
Previous Article
Existence of nontrivial solutions to systems of multipoint boundary value problems
 PROC Home
 This Issue

Next Article
Characterization of the spectral density function for a onesided tridiagonal Jacobi matrix operator
Regularization for illposed inhomogeneous evolution problems in a Hilbert space
1.  Division of Science and Engineering, Penn State Abington, 1600 Woodland Road, Abington, PA 19001, United States 
References:
[1] 
S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach space, Comm. Pure Appl. Math., 16 (1963), 121151. Google Scholar 
[2] 
K. A. Ames, "Comparison Results for Related Properly and Improperly Posed Problems, with Applications to Mechanics," Ph.D. Thesis, Cornell University, Ithaca, NY, 1980. Google Scholar 
[3] 
K. A. Ames and R. J. Hughes, Structural stability for illposed problems in Banach space, Semigroup Forum, 70 (2005), 127145. Google Scholar 
[4] 
B. Campbell Hetrick and R. J. Hughes, Continuous dependence results for inhomogeneous illposed problems in Banach space, J. Math. Anal. Appl., 331 (2007), 342357. Google Scholar 
[5] 
N. Dunford and J. Schwartz, "Linear Operators, Part II," John Wiley and Sons, Inc., New York, 1957. Google Scholar 
[6] 
M. Fury and R. J. Hughes, Continuous dependence of solutions for illposed evolution problems, Electron. J. Diff. Eqns., Conf. 19 (2010), 99121. Google Scholar 
[7] 
M. A. Fury and R. J. Hughes, Regularization for a class of illposed evolution problems in Banach space, Semigroup Forum, 85 (2012), 191212, (DOI) 10.1007/s0023301193533. Google Scholar 
[8] 
J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford Univ. Press, New York, 1985. Google Scholar 
[9] 
Y. Huang and Q. Zheng, Regularization for illposed Cauchy problems associated with generators of analytic semigroups, J. Differential Equations, 203 (2004), 3854. Google Scholar 
[10] 
Y. Huang and Q. Zheng, Regularization for a class of illposed Cauchy problems, Proc. Amer. Math. Soc., 13310 (2005), 30053012. Google Scholar 
[11] 
T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo, 25 (1970), 241258. Google Scholar 
[12] 
R. Lattes and J. L. Lions, "The Method of Quasireversibility, Applications to Partial Differential Equations," Amer. Elsevier, New York, 1969. Google Scholar 
[13] 
I. V. Mel'nikova, General theory of the illposed Cauchy problem, J. Inverse and Illposed Problems, 3 (1995), 149171. Google Scholar 
[14] 
I. V. Mel'nikova and A. I. Filinkov, "Abstract Cauchy Problems: Three Approaches," Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., vol. 120, Chapman & Hall, Boca Raton, FL, 2001. Google Scholar 
[15] 
K. Miller, Stabilized quasireversibility and other nearlybestpossible methods for nonwellposed problems, in "Symposium on NonWellPosed Problems and Logarithmic Convexity (HeriotWatt Univ., Edinburgh, 1972)," 161176, Springer Lecture Notes in Mathematics, Volume 316, Springer, Berlin, 1973. Google Scholar 
[16] 
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," SpringerVerlag, New York, 1983. Google Scholar 
[17] 
W. Rudin, "Real and Complex Analysis," $3^{rd}$ edition, McGrawHill Inc., New York, 1987. Google Scholar 
[18] 
R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl., 47 (1974), 563572. Google Scholar 
[19] 
D. D. Trong and N. H. Tuan, Regularization and error estimates for nonhomogeneous backward heat problems, Electron. J. Diff. Eqns., 2006 (2006) No. 4, 110. Google Scholar 
[20] 
D. D. Trong and N. H. Tuan, A nonhomogeneous backward heat problem: regularization and error estimates, Electron. J. Diff. Eqns., 2008 (2008) No. 33, 114. Google Scholar 
show all references
References:
[1] 
S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach space, Comm. Pure Appl. Math., 16 (1963), 121151. Google Scholar 
