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A Gaussian beam approach for computing Wigner measures in convex domains
1. | Aeroelasticity and Structural Dynamics Department, Onera-The French Aerospace Lab, F-92322 Châtillon, France |
2. | Department of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University, 200900 Shanghai, China |
3. | Mechanics, Structures and Materials Laboratory, École Centrale Paris, 92295 Châtenay-Malabry, France |
References:
[1] |
R. Alexandre, Oscillations in PDE with singularities of codimension one. Part I : review of the symbolic calculus and basic definitions,, preprint., ().
|
[2] |
G. Ariel, B. Engquist, N. M. Tanushev and R. Tsai, Gaussian beam decomposition of high frequency wave fields using expectation-maximization, J. Comput. Phys., 230 (2011), 2303-2321.
doi: 10.1016/j.jcp.2010.12.018. |
[3] |
V. M. Babič, Eigenfunctions concentrated in a neighborhood of a closed geodesic, (Russian), Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., 9 (1968), 15-63 . |
[4] |
S. Bougacha, J.-L. Akian and R. Alexandre, Gaussian beams summation for the wave equation in a convex domain, Commun. Math. Sci., 7 (2009), 973-1008. |
[5] |
N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, (French) [Exact controllability of waves in nonsmooth domains], Asympt. Anal., 14 (1997), 157-191. |
[6] |
N. Burq, Mesures semi-classiques et mesures de défaut, (French) [Semiclassical measures and defect measures], in "Séminaire Bourbaki," 1996/97, Astérisque, 245 (1997), 167-195. |
[7] |
N. Burq, Quantum ergodicity of boundary values of eigenfunctions: a control theory approach, Canad. Math. Bull., 48 (2005), 3-15.
doi: 10.4153/CMB-2005-001-3. |
[8] |
N. Burq and G. Lebeau, Mesures de défaut de compacité, application au systéme de Lamé, (French) [Microlocal defect measures and application to the Lame system], Ann. Sci. École Norm. Sup. (4), 34 (2001), 817-870. |
[9] |
F. Castella, The radiation condition at infinity for the high frequency Helmholtz equation with source term: a wave packet approach, J. Funct. Anal., 223 (2005), 204-257.
doi: 10.1016/j.jfa.2004.08.008. |
[10] |
V. Červený, M. M. Popov and I. Pšenčík, Computation of wave fields in inhomogeneous media-Gaussian beam approach, Geophys. J. R. Astr. Soc., 70 (1982), 109-128. |
[11] |
J. Chazarain, Paramétrix du problème mixte pour l'équation des ondes à l'intérieur d'un domaine convexe pour les bicaractéristiques, (French), in "Journées Équations aux Dérivées Partielles de Rennes," Astérisque, Soc. Math. France, (1976), 165-181. |
[12] |
M. Combescure, J. Ralston and D. Robert, A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition, Commun. Math. Phys., 202 (1999), 463-480.
doi: 10.1007/s002200050591. |
[13] |
T. Duyckaerts, Stabilization of the linear system of magnetoelasticity,, preprint, ().
|
[14] |
S. Filippas and G. N. Makrakis, Semiclassical Wigner function and geometrical optics, Multiscale Model. Simul., 1 (2003), 674-710.
doi: 10.1137/S1540345902409797. |
[15] |
E. Fouassier, High frequency limit of Helmholtz equations: refraction by sharp interfaces, J. Math. Pures Appl. (9), 87 (2007), 144-192.
doi: 10.1016/j.matpur.2006.11.002. |
[16] |
I. Gasser and P. A. Markowich, Quantum hydrodynamics, Wigner transform and the classical limit, Asympt. Anal., 14 (1997), 97-116. |
[17] |
P. Gérard, Mesures semi-classiques et ondes de Bloch, (French) [Semiclassical measures and Bloch waves], in "Séminaire sur les Équations aux Dérivéees Partielles," 1990-1991, École Polytech., Palaiseau, 1991. |
[18] |
P. Gérard, Microlocal defect measures, Commun. Partial Differential Equations, 16 (1991), 1761-1794.
doi: 10.1080/03605309108820822. |
[19] |
P. Gérard and E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J., 71 (1993), 559-607. |
[20] |
P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379.
doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C. |
[21] |
L. Hörmander, "The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis," Grundlehren der Mathematischen Wissenschaften [Fundamental Principle of Mathematical Sciences], 256, Springer-Verlag, Berlin, 1983. |
[22] |
V. Ivrii, "Microlocal Analysis and Precise Spectral Asymptotics," Springer Monographs in Mathematics, Springer Verlag, Berlin, 1998. |
[23] |
A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall/CRC, Boca Raton, 2001. |
[24] |
A. P. Katchalov and M. M. Popov, The application of the Gaussian beam summation method to the computation of high-frequency wave fields, Dokl. Akad. Nauk, 258 (1981), 1097-1100. |
[25] |
L. Klimeš, Expansion of a high-frequency time-harmonic wavefield given on an initial surface into Gaussian beams, Geophys. J. R. astr. Soc., 79 (1984), 105-118. |
[26] |
A. Laptev and I. M. Sigal, Global Fourier integral operators and semiclassical asymptotics, Rev. Math. Phys., 12 (2000), 749-766. |
[27] |
S. Leung and J. Qian, Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime, J. Comput. Phys., 228 (2009), 2951-2977.
doi: 10.1016/j.jcp.2009.01.007. |
[28] |
S. Leung and J. Qian, The backward phase flow and FBI-transform-based Eulerian Gaussian beams for the Schrödinger equation, J. Comput. Phys., 229 (2010), 8888-8917.
doi: 10.1016/j.jcp.2010.08.015. |
[29] |
P.-L. Lions and T. Paul, Sur les mesures de Wigner, (French) [On Wigner measures], Rev. Mat. Iberoamericana, 9 (1993), 553-618. |
[30] |
H. Liu and J. Ralston, Recovery of high frequency wave fields for the acoustic wave equation,, Multiscale Model. Simul., 8 (): 428.
doi: 10.1137/090761598. |
[31] |
H. Liu, O. Runborg and N. M. Tanushev, Error Estimates for Gaussian Beam Superpositions,, preprint, ().
|
[32] |
F. Macià and E. Zuazua, On the lack of observability for wave equations: a Gaussian beam approach, Asymptot. Anal., 32 (2002), 1-26. |
[33] |
P. A. Markowich and N. J. Mauser, The classical limit of a self-consistent Quantum-Vlasov equation in $3$D, Math. Models Methods Appl. Sci., 3 (1993), 109-124.
doi: 10.1142/S0218202593000072. |
[34] |
P. A. Markowich, N. J. Mauser and F. Poupaud, A Wigner-function approach to (semi)classical limits: electrons in a periodic potential, J. Math. Phys., 35 (1994), 1066-1094.
doi: 10.1063/1.530629. |
[35] |
P. A. Markowich, P. Pietra and C. Pohl., Weak limits of finite difference schemes of Schrödinger-type equations, Pubbl. Ian, 1035 (1997), 1-57. |
[36] |
A. Martinez, "An Introduction to Semiclassical and Microlocal Analysis," Universitext, Springer-Verlag, New York, 2002. |
[37] |
L. Miller, Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary, J. Math. Pures Appl. (9), 79 (2000), 227-269.
doi: 10.1016/S0021-7824(00)00158-6. |
[38] |
M. Motamed and O. Runborg, Taylor expansion and discretization errors in Gaussian beam superposition, Wave Motion, 47 (2010), 421-439.
