547
Views
0
CrossRef citations to date
0
Altmetric
Research Article

Data assimilation in 2D hyperbolic/parabolic systems using a stabilized explicit finite difference scheme run backward in time

Article: 2282641 | Received 24 May 2023, Accepted 07 Nov 2023, Published online: 08 Dec 2023

References

  • He Q, Barajas-Solano D, Tartakovsky G, et al. Physics-informed neural networks for multiphysics data assimilation with application to subsurface transport. Adv Water Resour. 2020;141:103610. doi: 10.1016/j.advwatres.2020.103610
  • Arcucci R, Zhu J, Hu S, et al. Deep data assimilation: integrating deep learning with data assimilation. Appl Sci. 2021;11:1114. doi: 10.3390/app11031114
  • Antil H, Lohner R, Price R. Data assimilation with deep neural nets informed by nudging. preprint, Nov 2021. arXiv:abs/2111.11505.
  • Chen C, Dou Y, Chen J, et al. A novel neural network training framework with data assimilation. J Supercomput. 2022;78:19020–19045. doi: 10.1007/s11227-04629-7
  • Lundvall J, Kozlov V, Weinerfelt P. Iterative methods for data assimilation for Burgers' equation. J Inverse Ill-Posed Probl. 2006;14:505–535. doi: 10.1515/156939406778247589
  • Auroux D, Blum J. A nudging-based data assimilation method for oceanographuc problems: the back and forth nudging (BFN) algorithm. Proc Geophys. 2008;15:305–319. doi: 10.5194/npg-15-305-2008
  • Ou K, Jameson A. Unsteady adjoint method for the optimal control of advection and Burgers' equation using high order spectral difference method. In: 49th AIAA Aerospace Science Meeting; 2011 Jan 4–7; Orlando, Florida.
  • Auroux D, Nodet M. The back and forth nudging algorithm for data assimilation problems: theoretical results on transport equations. ESAIM:COCV. 2012;18:318–342.
  • Auroux D, Bansart P, Blum J. An evolution of the back and forth nudging for geophysical data assimilation: application to Burgers equation and comparison. Inverse Probl Sci Eng. 2013;21:399–419. doi: 10.1080/17415977.2012.712528
  • Allahverdi N, Pozo A, Zuazua E. Numerical aspects of large-time optimal control of Burgers' equation. Esaim Math Model Numer Anal. 2016;50:1371–1401. doi: 10.1051/m2an/2015076
  • Gosse L, Zuazua E. Filtered gradient algorithms for inverse design problems of one-dimensional Burgers equation. In: Gosse L, Natalini R, editors. Innovative algorithms and analysis. Cham: Springer; 2017. p. 197–227. doi: 10.1007/978-3-319-49262-9
  • de Campos Velho HF, Barbosa VCF, Cocke S. Special issue on inverse problems in geosciences. Inverse Probl Sci Eng. 2013;21:355–356. doi: 10.1080/17415977.2012.712532
  • Gomez-Hernandez JJ, Xu T. Contaminant source identification in acquifers: a critical view. Math Geosci. 2022;54:437–458. doi: 10.1007/s11004-021-09976-4
  • Xu T, Zhang W, Gomez-Hernandez JJ, et al. Non-point contaminant source identification in an acquifer using the ensemble smoother with multiple data assimilation. J Hydrol. 2022;606:127405. doi: 10.1016/j.jhydrol.2021.127405
  • Vukicevic T, Steyskal M, Hecht M. Properties of advection algorithms in the context of variational data assimilation. Mon Weather Rev. 2001;129:1221–1231. doi: 10.1175/1520-0493(2001)129¡1221:POAAIT¿2.0.CO;2
  • Carasso AS. Data assimilation in 2D viscous Burgers equation using a stabilized explicit finite difference scheme run backward in time. Inverse Probl Sci Eng. 2021;29:3475–3489. doi: 10.1080/17415977.2021.200947
  • Carasso AS. Data assimilation in 2D nonlinear advection diffusion equations, using an explicit stabilized leapfrog scheme run backwrd in time. NIST Technical Note 2227, 2022 Jul 12.doi: 10.6028/NIST.TN2227
  • Carasso AS. Compensating operators and stable backward in time marching in nonlinear parabolic equations. Int J Geomath. 2014;5:1–16. doi: 10.1007/s13137-014-0057-1
  • Carasso AS. Stable explicit time-marching in well-posed or ill-posed nonlinear parabolic equations. Inverse Probl Sci Eng. 2016;24:1364–1384. doi: 10.1080/17415977.2015.1110150
  • Carasso AS. Stable explicit marching scheme in ill-posed time-reversed viscous wave equations. Inverse Probl Sci Eng. 2016;24:1454–1474. doi: 10.1080/17415977.2015.1124429
  • Carasso AS. Stabilized Richardson leapfrog scheme in explicit stepwise computation of forward or backward nonlinear parabolic equations. Inverse Probl Sci Eng. 2017;25:1–24. doi: 10.1080/17415977.2017.1281270
  • Carasso AS. Stabilized backward in time explicit marching schemes in the numerical computation of ill-posed time-reversed hyperbolic/parabolic systems. Inverse Probl Sci Eng. 2018;1:1–32. doi: 10.1080/17415977.2018.1446952
  • Carasso AS. Stable explicit stepwise marching scheme in ill- posed time-reversed 2D Burgers' equation. Inverse Probl Sci Eng. 2018;27(12):1–17. doi: 10.1080/17415977.2018.1523905
  • Carasso AS. Computing ill-posed time-reversed 2D Navier-Stokes equations, using a stabilized explicit finite difference scheme marching backward in time. Inverse Probl Sci Eng. 2020; 28:988–1010. doi: 10.1080/17415977.2019.1698564
  • Carasso AS. Stabilized leapfrog scheme run backward in time, and the explicit O(Δt)2 stepwise computation of ill-posed time-reversed 2D Navier-Stokes equations. Inverse Probl Sci Eng. 2021; 29:3062–3085. doi: 10.1080/17415977.2021.1972997
  • Richtmyer RD, Morton KW. Difference methods for initial value problems. 2nd ed., New York (NY): Wiley; 1967.
  • Liu Z, Renardy M. A note on the equations of a thermoelastic plate. Appl Math Lett. 1995;8:1–6. doi: 10.1016/0893-9659(95)00020-Q
  • Liu K, Liu Z. Exponential stability and analyticity of abstract linear thermoelastic systems. Z Angew Math Phys. 1997;48:885–904. doi: 10.1007/s000330050071
  • Liu Z, Zheng S. Semigroups associated with dissipative systems. New York: Chapman and Hall/CRC; 1999.
  • Lasiecka I, Triggiani R. Analyticity, and lack thereof, of thermoelastic semigroups. ESAIM Proc. 1998;4:199–222. doi: 10.1051/proc:1998029
  • Morse PM, Feshbach H. Methods of theoretical physics. New York (NY): McGraw-Hill; 1953.
  • Arendt W, Ter Elst AFM. Gaussian estimates for second order elliptic operators with boundary conditions. J Oper Theory. 1997;38:87–130.
  • Aronson DG. Bounds for the fundamental solution of a parabolic equation. Bull Am Math Soc. 1967;73:890–896. doi: 10.1090/bull/1967-73-06
  • Ouhabaz EM. Gaussian estimates and holomorphy of semigroups. Proc Am Math Soc. 1995;123:1465–1474. doi: 10.1090/proc/1995-123-05
  • Ouhabaz EM. Gaussian upper bounds for heat kernels of second order elliptic operators with complex coefficients on arbitrary domains. J Oper Theory. 2004;51:335–360.