American Institute of Mathematical Sciences

Advanced
October  2015, 8(5): 871-880. doi: 10.3934/dcdss.2015.8.871

Distance function and extension in normal direction for implicitly defined interfaces

Peter Frolkovič 1, , Karol Mikula 1, and  Jozef Urbán 1,

1. 

Department of Mathematics and Descriptive Geometry, Slovak University of Technology, Bratislava, Slovak Republic, Slovak Republic, Slovak Republic

Received  January 2014 Revised  June 2014 Published  July 2015

In this paper we present a novel application of extrapolation procedure for three popular numerical algorithms to compute the distance function for an interface that is given only implicitly. The methods include the fast marching method [8], the fast sweeping method [10] and the linearization method [10]. The extrapolation procedure removes the necessity of a special initialization procedure for the grid nodes next to the interface that is used so far with the methods, thus it represents a natural extension of these methods. The extrapolation procedure can be used also for an extension of a function that is defined only locally on the interface in the direction given by the gradient of distance function [2].
Keywords: extrapolation, Distance function, fast marching method..
Mathematics Subject Classification: Primary: 35L50, 76M12; Secondary: 65L2.
Citation: Peter Frolkovič, Karol Mikula, Jozef Urbán. Distance function and extension in normal direction for implicitly defined interfaces. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 871-880. doi: 10.3934/dcdss.2015.8.871
References:
[1]

D. Adalsteinsson and J. Sethian, The fast construction of extension velocities in level set methods,, J. Comput. Phys., 148 (1999), 2.  doi: 10.1006/jcph.1998.6090.  Google Scholar

[2]

T. D. Aslam, A partial differential equation approach to multidimensional extrapolation,, J. Comput. Phys., 193 (2004), 349.  doi: 10.1016/j.jcp.2003.08.001.  Google Scholar

[3]

S. Fomel, Traveltime Computation with the Linearized Eikonal Equation,, Technical report, (1997).   Google Scholar

[4]

P. Frolkovič, Flux-based level set method for extrapolation along characteristics using immersed interface formulation,, In P. Struk, (2010), 15.   Google Scholar

[5]

S. Hysing and S. Turek, The Eikonal equation: numerical efficiency vs. algorithmic complexity on quadrilateral grids,, In Proceedings of Algoritmy, (2005), 22.   Google Scholar

[6]

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces,, Springer, (2003).  doi: 10.1007/b98879.  Google Scholar

[7]

E. Rouy and A. Tourin, A viscosity solutions approach to shape-from-shading,, SIAM J. Num. Anal., 29 (1992), 867.  doi: 10.1137/0729053.  Google Scholar

[8]

J. Sethian, A fast marching level set method for monotonically advancing fronts,, Proc. Nat. Acad. Sci., 93 (1996), 1591.  doi: 10.1073/pnas.93.4.1591.  Google Scholar

[9]

J. Sethian, Level Set Methods and Fast Marching Methods,, Cambridge University Press, (1999).   Google Scholar

[10]

H. Zhao, A fast sweeping method for eikonal equations,, Math. Comput., 74 (2005), 603.  doi: 10.1090/S0025-5718-04-01678-3.  Google Scholar

[11]

H. Zhao, T. Chan, B. Merriman and S. Osher, A variational level set approach to multiphase motion,, J. Comput. Phys., 127 (1996), 179.  doi: 10.1006/jcph.1996.0167.  Google Scholar

show all references

References:
[1]

D. Adalsteinsson and J. Sethian, The fast construction of extension velocities in level set methods,, J. Comput. Phys., 148 (1999), 2.  doi: 10.1006/jcph.1998.6090.  Google Scholar

[2]

T. D. Aslam, A partial differential equation approach to multidimensional extrapolation,, J. Comput. Phys., 193 (2004), 349.  doi: 10.1016/j.jcp.2003.08.001.  Google Scholar

[3]

S. Fomel, Traveltime Computation with the Linearized Eikonal Equation,, Technical report, (1997).   Google Scholar

[4]

P. Frolkovič, Flux-based level set method for extrapolation along characteristics using immersed interface formulation,, In P. Struk, (2010), 15.   Google Scholar

[5]

S. Hysing and S. Turek, The Eikonal equation: numerical efficiency vs. algorithmic complexity on quadrilateral grids,, In Proceedings of Algoritmy, (2005), 22.   Google Scholar

[6]

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces,, Springer, (2003).  doi: 10.1007/b98879.  Google Scholar

[7]

E. Rouy and A. Tourin, A viscosity solutions approach to shape-from-shading,, SIAM J. Num. Anal., 29 (1992), 867.  doi: 10.1137/0729053.  Google Scholar