doi: 10.1016/j.wavemoti.2010.02.001. |
[39] |
A. N. Norris, Elastic Gaussian wave packets in isotropic media, Acta Mech., 71 (1988), 95-114.
doi: 10.1007/BF01173940. |
[40] |
G. Papanicolaou and L. Ryzhik, Waves and Transport, in "Hyperbolic Equations and Frequency Interactions" (Park City, UT, 1995), IAS/Park City Math. Ser., 5, Amer. Math. Soc., Providence, RI, (1999), 305-382. |
[41] |
T. Paul and A. Uribe, On the pointwise behavior of semi-classical measures, Comm. Math. Phys., 175 (1996), 229-258.
doi: 10.1007/BF02102407. |
[42] |
M. Pulvirenti, Semiclassical expansion of Wigner functions, J. Math. Phys., 47 (2006), 12 pp. |
[43] |
J. Qian and L. Ying, Fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beams for the wave equation, Multiscale Model. Simul., 8 (2010), 1803-1837.
doi: 10.1137/100787313. |
[44] |
J. Ralston, Gaussian beams and the propagation of singularities, in "Studies in Partial Differential Equations," MAA Stud. Math., 23, Math. Assoc. America, (1982), 206-248. |
[45] |
S. L. Robinson, Semiclassical mechanics for time-dependent Wigner functions, J. Math. Phys., 34 (1993), 2185-2205.
doi: 10.1063/1.530112. |
[46] |
N. M. Tanushev, Superpositions and higher order Gaussian beams, Commun. Math. Sci., 6 (2008), 449-475. |
[47] |
N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields, J. Comput. Phys., 228 (2009), 8856-8871.
doi: 10.1016/j.jcp.2009.08.028. |
[48] |
N. M. Tanushev, J. Qian and J. V. Ralston, Mountain waves and Gaussian beams, Multiscale Model. Simul., 6 (2007), 688-709.
doi: 10.1137/060673667. |
[49] |
L. Tartar, $H$-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 193-230. |
[50] |
E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759.
doi: 10.1103/PhysRev.40.749. |
[51] |
M. Wilkinson, A semiclassical sum rule for matrix elements of classically chaotic systems, J. Phys. A, 20 (1987), 2415-2423.
doi: 10.1088/0305-4470/20/9/028. |
[52] |
S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J., 55 (1987), 919-941.
doi: 10.1215/S0012-7094-87-05546-3. |
show all references
References:
[1] |
R. Alexandre, Oscillations in PDE with singularities of codimension one. Part I : review of the symbolic calculus and basic definitions,, preprint., ().
|
[2] |
G. Ariel, B. Engquist, N. M. Tanushev and R. Tsai, Gaussian beam decomposition of high frequency wave fields using expectation-maximization, J. Comput. Phys., 230 (2011), 2303-2321.
doi: 10.1016/j.jcp.2010.12.018. |
[3] |
V. M. Babič, Eigenfunctions concentrated in a neighborhood of a closed geodesic, (Russian), Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., 9 (1968), 15-63 . |
[4] |
S. Bougacha, J.-L. Akian and R. Alexandre, Gaussian beams summation for the wave equation in a convex domain, Commun. Math. Sci., 7 (2009), 973-1008. |
[5] |
N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, (French) [Exact controllability of waves in nonsmooth domains], Asympt. Anal., 14 (1997), 157-191. |
[6] |
N. Burq, Mesures semi-classiques et mesures de défaut, (French) [Semiclassical measures and defect measures], in "Séminaire Bourbaki," 1996/97, Astérisque, 245 (1997), 167-195. |
[7] |
N. Burq, Quantum ergodicity of boundary values of eigenfunctions: a control theory approach, Canad. Math. Bull., 48 (2005), 3-15.
doi: 10.4153/CMB-2005-001-3. |
[8] |
N. Burq and G. Lebeau, Mesures de défaut de compacité, application au systéme de Lamé, (French) [Microlocal defect measures and application to the Lame system], Ann. Sci. École Norm. Sup. (4), 34 (2001), 817-870. |
[9] |
F. Castella, The radiation condition at infinity for the high frequency Helmholtz equation with source term: a wave packet approach, J. Funct. Anal., 223 (2005), 204-257.
