# American Institute of Mathematical Sciences

December  2010, 3(4): 529-644. doi: 10.3934/krm.2010.3.529

## Kinetic limits for waves in a random medium

 1 Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027 2 Institute of Mathematics, UMCS, pl. Marii Curie-Skłodowskiej 1, 20-031, Lubin, IMPAN, ul. Śniadeckich 8, 00-956 Warsaw, Poland 3 Department of Mathematics, Stanford University, Stanford, CA 94305, United States

Received  September 2010 Revised  October 2010 Published  October 2010

N/A
Citation: Guillaume Bal, Tomasz Komorowski, Lenya Ryzhik. Kinetic limits for waves in a random medium. Kinetic and Related Models, 2010, 3 (4) : 529-644. doi: 10.3934/krm.2010.3.529
##### References:
 [1] F. Bailly, J. F. Clouet and J.-P. Fouque, Parabolic and gaussian white noise approximation for wave propagation in random media, SIAM J. Appl. Math, 56 (1996), 1445-1470. [2] G. Bal, On the self-averaging of wave energy in random media, Multiscale Model. Simul., 2 (2004), 398-420. [3] G. Bal, Kinetics of scalar wave fields in random media, Wave Motion, 43 (2005), 132-157. [4] G. Bal, Inverse problems in random media: A kinetic approach, J. Phys. Conf. Series, 124 (2008), 012001. [5] G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001. [6] G. Bal, L. Carin, D. Liu and K. Ren, Experimental validation of a transport-based imaging method in highly scattering environments, Inverse Problems, 26 (2007), 2527-2539. [7] G. Bal, D. Liu, S. Vasudevan, J. Krolik and L. Carin, Electromagnetic time-reversal imaging in changing media: Experiment and analysis, IEEE Trans. Anten. and Prop., 55 (2007), 344-354. [8] G. Bal, T. Komorowski and L. Ryzhik, Self-averaging of Wigner transforms in random media, Comm. Math. Phys., 242 (2003), 81-135. [9] G. Bal, T. Komorowski and L. Ryzhik, Asymptotics of the solutions of the random Schródinger equation, to appear in Arch. Rat. Mech., (2010). [10] G. Bal, G. Papanicolaou and L. Ryzhik, Radiative transport limit for the random Schrödinger equations, Nonlinearity, 15 (2002), 513-529. [11] G. Bal, G. Papanicolaou and L. Ryzhik, Self-averaging in time reversal for the parabolic wave equation, Stochastics and Dynamics, 4 (2002), 507-531. [12] G. Bal and O. Pinaud, Time reversal-based imaging in random media, Inverse Problems, 21 (2005), 1593-1620. [13] G. Bal and O. Pinaud, Accuracy of transport models for waves in random media, Wave Motion, 43 (2006), 561-578. [14] G. Bal and O. Pinaud, Kinetic models for imaging in random media, Multiscale Model. Simul., 6 (2007), 792-819. [15] G. Bal and O. Pinaud, Self-averaging of kinetic models for waves in random media, Kinetic Related Models, 1 (2008), 85-100. [16] G. Bal and O. Pinaud, Imaging using transport models for wave-wave correlations, to appear in M3AS, (2011). [17] G. Bal and K. Ren, Transport-based imaging in random media, SIAM J. Applied Math., 68 (2008), 1738-1762. [18] G. Bal and L. Ryzhik, Time reversal and refocusing in random media, SIAM J. Appl. Math., 63 (2003), 1475-1498. [19] G. Bal and L. Ryzhik, Time splitting for wave equations in random media, preprint, (2004). [20] G. Bal and L. Ryzhik, Stability of time reversed waves in changing media, Discrete Contin. Dyn. Syst., 12 (2005), 793-815. [21] G. Bal and R. Verástegui, Time reversal in changing environment, Multiscale Model. Simul., 2 (2004), 639-661. [22] W. Bao, S. Jin and P. A. Markowich, On Time-Splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comp. Phys., 175 (2002), 487-524. [23] P. Billingsley, "Convergence of Probability Measures," Wiley, 1999. [24] P. Blankenship and G. C. Papanicolaou, Stability and control of stochastic systems