[2] 
K. A. Ames, "Comparison Results for Related Properly and Improperly Posed Problems, with Applications to Mechanics," Ph.D. Thesis, Cornell University, Ithaca, NY, 1980. Google Scholar 
[3] 
K. A. Ames and R. J. Hughes, Structural stability for illposed problems in Banach space, Semigroup Forum, 70 (2005), 127145. Google Scholar 
[4] 
B. Campbell Hetrick and R. J. Hughes, Continuous dependence results for inhomogeneous illposed problems in Banach space, J. Math. Anal. Appl., 331 (2007), 342357. Google Scholar 
[5] 
N. Dunford and J. Schwartz, "Linear Operators, Part II," John Wiley and Sons, Inc., New York, 1957. Google Scholar 
[6] 
M. Fury and R. J. Hughes, Continuous dependence of solutions for illposed evolution problems, Electron. J. Diff. Eqns., Conf. 19 (2010), 99121. Google Scholar 
[7] 
M. A. Fury and R. J. Hughes, Regularization for a class of illposed evolution problems in Banach space, Semigroup Forum, 85 (2012), 191212, (DOI) 10.1007/s0023301193533. Google Scholar 
[8] 
J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford Univ. Press, New York, 1985. Google Scholar 
[9] 
Y. Huang and Q. Zheng, Regularization for illposed Cauchy problems associated with generators of analytic semigroups, J. Differential Equations, 203 (2004), 3854. Google Scholar 
[10] 
Y. Huang and Q. Zheng, Regularization for a class of illposed Cauchy problems, Proc. Amer. Math. Soc., 13310 (2005), 30053012. Google Scholar 
[11] 
T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo, 25 (1970), 241258. Google Scholar 
[12] 
R. Lattes and J. L. Lions, "The Method of Quasireversibility, Applications to Partial Differential Equations," Amer. Elsevier, New York, 1969. Google Scholar 
[13] 
I. V. Mel'nikova, General theory of the illposed Cauchy problem, J. Inverse and Illposed Problems, 3 (1995), 149171. Google Scholar 
[14] 
I. V. Mel'nikova and A. I. Filinkov, "Abstract Cauchy Problems: Three Approaches," Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., vol. 120, Chapman & Hall, Boca Raton, FL, 2001. Google Scholar 
[15] 
K. Miller, Stabilized quasireversibility and other nearlybestpossible methods for nonwellposed problems, in "Symposium on NonWellPosed Problems and Logarithmic Convexity (HeriotWatt Univ., Edinburgh, 1972)," 161176, Springer Lecture Notes in Mathematics, Volume 316, Springer, Berlin, 1973. Google Scholar 
[16] 
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," SpringerVerlag, New York, 1983. Google Scholar 
[17] 
W. Rudin, "Real and Complex Analysis," $3^{rd}$ edition, McGrawHill Inc., New York, 1987. Google Scholar 
[18] 
R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl., 47 (1974), 563572. Google Scholar 
[19] 
D. D. Trong and N. H. Tuan, Regularization and error estimates for nonhomogeneous backward heat problems, Electron. J. Diff. Eqns., 2006 (2006) No. 4, 110. Google Scholar 
[20] 
D. D. Trong and N. H. Tuan, A nonhomogeneous backward heat problem: regularization and error estimates, Electron. J. Diff. Eqns., 2008 (2008) No. 33, 114. Google Scholar 
[1] 
Paola Favati, Grazia Lotti, Ornella Menchi, Francesco Romani. An innerouter regularizing method for illposed problems. Inverse Problems & Imaging, 2014, 8 (2) : 409420. doi: 10.3934/ipi.2014.8.409 
[2] 
Markus Haltmeier, Richard Kowar, Antonio Leitão, Otmar Scherzer. Kaczmarz methods for regularizing nonlinear illposed equations II: Applications. Inverse Problems & Imaging, 2007, 1 (3) : 507523. doi: 10.3934/ipi.2007.1.507 
[3] 
Zonghao Li, Caibin Zeng. Center manifolds for illposed stochastic evolution equations. Discrete & Continuous Dynamical Systems  B, 2021 doi: 10.3934/dcdsb.2021142 
[4] 
Markus Haltmeier, Antonio Leitão, Otmar Scherzer. Kaczmarz methods for regularizing nonlinear illposed equations I: convergence analysis. Inverse Problems & Imaging, 2007, 1 (2) : 289298. doi: 10.3934/ipi.2007.1.289 