doi: 10.1016/j.jfa.2004.08.008. |
[10] |
V. Červený, M. M. Popov and I. Pšenčík, Computation of wave fields in inhomogeneous media-Gaussian beam approach, Geophys. J. R. Astr. Soc., 70 (1982), 109-128. |
[11] |
J. Chazarain, Paramétrix du problème mixte pour l'équation des ondes à l'intérieur d'un domaine convexe pour les bicaractéristiques, (French), in "Journées Équations aux Dérivées Partielles de Rennes," Astérisque, Soc. Math. France, (1976), 165-181. |
[12] |
M. Combescure, J. Ralston and D. Robert, A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition, Commun. Math. Phys., 202 (1999), 463-480.
doi: 10.1007/s002200050591. |
[13] |
T. Duyckaerts, Stabilization of the linear system of magnetoelasticity,, preprint, ().
|
[14] |
S. Filippas and G. N. Makrakis, Semiclassical Wigner function and geometrical optics, Multiscale Model. Simul., 1 (2003), 674-710.
doi: 10.1137/S1540345902409797. |
[15] |
E. Fouassier, High frequency limit of Helmholtz equations: refraction by sharp interfaces, J. Math. Pures Appl. (9), 87 (2007), 144-192.
doi: 10.1016/j.matpur.2006.11.002. |
[16] |
I. Gasser and P. A. Markowich, Quantum hydrodynamics, Wigner transform and the classical limit, Asympt. Anal., 14 (1997), 97-116. |
[17] |
P. Gérard, Mesures semi-classiques et ondes de Bloch, (French) [Semiclassical measures and Bloch waves], in "Séminaire sur les Équations aux Dérivéees Partielles," 1990-1991, École Polytech., Palaiseau, 1991. |
[18] |
P. Gérard, Microlocal defect measures, Commun. Partial Differential Equations, 16 (1991), 1761-1794.
doi: 10.1080/03605309108820822. |
[19] |
P. Gérard and E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J., 71 (1993), 559-607. |
[20] |
P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379.
doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C. |
[21] |
L. Hörmander, "The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis," Grundlehren der Mathematischen Wissenschaften [Fundamental Principle of Mathematical Sciences], 256, Springer-Verlag, Berlin, 1983. |
[22] |
V. Ivrii, "Microlocal Analysis and Precise Spectral Asymptotics," Springer Monographs in Mathematics, Springer Verlag, Berlin, 1998. |
[23] |
A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall/CRC, Boca Raton, 2001. |
[24] |
A. P. Katchalov and M. M. Popov, The application of the Gaussian beam summation method to the computation of high-frequency wave fields, Dokl. Akad. Nauk, 258 (1981), 1097-1100. |
[25] |
L. Klimeš, Expansion of a high-frequency time-harmonic wavefield given on an initial surface into Gaussian beams, Geophys. J. R. astr. Soc., 79 (1984), 105-118. |
[26] |
A. Laptev and I. M. Sigal, Global Fourier integral operators and semiclassical asymptotics, Rev. Math. Phys., 12 (2000), 749-766. |
[27] |
S. Leung and J. Qian, Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime, J. Comput. Phys., 228 (2009), 2951-2977.
doi: 10.1016/j.jcp.2009.01.007. |
[28] |
S. Leung and J. Qian, The backward phase flow and FBI-transform-based Eulerian Gaussian beams for the Schrödinger equation, J. Comput. Phys., 229 (2010), 8888-8917.
doi: 10.1016/j.jcp.2010.08.015. |
[29] |
P.-L. Lions and T. Paul, Sur les mesures de Wigner, (French) [On Wigner measures], Rev. Mat. Iberoamericana, 9 (1993), 553-618. |
[30] |
H. Liu and J. Ralston, Recovery of high frequency wave fields for the acoustic wave equation,, Multiscale Model. Simul., 8 (): 428.
doi: 10.1137/090761598. |
[31] |
H. Liu, O. Runborg and N. M. Tanushev, Error Estimates for Gaussian Beam Superpositions,, preprint, ().
|
[32] |
F. Macià and E. Zuazua, On the lack of observability for wave equations: a Gaussian beam approach, Asymptot. Anal., 32 (2002), 1-26. |
[33] |
P. A. Markowich and N. J. Mauser, The classical limit of a self-consistent Quantum-Vlasov equation in $3$D, Math. Models Methods Appl. Sci., 3 (1993), 109-124.
doi: 10.1142/S0218202593000072. |
[34] |
P. A. Markowich, N. J. Mauser and F. Poupaud, A Wigner-function approach to (semi)classical limits: electrons in a periodic potential, J. Math. Phys., 35 (1994), 1066-1094.
doi: 10.1063/1.530629. |
[35] |
P. A. Markowich, P. Pietra and C. Pohl., Weak limits of finite difference schemes of Schrödinger-type equations, Pubbl. Ian, 1035 (1997), 1-57. |
[36] |
A. Martinez, "An Introduction to Semiclassical and Microlocal Analysis," Universitext, Springer-Verlag, New York, 2002. |
[37] |
L. Miller, Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary, J. Math. Pures Appl. (9), 79 (2000), 227-269.
doi: 10.1016/S0021-7824(00)00158-6. |
[38] |
M. Motamed and O. Runborg, Taylor expansion and discretization errors in Gaussian beam superposition, Wave Motion, 47 (2010), 421-439.
doi: 10.1016/j.wavemoti.2010.02.001. |
[39] |
A. N. Norris, Elastic Gaussian wave packets in isotropic media, Acta Mech., 71 (1988), 95-114.
doi: 10.1007/BF01173940. |
[40] |
G. Papanicolaou and L. Ryzhik, Waves and Transport, in "Hyperbolic Equations and Frequency Interactions" (Park City, UT, 1995), IAS/Park City Math. Ser., 5, Amer. Math. Soc., Providence, RI, (1999), 305-382. |
[41] |
T. Paul and A. Uribe, On the pointwise behavior of semi-classical measures, Comm. Math. Phys., 175 (1996), 229-258.
doi: 10.1007/BF02102407. |
[42] |
M. Pulvirenti, Semiclassical expansion of Wigner functions, J. Math. Phys., 47 (2006), 12 pp. |
[43] |
J. Qian and L. Ying, Fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beams for the wave equation, Multiscale Model. Simul., 8 (2010), 1803-1837.
doi: 10.1137/100787313. |
[44] |
J. Ralston, Gaussian beams and the propagation of singularities, in "Studies in Partial Differential Equations," MAA Stud. Math., 23, Math. Assoc. America, (1982), 206-248. |
[45] |
S. L. Robinson, Semiclassical mechanics for time-dependent Wigner functions, J. Math. Phys., 34 (1993), 2185-2205.
doi: 10.1063/1.530112. |
[46] |
N. M. Tanushev, Superpositions and higher order Gaussian beams, Commun. Math. Sci., 6 (2008), 449-475. |
[47] |
N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields, J. Comput. Phys., 228 (2009), 8856-8871.
doi: 10.1016/j.jcp.2009.08.028. |
[48] |
N. M. Tanushev, J. Qian and J. V. Ralston, Mountain waves and Gaussian beams, Multiscale Model. Simul., 6 (2007), 688-709.
doi: 10.1137/060673667. |
[49] |
L. Tartar, $H$-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 193-230. |
[50] |
E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759.
doi: 10.1103/PhysRev.40.749. |
[51] |
M. Wilkinson, A semiclassical sum rule for matrix elements of classically chaotic systems, J. Phys. A, 20 (1987), 2415-2423.
doi: 10.1088/0305-4470/20/9/028. |
[52] |
S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J., 55 (1987), 919-941.
doi: 10.1215/S0012-7094-87-05546-3. |
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