Gaussian beam theory which adopts global Cartesian coordinates, level sets, and… (More)

Ptychography promises diffraction limited resolution without the need for high resolution lenses. To achieve high resolution one has to solve the phase problem for many partially overlapping frames. Here we review some of the existing methods for solving ptychographic phase retrieval problem from a numerical analysis point of view, and propose alternative… (More)

- Jianliang Qian, Yong-Tao Zhang, Hongkai Zhao
- J. Sci. Comput.
- 2007

We develop a fast sweeping method for static Hamilton-Jacobi equations with convex Hamiltonians. Local solvers and fast sweeping strategies apply to structured and unstructured meshes. With causality correctly enforced during sweepings numerical evidence indicates that the fast sweeping method converges in a finite number of iterations independent of mesh… (More)

- Jianliang Qian, Yong-Tao Zhang, Hongkai Zhao
- SIAM J. Numerical Analysis
- 2007

The original fast sweeping method, which is an efficient iterative method for stationary Hamilton-Jacobi equations, relies on natural ordering provided by a rectangular mesh. We propose novel ordering strategies so that the fast sweeping method can be extended efficiently and easily to any unstructured mesh. To that end we introduce multiple reference… (More)

- Shingyu Leung, Jianliang Qian
- J. Comput. Physics
- 2009

We propose Gaussian-beam based Eulerian methods to compute semi-classical solutions of the Schrödinger equation. Traditional Gaussian beam type methods for the Schrödinger equation are based on the Lagrangian ray tracing. We develop a new Eulerian framework which uses global Cartesian coordinates, level-set based implicit representation and Liouville… (More)

- Jianliang Qian, Plamen Stefanov, Gunther Uhlmann, Hongkai Zhao
- SIAM J. Imaging Sciences
- 2011

We present an efficient algorithm for reconstructing an unknown source in Thermoacoustic and Photoacoustic Tomography based on the recent advances in understanding the theoretical nature of the problem. We work with variable sound speeds that might be also discontinuous across some surface. The latter problem arises in brain imaging. The algorithmic… (More)

The geometric optics approximation to high frequency anisotropic wave propagation reduces the anisotropic wave equation to a static Hamilton–Jacobi equation. This equation is known as the anisotropic eikonal equation and has three different coupled wave modes as solutions. We introduce here a level set-based Eulerian approach that captures all three of… (More)

Traditional transmission travel-time tomography hinges on ray tracing techniques. We propose a PDE-based Eulerian approach to travel-time tomography so that we can avoid the cumbersome ray-tracing. We start from the eikonal equation, define a mismatching functional and derive the gradient of the nonlinear functional by an adjoint state method. The resulting… (More)