5 research outputs found
Deep Eikonal Solvers
A deep learning approach to numerically approximate the solution to the
Eikonal equation is introduced. The proposed method is built on the fast
marching scheme which comprises of two components: a local numerical solver and
an update scheme. We replace the formulaic local numerical solver with a
trained neural network to provide highly accurate estimates of local distances
for a variety of different geometries and sampling conditions. Our learning
approach generalizes not only to flat Euclidean domains but also to curved
surfaces enabled by the incorporation of certain invariant features in the
neural network architecture. We show a considerable gain in performance,
validated by smaller errors and higher orders of accuracy for the numerical
solutions of the Eikonal equation computed on different surfaces The proposed
approach leverages the approximation power of neural networks to enhance the
performance of numerical algorithms, thereby, connecting the somewhat disparate
themes of numerical geometry and learning.Comment: Accepted for oral presentation at Seventh International Conference on
Scale Space and Variational Methods in Computer Vision (SSVM) 201
Equivalent extensions of Hamilton-Jacobi-Bellman equations on hypersurfaces
We present a new formulation for the computation of solutions of a class of
Hamilton Jacobi Bellman (HJB) equations on closed smooth surfaces of
co-dimension one. For the class of equations considered in this paper, the
viscosity solution of the HJB equation is equivalent to the value function of a
corresponding optimal control problem. In this work, we extend the optimal
control problem given on the surface to an equivalent one defined in a
sufficiently thin narrow band of the co-dimensional one surface. The extension
is done appropriately so that the corresponding HJB equation, in the narrow
band, has a unique viscosity solution which is identical to the constant normal
extension of the value function of the original optimal control problem. With
this framework, one can easily use existing (high order) numerical methods
developed on Cartesian grids to solve HJB equations on surfaces, with a
computational cost that scales with the dimension of the surfaces. This
framework also provides a systematic way for solving HJB equations on the
unstructured point clouds that are sampled from the surface
Absolutely convergent fixed-point fast sweeping WENO methods for steady state of hyperbolic conservation laws
Fixed-point iterative sweeping methods were developed in the literature to
efficiently solve steady state solutions of Hamilton-Jacobi equations and
hyperbolic conservation laws. Similar as other fast sweeping schemes, the key
components of this class of methods are the Gauss-Seidel iterations and
alternating sweeping strategy to achieve fast convergence rate. Furthermore,
good properties of fixed-point iterative sweeping methods include that they
have explicit forms and do not involve inverse operation of nonlinear local
systems, and they can be applied to general hyperbolic equations using any
monotone numerical fluxes and high order approximations easily. In [L. Wu,
Y.-T. Zhang, S. Zhang and C.-W. Shu, Commun. Comput. Phys., 20 (2016)], a fifth
order fixed-point sweeping WENO scheme was designed and it was shown that the
scheme converges much faster than the total variation diminishing (TVD)
Runge-Kutta approach by stability improvement of high order schemes with a
forward Euler time-marching. An open problem is that for some benchmark
numerical examples, the iteration residue of the fixed-point sweeping WENO
scheme hangs at a truncation error level instead of settling down to machine
zero. This issue makes it difficult to determine the convergence criterion for
the iteration and challenging to apply the method to complex problems. To solve
this issue, in this paper we apply the multi-resolution WENO scheme developed
in [J. Zhu and C.-W. Shu, J. Comput. Phys., 375 (2018)] to the fifth order
fixed-point sweeping WENO scheme and obtain an absolutely convergent
fixed-point fast sweeping method for steady state of hyperbolic conservation
laws, i.e., the residue of the fast sweeping iterations converges to machine
zero / round off errors for all benchmark problems tested.Comment: 35 page
A Second Order Discontinuous Galerkin Fast Sweeping Method for Eikonal Equations
In this paper, we construct a second order fast sweeping method with a discontinuous Galerkin (DG) local solver for computing viscosity solutions of a class of static Hamilton-Jacobi equations, namely the Eikonal equations. Our piecewise linear DG local solver is built on a DG method developed recently [Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations, Journal of Computational Physics, 223 (2007), 398-415] for the time-dependent Hamilton-Jacobi equations. The causality property of Eikonal equations is incorporated into the design of this solver. The resulting local nonlinear system in the Gauss-Seidel iterations is a simple quadratic system and can be solved explicitly. The compactness of the DG method and the fast sweeping strategy lead to fast convergence of the new scheme for Eikonal equations. Extensive numerical examples verify efficiency, convergence and second order accuracy of the proposed method. Key Words: fast sweeping methods, discontinuous Galerkin finite element methods, second order accuracy, static Hamilton-Jacobi equations, Eikonal equation