A novel stochastic method for the solution of direct and inverse exterior elliptic problems
Authors:
Antonios Charalambopoulos and Leonidas N. Gergidis
Journal:
Quart. Appl. Math. 76 (2018), 65-111
MSC (2010):
Primary 35J25, 60H10; Secondary 78A46
DOI:
https://doi.org/10.1090/qam/1480
Published electronically:
October 2, 2017
MathSciNet review:
3733095
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Abstract: A new method, in the interface of stochastic differential equations with boundary value problems, is developed in this work, aiming at representing solutions of exterior boundary value problems in terms of stochastic processes. The main effort concerns exterior harmonic problems but furthermore special attention has been paid to the investigation of time-reduced scattering processes (involving the Helmholtz operator) in the realm of low frequencies. The method, in principle, faces the construction of the solution of the direct versions of the aforementioned boundary value problems but the special features of the method assure definitely the usefulness of the approach to the solution of the corresponding inverse problems as clearly indicated herein.
References
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- Bernt Øksendal, Stochastic differential equations, 6th ed., Universitext, Springer-Verlag, Berlin, 2003. An introduction with applications. MR 2001996
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- Tu Sheng Zhang, Probabilistic approach to the Neumann problem, Acta Math. Appl. Sinica (English Ser.) 6 (1990), no. 2, 126–134. MR 1056122, DOI https://doi.org/10.1007/BF02006749
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- Fred Espen Benth, Option theory with stochastic analysis, Universitext, Springer-Verlag, Berlin, 2004. An introduction to mathematical finance; Revised edition of the 2001 Norwegian original. MR 2043196
- A. Charalambopoulos, G. Dassios, and M. Hadjinicolaou, An analytic solution for low-frequency scattering by two soft spheres, SIAM J. Appl. Math. 58 (1998), no. 2, 370–386. MR 1617658, DOI https://doi.org/10.1137/S0036139996304081
- N. N. Lebedev, Special functions and their applications, Revised English edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. Translated and edited by Richard A. Silverman. MR 0174795
- Peter E. Kloeden, Eckhard Platen, and Henri Schurz, Numerical solution of SDE through computer experiments, Universitext, Springer-Verlag, Berlin, 1994. With 1 IBM-PC floppy disk (3.5 inch; HD). MR 1260431
- A. Charalambopoulos and G. Dassios, Scattering of a spherical wave by a small ellipsoid, IMA J. Appl. Math. 62 (1999), no. 2, 117–136. MR 1694639, DOI https://doi.org/10.1093/imamat/62.2.117
- George Dassios, Ellipsoidal harmonics, Encyclopedia of Mathematics and its Applications, vol. 146, Cambridge University Press, Cambridge, 2012. Theory and applications. MR 2977792
- A. Charalambopoulos and N. L. Gergidis, On the investigation of exterior boundary value problems involving Helmholtz operator via stochastic calculus., In preparation.
References
- William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312
- R. Kress, Linear Integral Equations, Springer-Verlag Berlin Heidelberg, 1989
- Frank Ihlenburg, Finite element analysis of acoustic scattering, Applied Mathematical Sciences, vol. 132, Springer-Verlag, New York, 1998. MR 1639879
- Robert John Strutt Rayleigh Fourth Baron, Life of John William Strutt, Third Baron Rayleigh, O.M., F.R.S, An augmented edition with annotations by the author and foreword by John N. Howard, The University of Wisconsin Press, Madison, Wis.-London, 1968. MR 0224421
- R. Courant, K. Friedrichs, and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100 (1928), no. 1, 32–74 (German). MR 1512478, DOI https://doi.org/10.1007/BF01448839
- Mandar K. Chati, Mircea D. Grigoriu, Salil S. Kulkarni, and Subrata Mukherjee, Random walk method for the two- and three-dimensional Laplace, Poisson and Helmholtz’s equations, Internat. J. Numer. Methods Engrg. 51 (2001), no. 10, 1133–1156. MR 1842723, DOI https://doi.org/10.1002/nme.178
- Bernt Øksendal, Stochastic differential equations, 6th ed., Universitext, Springer-Verlag, Berlin, 2003. An introduction with applications. MR 2001996
- J. Lamperti, Stochastic Properties, Springer-Verlag, 1977
- R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR 0264757
- J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258
- George Papanicolau and Joseph B. Keller, Stochastic differential equations with applications to random harmonic oscillators and wave propagation in random media, SIAM J. Appl. Math. 21 (1971), 287–305. MR 0309194, DOI https://doi.org/10.1137/0121032
- G. C. Papanicolaou, Stochastic equations and their applications, Amer. Math. Monthly 80 (1973), 526–545. MR 0317407, DOI https://doi.org/10.2307/2319609
- Richard F. Bass and Pei Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains, Ann. Probab. 19 (1991), no. 2, 486–508. MR 1106272
- Tu Sheng Zhang, Probabilistic approach to the Neumann problem, Acta Math. Appl. Sinica (English Ser.) 6 (1990), no. 2, 126–134. MR 1056122, DOI https://doi.org/10.1007/BF02006749
- Pei Hsu, Probabilistic approach to the Neumann problem, Comm. Pure Appl. Math. 38 (1985), no. 4, 445–472. MR 792399, DOI https://doi.org/10.1002/cpa.3160380406
- Bair V. Budaev and David B. Bogy, Novel solutions of the Helmholtz equation and their application to diffraction, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), no. 2080, 1005–1027. MR 2310134, DOI https://doi.org/10.1098/rspa.2006.1809
- Fred Espen Benth, Option theory with stochastic analysis, Universitext, Springer-Verlag, Berlin, 2004. An introduction to mathematical finance; Revised edition of the 2001 Norwegian original. MR 2043196
- A. Charalambopoulos, G. Dassios, and M. Hadjinicolaou, An analytic solution for low-frequency scattering by two soft spheres, SIAM J. Appl. Math. 58 (1998), no. 2, 370–386. MR 1617658, DOI https://doi.org/10.1137/S0036139996304081
- N. N. Lebedev, Special functions and their applications, Revised English edition. Translated and edited by Richard A. Silverman, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0174795
- Peter E. Kloeden, Eckhard Platen, and Henri Schurz, Numerical solution of SDE through computer experiments, Universitext, Springer-Verlag, Berlin, 1994. With 1 IBM-PC floppy disk (3.5 inch; HD). MR 1260431
- A. Charalambopoulos and G. Dassios, Scattering of a spherical wave by a small ellipsoid, IMA J. Appl. Math. 62 (1999), no. 2, 117–136. MR 1694639, DOI https://doi.org/10.1093/imamat/62.2.117
- George Dassios, Ellipsoidal harmonics, Encyclopedia of Mathematics and its Applications, vol. 146, Cambridge University Press, Cambridge, 2012. Theory and applications. MR 2977792
- A. Charalambopoulos and N. L. Gergidis, On the investigation of exterior boundary value problems involving Helmholtz operator via stochastic calculus., In preparation.
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Additional Information
Antonios Charalambopoulos
Affiliation:
Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou Campus, 15780, Greece
MR Author ID:
328137
Email:
acharala@math.ntua.gr
Leonidas N. Gergidis
Affiliation:
Department of Materials Science and Engineering, University of Ioannina, 45110,Greece
MR Author ID:
823867
Email:
lgergidi@uoi.gr
Keywords:
Exterior boundary value problems,
direct and inverse problems,
stochastic differential equations
Received by editor(s):
April 21, 2017
Published electronically:
October 2, 2017
Article copyright:
© Copyright 2017
Brown University