with wide-band noise disturbances. i, SIAM J. Appl. Math., 34 (1978), 437-476. [25] P. Blomgren, G. C. Papanicolaou and H. Zhao, Super-Resolution in Time-Reversal Acoustics, J. Acoust. Soc. Am., 111 (2002), 230-248. [26] B. Borcea, G. C. Papanicolaou and C. Tsogka, Theory and applications of time reversal and interferometric imaging, Inverse Problems, 19 (2003), S139-S164. [27] B. Borcea, G. C. Papanicolaou and C. Tsogka, Interferometric array imaging in clutter, Inverse Problems, 21 (2005), 1419-1460. [28] B. Borcea, G. C. Papanicolaou and C. Tsogka, Adaptive interferometric imaging in clutter and optimal illumination, Inverse Problems, 22 (2006), 1405-1436. [29] A. N. Borodin, A limit theorem for solutions of differential equations with random right hand side, Teor. Veroyatn. Ee Primen, 22 (1977), 498-512. [30] R. A. Carmona and S. A. Molchanov, Parabolic Anderson problem and intermittency, Mem. Amer. Math. Soc, 108 (1994), viii+125. [31] S. Chandrasekhar, "Radiative Transfer," Dover Publications, New York, 1960. [32] G. C. Cohen, "Higher-Order Numerical Methods for Transient Wave Equations," Scientific Computation, Springer Verlag, Berlin, 2002. [33] D. A. Dawson and G. C. Papanicolaou, A random wave process, Appl. Math. Optim., 12 (1984), 97-114. [34] D. Dürr, S. Goldstein and J. Lebowitz, Asymptotic motion of a classical particle in a random potential in two dimensions: Landau model, Comm. Math. Phys., 113 (1987), 209-230. [35] D. R. Durran, "Numerical Methods for Wave equations in Geophysical Fluid Dynamics," Springer, New York, 1999. [36] G. F. Edelmann, T. Akal, W. S. Hodgkiss, S. Kim, W. A. Kuperman and H. C. Song, An initial demonstration of underwater acoustic communication using time reversal, IEEE J. Oceanic Eng., 27 (2002), 602-609. [37] L. Erdös and H. T. Yau, Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation, Comm. Pure Appl. Math., 53 (2000), 667-735. [38] L. Evans and M. Zworski, Lectures on semiclassical analysis,, Berkeley., (). [39] A. Fannjiang, Self-averaging in scaling limits for random high-frequency parabolic waves, Archives of Rational Mechanics and Analysis, 175 (2005), 343-387. [40] J. P. Fouque, La convergence en loi pour les processus à valeur dans un espace nucléaire, Ann. Inst. H. Poincaré Prob. Stat, 20 (1984), 225-245. [41] J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, "Wave Propagation and Time Reversal in Randomly Layered Media," Springer Verlag, New York, 2007. [42] P. Gérard, Microlocal defect measures, Comm. PDEs, 16 (1991), 1761-1794. [43] P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-380. [44] F. Golse, S. Jin and C. D. Levermore, The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method, SIAM J. Numer. Anal., 36 (1999), 1333-1369. [45] T. G. Ho, L. J. Landau and A. J. Wilkins, On the weak coupling limit for a Fermi gas in a random potential, Rev. Math. Phys., 5 (1992), 209-298. [46] H. Hochstadt, "The Functions of Mathematical Physics," Dover Publications, New York, 1986. [47] T. Y. Hou, X. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 227 (1999), 913-943. [48] I. A. Ibragimov and Yu. V. Linnik, "Independent and Stationary Sequences of Random Variables," Wolters-Noordhoff Publishing, Groningen, 1971. [49] A. Ishimaru, "Wave Propagation and Scattering in Random Media," New York, Academics, 1978. [50] J. Jacod and A. N. Shiryaev, "Limit Theorems for Stochastic Processes," Grundlehren der mathematischen Wissenschaft 288, Springer-Verlag, 2003. [51] A. Jakubowski, A non-Skorohod topology on the Skorohod space, Electron. J. Probability, 2 (1997), 1-21. [52] H. Kesten and G. Papanicolaou, A limit theorem