Walk On Spheres Algorithm for Helmholtz and Yukawa Equations via Duffin Correspondence

  • Xuxin Yang
  • Antti RasilaEmail author
  • Tommi Sottinen


We show that a constant-potential time-independent Schrödinger equation with Dirichlet boundary data can be reformulated as a Laplace equation with Dirichlet boundary data. With this reformulation, which we call the Duffin correspondence, we provide a classical Walk On Spheres (WOS) algorithm for Monte Carlo simulation of the solutions of the boundary value problem. We compare the obtained Duffin WOS algorithm with existing modified WOS algorithms.


Brownian motion Helmholtz equation Linearized Poisson–Boltzmann equation Monte Carlo simulation Numerical algorithm Walk On Spheres algorithm Yukawa equation 

Mathematics Subject Classification (2010)

65C05 68U20 35Q40 


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  1. Chung KL, Zhao Z (2001) From Brownian motion to Schrödinger’s equation. 2nd Printing. SpringerGoogle Scholar
  2. Ciesielski Z, Taylor SJ (1962) First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans Am Math Soc 103:434–450MathSciNetCrossRefzbMATHGoogle Scholar
  3. Deaconu M, Herrmann S (2013) Hitting time for bessel processes—walk on moving spheres algorithm. Ann Appl Probab 23(6):2259–2289MathSciNetCrossRefzbMATHGoogle Scholar
  4. Deaconu M, Herrmann S, Maire S (2015) The walk on moving spheres: a new tool for simulating Brownian motion’s exit time from a domain. Math Comput Simul, article in pressGoogle Scholar
  5. Duffin RJ (1971) Yukawan potential theory. J Math Anal Appl 35:105–130MathSciNetCrossRefzbMATHGoogle Scholar
  6. Durrett R (1996) Stochastic calculus: a practical introduction, CRC Press, Boca RatonGoogle Scholar
  7. Elepov BS, Mihailov GA (1973) The “Walk On Spheres” algorithm for the equation Δuc u=−g. Soviet Math Dokl 14:1276–1280Google Scholar
  8. Hwang C-O, Mascagni M (2001) Efficient modified “Walk On Spheres” algorithm for the linearized Poisson-Boltzmann equation. Appl Phys Lett 78(6):787–789CrossRefGoogle Scholar
  9. Kakutani S (1944) On Brownian motion in n-space. Proc Imp Acad Tokyo 20 (9):648–652MathSciNetCrossRefzbMATHGoogle Scholar
  10. Kent JT (1980) Eigenvalue expansion for diffusion hitting times. Z Wahr Ver Gebiete 52:309–319MathSciNetCrossRefzbMATHGoogle Scholar
  11. Krahn E (1926) Über minimaleigenschaft kerkugel in drei und mehr Dimensionen. Acta Comm Univ Tartu (Dorpat) A9:1–44Google Scholar
  12. Mascagni C-O, Hwang M, Given JA (2003) A Feynman–Kac path-integral implementation for Poisson’s equation using an h-conditioned Green’s function. Math Comput Simul 62:347–355MathSciNetCrossRefzbMATHGoogle Scholar
  13. Muller ME (1956) Some continuous Monte Carlo methods for the Dirichlet problem. Ann Math Stat 27:569–589MathSciNetCrossRefzbMATHGoogle Scholar
  14. Rasila A, Sottinen T (2015) Yukawa potential, panharmonic measure and Brownian motion. Preprint. arXiv:1310.2167
  15. Yang X, Rasila A, Sottinen T (2015) Efficient simulation of Schrödinger equation with piecewise constant positive potential. Preprint. arXiv:1512.01306
  16. Wendel JG (1980) Hitting spheres with Brownian motion. Ann Probab 8(1):164–169MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsHunan First Normal UniversityChangshaPeople’s Republic of China
  2. 2.Department of Mathematics and Systems AnalysisAalto UniversityEspooFinland
  3. 3.Faculty of Technology, Department of Mathematics and StatisticsUniversity of VaasaVaasaFinland

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