[5] 
Eliane Bécache, Laurent Bourgeois, Lucas Franceschini, Jérémi Dardé. Application of mixed formulations of quasireversibility to solve illposed problems for heat and wave equations: The 1D case. Inverse Problems & Imaging, 2015, 9 (4) : 9711002. doi: 10.3934/ipi.2015.9.971 
[6] 
Stefan Kindermann. Convergence of the gradient method for illposed problems. Inverse Problems & Imaging, 2017, 11 (4) : 703720. doi: 10.3934/ipi.2017033 
[7] 
Sergiy Zhuk. Inverse problems for linear illposed differentialalgebraic equations with uncertain parameters. Conference Publications, 2011, 2011 (Special) : 14671476. doi: 10.3934/proc.2011.2011.1467 
[8] 
Matthew A. Fury. Estimates for solutions of nonautonomous semilinear illposed problems. Conference Publications, 2015, 2015 (special) : 479488. doi: 10.3934/proc.2015.0479 
[9] 
Misha Perepelitsa. An illposed problem for the NavierStokes equations for compressible flows. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 609623. doi: 10.3934/dcds.2010.26.609 
[10] 
Felix Lucka, Katharina Proksch, Christoph Brune, Nicolai Bissantz, Martin Burger, Holger Dette, Frank Wübbeling. Risk estimators for choosing regularization parameters in illposed problems  properties and limitations. Inverse Problems & Imaging, 2018, 12 (5) : 11211155. doi: 10.3934/ipi.2018047 
[11] 
Ye Zhang, Bernd Hofmann. Two new nonnegativity preserving iterative regularization methods for illposed inverse problems. Inverse Problems & Imaging, 2021, 15 (2) : 229256. doi: 10.3934/ipi.2020062 
[12] 
Olha P. Kupenko, Rosanna Manzo. On optimal controls in coefficients for illposed nonLinear elliptic Dirichlet boundary value problems. Discrete & Continuous Dynamical Systems  B, 2018, 23 (4) : 13631393. doi: 10.3934/dcdsb.2018155 
[13] 
Guozhi Dong, Bert Jüttler, Otmar Scherzer, Thomas Takacs. Convergence of Tikhonov regularization for solving illposed operator equations with solutions defined on surfaces. Inverse Problems & Imaging, 2017, 11 (2) : 221246. doi: 10.3934/ipi.2017011 
[14] 
Lianwang Deng. Local integral manifolds for nonautonomous and illposed equations with sectorially dichotomous operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 145174. doi: 10.3934/cpaa.2020009 
[15] 
Johann Baumeister, Barbara Kaltenbacher, Antonio Leitão. On LevenbergMarquardtKaczmarz iterative methods for solving systems of nonlinear illposed equations. Inverse Problems & Imaging, 2010, 4 (3) : 335350. doi: 10.3934/ipi.2010.4.335 
[16] 
Adriano De Cezaro, Johann Baumeister, Antonio Leitão. Modified iterated Tikhonov methods for solving systems of nonlinear illposed equations. Inverse Problems & Imaging, 2011, 5 (1) : 117. doi: 10.3934/ipi.2011.5.1 
[17] 
Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an illposed strongly nonlinear elliptic equation with $p$Laplace operator and $L^1$type of nonlinearity. Discrete & Continuous Dynamical Systems  B, 2019, 24 (3) : 12731295. doi: 10.3934/dcdsb.2019016 
[18] 
Youri V. Egorov, Evariste SanchezPalencia. Remarks on certain singular perturbations with illposed limit in shell theory and elasticity. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 12931305. doi: 10.3934/dcds.2011.31.1293 
[19] 
Alfredo Lorenzi, Luca Lorenzi. A strongly illposed integrodifferential singular parabolic problem in the unit cube of $\mathbb{R}^n$. Evolution Equations & Control Theory, 2014, 3 (3) : 499524. doi: 10.3934/eect.2014.3.499 
[20] 
Faker Ben Belgacem. Uniqueness for an illposed reactiondispersion model. Application to organic pollution in streamwaters. Inverse Problems & Imaging, 2012, 6 (2) : 163181. doi: 10.3934/ipi.2012.6.163 
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]