for turbulent diffusion, Comm. Math. Phys., 65 (1979), 97-128. [53] H. Kesten and G. C. Papanicolaou, A limit theorem for stochastic acceleration, Comm. Math. Phys., 78 (1980), 19-63. [54] R. Khasminskii, A limit theorem for solutions of differential equations with a random right hand side, Theory Probab. Appl., 11 (1966), 390-406. [55] T. Komorowski, Diffusion approximation for the advection of particles in a a strongly turbulent random environment, Ann. Probab., 24 (1996), 346-376. [56] T. Komorowski, Sz. Peszat and L. Ryzhik, Limit of fluctuations of solutions of Wigner equation, Comm. Math. Phys., 292 (2009), 479-510. [57] T. Komorowski and L. Ryzhik, Diffusion in a weakly random Hamiltonian flow, Comm. Math. Phys., 263 (2006), 277-323. [58] T. Komorowski and L. Ryzhik, The stochastic acceleration problem in two dimensions, Israel Jour.Math., 155 (2006), 157-204. [59] T. Komorowski and L. Ryzhik, Asymptotics of the phase of the solutions of the random Schrödinger equation, preprint, (2010). [60] P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618. [61] D. Liu, S. Vasudevan, J. Krolik, G. Bal and L. Carin, Electromagnetic time-reversal imaging in changing media: Experiment and analysis, IEEE Trans. Antennas and Prop., 55 (2007), 344-354. [62] J. Lukkarinen and H. Spohn, Kinetic limit for wave propagation in a random medium, Arch. Ration. Mech. Anal., 183 (2007), 93-162. [63] P. Markowich, P. Pietra and C. Pohl, Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit, Numer. Math., 81 (1999), 595-630. [64] P. Markowich, P. Pietra, C. Pohl and H. P. Stimming, A Wigner-measure analysis of the Dufort-Frankel scheme for the Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), 1281-1310. [65] I. Mitoma, On the sample continuity of $\mathcal S'$ processes, J. Math. Soc. Japan, 35 (1983), 629-636. [66] B. Øksendal, "Stochastic Differential Equations," Springer-Verlag, Berlin, 2000. [67] G. Papanicolaou and W. Kohler, Asymptotic theory of mixing stochastic ordinary differential equations, Comm. Pure Appl. Math., 27 (1974), 641-668. [68] G. Papanicolaou, L. Ryzhik and K. Sølna, The parabolic wave approximation and time reversal, Matematica Contemporanea, 23 (2002), 139-159. [69] G. C. Papanicolaou, L. Ryzhik and K. Sølna, Self-averaging from lateral diversity in the Ito-Schroedinger equation, Multiscale Model. Simul., 6 (2007), 468-492. [70] F. Poupaud and A. Vasseur, Classical and quantum transport in random media, J. Math. Pures Appl., 6 (2003), 711-748. [71] L. Ryzhik, G. Papanicolaou and J. B. Keller, Transport equations for elastic and other waves in random media, Wave Motion, 24 (1996), 327-370. [72] H. Sato and M. C. Fehler, "Seismic Wave Propagation and Scattering in the Heterogeneous Earth," AIP series in modern acoustics and signal processing, AIP Press, Springer, New York, 1998. [73] P. Sheng, "Introduction to Wave Scattering, Localization and Mesoscopic Phenomena," Academic Press, New York, 1995. [74] H. Spohn, Derivation of the transport equation for electrons moving through random impurities, Jour. Stat. Phys., 17 (1977), 385-412. [75] G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 507-517. [76] D. W. Stroock and S. R. S. Varadhan, "Multidimensional Diffusion Processes," Grundlehren der mathematischen Wissenschaften 233, Berlin, Heidelberg, New York, 1979, 1997. [77] C. R. Vogel, "Computational Methods for Inverse Problems," Frontiers Appl. Math., SIAM, Philadelphia, 2002. [78] E. Wigner, On the quantum correction for thermodynamic equilibrium, Physical Rev., 40 (1932), 749-759. [79] B. White, The stochastic caustic, SIMA Jour. Appl. Math., 44 (1984), 127-149.

show all references

##### References:
 [1] F. Bailly, J. F. Clouet and J.-P. Fouque, Parabolic and gaussian white noise approximation for wave propagation in random media, SIAM J. Appl. Math, 56 (1996), 1445-1470. [2] G. Bal, On the self-averaging of wave energy in random media, Multiscale Model. Simul., 2 (2004), 398-420. [3] G. Bal, Kinetics of scalar wave fields in random media, Wave Motion, 43 (2005), 132-157. [4] G. Bal, Inverse problems in random media: A kinetic approach, J. Phys. Conf. Series, 124 (2008), 012001. [5] G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001. [6] G. Bal, L. Carin, D. Liu and K. Ren, Experimental validation of a transport-based imaging method in highly scattering environments, Inverse Problems, 26 (2007), 2527-2539. [7] G. Bal, D. Liu, S. Vasudevan, J. Krolik and L. Carin, Electromagnetic time-reversal imaging in changing media: Experiment and analysis, IEEE Trans. Anten. and Prop., 55 (2007), 344-354. [8] G. Bal, T. Komorowski and L. Ryzhik, Self-averaging of Wigner transforms in random media, Comm. Math. Phys., 242 (2003), 81-135. [9] G. Bal, T. Komorowski and L. Ryzhik, Asymptotics of the solutions of the random Schródinger equation, to appear in Arch. Rat. Mech., (2010). [10] G. Bal, G. Papanicolaou and L. Ryzhik, Radiative transport limit for the random Schrödinger equations, Nonlinearity, 15 (2002), 513-529. [11] G. Bal, G. Papanicolaou and L. Ryzhik, Self-averaging in time reversal for the parabolic wave equation, Stochastics and Dynamics, 4 (2002), 507-531. [12] G. Bal and O. Pinaud, Time reversal-based imaging in random media, Inverse Problems, 21 (2005), 1593-1620. [13] G. Bal and O. Pinaud, Accuracy of transport models for waves in random media, Wave Motion, 43 (2006), 561-578. [14] G. Bal and O. Pinaud, Kinetic models for imaging in random media, Multiscale Model. Simul., 6 (2007), 792-819. [15] G. Bal and O. Pinaud, Self-averaging of kinetic models for waves in random media, Kinetic Related Models, 1 (2008), 85-100. [16] G. Bal and O. Pinaud, Imaging using transport models for wave-wave correlations, to appear in M3AS, (2011). [17] G. Bal and K. Ren, Transport-based imaging in random media, SIAM J. Applied Math., 68 (2008), 1738-1762. [18] G. Bal and L. Ryzhik, Time reversal and refocusing in random media, SIAM J. Appl. Math., 63 (2003), 1475-1498. [19] G. Bal and L. Ryzhik, Time splitting for wave equations in random media, preprint, (2004). [20] G. Bal and L. Ryzhik, Stability of time reversed waves in changing media, Discrete Contin. Dyn. Syst., 12 (2005), 793-815. [21] G. Bal and R. Verástegui, Time reversal in changing environment, Multiscale Model. Simul., 2 (2004), 639-661. [22] W. Bao, S. Jin and P. A. Markowich, On Time-Splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comp. Phys., 175 (2002), 487-524. [23] P. Billingsley, "Convergence of Probability Measures," Wiley, 1999. [24] P. Blankenship and G. C. Papanicolaou, Stability and control of stochastic systems with wide-band noise disturbances. i, SIAM J. Appl. Math., 34 (1978), 437-476. [25] P. Blomgren, G. C. Papanicolaou and H. Zhao, Super-Resolution in Time-Reversal Acoustics, J. Acoust. Soc. Am., 111 (2002), 230-248. [26] B. Borcea, G. C. Papanicolaou and C. Tsogka, Theory and applications of time reversal and interferometric imaging, Inverse Problems, 19 (2003), S139-S164. [27] B. Borcea, G. C. Papanicolaou and C. Tsogka, Interferometric array imaging in clutter, Inverse Problems, 21 (2005), 1419-1460. [28] B. Borcea, G. C. Papanicolaou and C. Tsogka, Adaptive interferometric imaging in clutter and optimal illumination, Inverse Problems, 22 (2006), 1405-1436. [29] A. N. Borodin, A limit theorem for solutions of differential equations with random right hand side, Teor. Veroyatn. Ee Primen, 22 (1977), 498-512. [30] R. A. Carmona and S. A. Molchanov, Parabolic Anderson problem and intermittency, Mem. Amer. Math. Soc, 108 (1994), viii+125. [31] S. Chandrasekhar, "Radiative Transfer," Dover Publications, New York, 1960. [32] G. C. Cohen, "Higher-Order Numerical Methods for Transient Wave Equations," Scientific Computation, Springer Verlag, Berlin, 2002. [33] D. A. Dawson and G. C. Papanicolaou, A random wave process, Appl. Math. Optim., 12 (1984), 97-114. [34] D. Dürr, S. Goldstein and J. Lebowitz, Asymptotic motion of a classical particle in a random potential in two dimensions: Landau model, Comm. Math. Phys., 113 (1987), 209-230. [35] D. R. Durran, "Numerical Methods for Wave equations in Geophysical Fluid Dynamics," Springer, New York, 1999. [36] G. F. Edelmann, T. Akal, W. S. Hodgkiss, S. Kim, W. A. Kuperman and H. C. Song, An initial demonstration of underwater acoustic communication using time reversal, IEEE J. Oceanic Eng., 27 (2002), 602-609. [37] L. Erdös and H. T. Yau, Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation, Comm. Pure Appl. Math., 53 (2000), 667-735. [38] L. Evans and M. Zworski, Lectures on semiclassical analysis,, Berkeley., (). [39] A. Fannjiang, Self-averaging in scaling limits for random high-frequency parabolic waves, Archives of Rational Mechanics and Analysis, 175 (2005), 343-387. [40] J. P. Fouque, La convergence en loi pour les processus à valeur dans un espace nucléaire, Ann. Inst. H. Poincaré Prob. Stat, 20 (1984), 225-245. [41] J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, "Wave Propagation and Time Reversal in Randomly Layered Media," Springer Verlag, New York, 2007. [42] P. Gérard, Microlocal defect measures, Comm. PDEs, 16 (1991), 1761-1794. [43] P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-380. [44] F. Golse, S. Jin and C. D. Levermore, The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method, SIAM J. Numer. Anal., 36 (1999), 1333-1369. [45] T. G. Ho, L. J. Landau and A. J. Wilkins, On the weak coupling limit for a Fermi gas in a random potential, Rev. Math. Phys., 5 (1992), 209-298. [46] H. Hochstadt, "The Functions of Mathematical Physics," Dover Publications, New York, 1986. [47] T. Y. Hou, X. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 227 (1999), 913-943. [48] I. A. Ibragimov and Yu. V. Linnik, "Independent and Stationary Sequences of Random Variables," Wolters-Noordhoff Publishing, Groningen, 1971. [49] A. Ishimaru, "Wave Propagation and Scattering in Random Media," New York, Academics, 1978. [50] J. Jacod and A. N. Shiryaev, "Limit Theorems for Stochastic Processes," Grundlehren der mathematischen Wissenschaft 288, Springer-Verlag, 2003. [51] A. Jakubowski, A non-Skorohod topology on the Skorohod space, Electron. J. Probability, 2 (1997), 1-21. [52] H. Kesten and G. Papanicolaou, A limit theorem for turbulent diffusion, Comm. Math. Phys., 65 (1979), 97-128. [53] H. Kesten and G. C. Papanicolaou, A limit theorem for stochastic acceleration, Comm. Math. Phys., 78 (1980), 19-63. [54] R. Khasminskii, A limit theorem for solutions of differential equations