[8]

J. Sethian, A fast marching level set method for monotonically advancing fronts,, Proc. Nat. Acad. Sci., 93 (1996), 1591.  doi: 10.1073/pnas.93.4.1591.  Google Scholar

[9]

J. Sethian, Level Set Methods and Fast Marching Methods,, Cambridge University Press, (1999).   Google Scholar

[10]

H. Zhao, A fast sweeping method for eikonal equations,, Math. Comput., 74 (2005), 603.  doi: 10.1090/S0025-5718-04-01678-3.  Google Scholar

[11]

H. Zhao, T. Chan, B. Merriman and S. Osher, A variational level set approach to multiphase motion,, J. Comput. Phys., 127 (1996), 179.  doi: 10.1006/jcph.1996.0167.  Google Scholar

[1]

Y. Goto, K. Ishii, T. Ogawa. Method of the distance function to the Bence-Merriman-Osher algorithm for motion by mean curvature. Communications on Pure & Applied Analysis, 2005, 4 (2) : 311-339. doi: 10.3934/cpaa.2005.4.311
[2]

Van Hieu Dang. An extension of hybrid method without extrapolation step to equilibrium problems. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1723-1741. doi: 10.3934/jimo.2017015
[3]

Laurent Imbert, Michael J. Jacobson, Jr., Arthur Schmidt. Fast ideal cubing in imaginary quadratic number and function fields. Advances in Mathematics of Communications, 2010, 4 (2) : 237-260. doi: 10.3934/amc.2010.4.237
[4]

Sohana Jahan. Supervised distance preserving projection using alternating direction method of multipliers. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019029
[5]

Zhendong Luo. A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1189-1212. doi: 10.3934/dcdsb.2015.20.1189
[6]

Huan Gao, Zhibao Li, Haibin Zhang. A fast continuous method for the extreme eigenvalue problem. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1587-1599. doi: 10.3934/jimo.2017008
[7]

Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091
[8]

Zhiyou Wu, Fusheng Bai, Guoquan Li, Yongjian Yang. A new auxiliary function method for systems of nonlinear equations. Journal of Industrial & Management Optimization, 2015, 11 (2) : 345-364. doi: 10.3934/jimo.2015.11.345
[9]

Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353
[10]

Regina S. Burachik, C. Yalçın Kaya. An update rule and a convergence result for a penalty function method. Journal of Industrial & Management Optimization, 2007, 3 (2) : 381-398. doi: 10.3934/jimo.2007.3.381
[11]

Liuyang Yuan, Zhongping Wan, Jingjing Zhang, Bin Sun. A filled function method for solving nonlinear complementarity problem. Journal of Industrial & Management Optimization, 2009, 5 (4) : 911-928. doi: 10.3934/jimo.2009.5.911
[12]

Jingzhi Li, Hongyu Liu, Qi Wang. Fast imaging of electromagnetic scatterers by a two-stage multilevel sampling method. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 547-561. doi: 10.3934/dcdss.2015.8.547
[13]

Hyun Geun Lee, Yangjin Kim, Junseok Kim. Mathematical model and its fast numerical method for the tumor growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1173-1187. doi: 10.3934/mbe.2015.12.1173
[14]

Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112
[15]

Yangyang Xu, Wotao Yin. A fast patch-dictionary method for whole image recovery. Inverse Problems & Imaging, 2016, 10 (2) : 563-583. doi: 10.3934/ipi.2016012
[16]

Cheng Ma, Xun Li, Ka-Fai Cedric Yiu, Yongjian Yang, Liansheng Zhang. On an exact penalty function method for semi-infinite programming problems. Journal of Industrial & Management Optimization, 2012, 8 (3) : 705-726. doi: 10.3934/jimo.2012.8.705
[17]

Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. A new exact penalty function method for continuous inequality constrained optimization problems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 895-910. doi: 10.3934/jimo.2010.6.895
[18]

Zhichuan Zhu, Bo Yu, Li Yang. Globally convergent homotopy method for designing piecewise linear deterministic contractual function. Journal of Industrial & Management Optimization, 2014, 10 (3) : 717-741. doi: 10.3934/jimo.2014.10.717
[19]

Qinghua Ma, Zuoliang Xu, Liping Wang. Recovery of the local volatility function using regularization and a gradient projection method. Journal of Industrial & Management Optimization, 2015, 11 (2) : 421-437. doi: 10.3934/jimo.2015.11.421
[20]

Yanfei Wang, Qinghua Ma. A gradient method for regularizing retrieval of aerosol particle size distribution function. Journal of Industrial & Management Optimization, 2009, 5 (1) : 115-126. doi: 10.3934/jimo.2009.5.115

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences