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Articles

Determination of the initial condition in parabolic equations from integral observations

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Pages 1138-1167 | Received 26 Jun 2016, Accepted 19 Aug 2016, Published online: 16 Sep 2016

References

  • Agoshkov VI. Optimal control methods and the method of adjoint equations in problems of mathematical physics. Moscow: Russian Academy of Sciences, Institute for Numerical Mathematics; 2003. 257pp. Russian.
  • Shutyaev VP. Control operators and iterative algorithms in variational data assimilation problems. Moscow: Nauka; 2001. 240pp. Russian.
  • Aubert G, Kornprobst P. Mathematical problems in image processing. New York (NY): Springer; 2006.
  • Hào DN. Methods for inverse heat conduction problems. Frankfurt/Main, Bern, New York (NY), Paris: Peter Lang Verlag; 1998.
  • Boussetila N, Rebbani F. Optimal regularization method for ill-posed Cauchy problems. Electron. J. Differ. Equ. 2006;147:1–15.
  • Hào DN, Duc NV. Stability results for backward parabolic equations with time dependent coefficients. Inverse Prob. 2011;27:025003, 20pp.
  • Li J, Yamamoto M, Zou J. Conditional stability and numerical reconstruction of initial temperature. Commun. Pure Appl. Anal. 2009;8:361–382.
  • Tröltzsch F. Optimal control of partial differential equations: theory, methods and applications. Providence (RI): American Mathematical Society; 2010.
  • Ladyzhenskaya OA, Solonnikov VA, Ural’ceva NN. Linear and quasilinear equations of parabolic type. Providence (RI): American Mathematical Society; 1968.
  • Wloka J. Partial differential equations. Cambridge: Cambridge University Press; 1987.
  • Klibanov MV. Carleman estimates for the regularization of ill-posed Cauchy problems. Appl. Numer. Math. 2015;94:46–74.
  • Klibanov MV, Kuzhuget AV, Golubnichiy KV. An ill-posed problem for the Black-Scholes equation for a profitable forecast of prices of stock options on real market data. Inverse Prob. 2016;32:015010, 16pp.
  • Oanh NTN. A splitting method for a backward parabolic equation with time-dependent coefficients. Comput. Math. Appl. 2013;65:17–28.
  • Trefethen LN, Bau D III. Numerical linear algebra. Philadelphia: SIAM; 1997.
  • Hào DN, Oanh NTN. Determination of the initial condition in parabolic equations from boundary observations. J. Inverse Ill-Posed Prob. 2016;24:195–220.
  • Nocedal J, Wright SJ. Numerical optimization. 2nd ed. New York (NY): Springer; 2006.
  • Hinze M. A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optimiz. Appl. 2005;30:45–61.
  • Marchuk GI. Methods of numerical mathematics. New York (NY): Springer-Verlag; 1975.
  • Marchuk GI. Splitting and alternating direction methods. In Ciarlet PG, Lions JL, editors. Handbook of numerical mathematics. Volume 1: finite difference methods. North-Holland, Amsterdam: Elsevier Science Publisher B.V.; 1990.
  • Yanenko NN. The method of fractional steps. Berlin: Springer-Verlag; 1971.
  • Hào DN, Thành NT, Sahli H. Splitting-based gradient method for multi-dimensional inverse conduction problems. J. Comput. Appl. Math. 2009;232:361–377.
  • Oanh NTN, Huong BV. Determination of a time-dependent term in the right hand side of linear parabolic equations. Acta Math. Vietnam. 2016;41:313–335.
  • Thành NT. Infrared thermography for the detection and characterization of buried objects [PhD thesis]. Brussels, Belgium: Vrije Universiteit Brussel; 2007.
  • Ladyzhenskaya OA. The boundary value problems of mathematical physics. New York (NY): Springer-Verlag; 1985.
  • Hào DN, Thanh PX, Lesnic D, Johansson BT. A boundary element method for a multi-dimensional inverse heat conduction problem. Int. J. Comput. Math. 2012;89:1540–1554.
  • Hào DN. A noncharacteristic Cauchy problem for linear parabolic equations II: a variational method. Numer. Funct. Anal. Optim. 1992;13:541–564.
  • Hào DN. A noncharacteristic Cauchy problem for linear parabolic equations III: a variational method and its approximation schemes. Numer. Funct. Anal. Optim. 1992;13:565–583.
  • Nemirovskii AS. The regularizing properties of the adjoint gradient method in ill-posed problems. Zh. vychisl. Mat. Mat. Fiz. 1986;26:332–347; Engl. Transl. in U.S.S.R. Comput. Maths. Math. Phys. 1986;26:7–16.

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