with a random right hand side, Theory Probab. Appl., 11 (1966), 390-406. [55] T. Komorowski, Diffusion approximation for the advection of particles in a a strongly turbulent random environment, Ann. Probab., 24 (1996), 346-376. [56] T. Komorowski, Sz. Peszat and L. Ryzhik, Limit of fluctuations of solutions of Wigner equation, Comm. Math. Phys., 292 (2009), 479-510. [57] T. Komorowski and L. Ryzhik, Diffusion in a weakly random Hamiltonian flow, Comm. Math. Phys., 263 (2006), 277-323. [58] T. Komorowski and L. Ryzhik, The stochastic acceleration problem in two dimensions, Israel Jour.Math., 155 (2006), 157-204. [59] T. Komorowski and L. Ryzhik, Asymptotics of the phase of the solutions of the random Schrödinger equation, preprint, (2010). [60] P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618. [61] D. Liu, S. Vasudevan, J. Krolik, G. Bal and L. Carin, Electromagnetic time-reversal imaging in changing media: Experiment and analysis, IEEE Trans. Antennas and Prop., 55 (2007), 344-354. [62] J. Lukkarinen and H. Spohn, Kinetic limit for wave propagation in a random medium, Arch. Ration. Mech. Anal., 183 (2007), 93-162. [63] P. Markowich, P. Pietra and C. Pohl, Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit, Numer. Math., 81 (1999), 595-630. [64] P. Markowich, P. Pietra, C. Pohl and H. P. Stimming, A Wigner-measure analysis of the Dufort-Frankel scheme for the Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), 1281-1310. [65] I. Mitoma, On the sample continuity of $\mathcal S'$ processes, J. Math. Soc. Japan, 35 (1983), 629-636. [66] B. Øksendal, "Stochastic Differential Equations," Springer-Verlag, Berlin, 2000. [67] G. Papanicolaou and W. Kohler, Asymptotic theory of mixing stochastic ordinary differential equations, Comm. Pure Appl. Math., 27 (1974), 641-668. [68] G. Papanicolaou, L. Ryzhik and K. Sølna, The parabolic wave approximation and time reversal, Matematica Contemporanea, 23 (2002), 139-159. [69] G. C. Papanicolaou, L. Ryzhik and K. Sølna, Self-averaging from lateral diversity in the Ito-Schroedinger equation, Multiscale Model. Simul., 6 (2007), 468-492. [70] F. Poupaud and A. Vasseur, Classical and quantum transport in random media, J. Math. Pures Appl., 6 (2003), 711-748. [71] L. Ryzhik, G. Papanicolaou and J. B. Keller, Transport equations for elastic and other waves in random media, Wave Motion, 24 (1996), 327-370. [72] H. Sato and M. C. Fehler, "Seismic Wave Propagation and Scattering in the Heterogeneous Earth," AIP series in modern acoustics and signal processing, AIP Press, Springer, New York, 1998. [73] P. Sheng, "Introduction to Wave Scattering, Localization and Mesoscopic Phenomena," Academic Press, New York, 1995. [74] H. Spohn, Derivation of the transport equation for electrons moving through random impurities, Jour. Stat. Phys., 17 (1977), 385-412. [75] G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 507-517. [76] D. W. Stroock and S. R. S. Varadhan, "Multidimensional Diffusion Processes," Grundlehren der mathematischen Wissenschaften 233, Berlin, Heidelberg, New York, 1979, 1997. [77] C. R. Vogel, "Computational Methods for Inverse Problems," Frontiers Appl. Math., SIAM, Philadelphia, 2002. [78] E. Wigner, On the quantum correction for thermodynamic equilibrium, Physical Rev., 40 (1932), 749-759. [79] B. White, The stochastic caustic, SIMA Jour. Appl. Math., 44 (1984), 127-149.
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