On the second-order asymptotical regularization of linear ill-posed inverse problems

References

• Tikhonov A, Leonov A, Yagola A. Nonlinear ill-posed problems. Vols. I and II. London: Chapman and Hall; 1998.
   Google Scholar
• Engl H, Hanke M, Neubauer A. Regularization of inverse problems. New York (NY): Springer; 1996.
   Google Scholar
• Kaltenbacher B, Neubauer A, Scherzer O. Iterative regularization methods for nonlinear ill-posed problems. Berlin: Walter de Gruyter GmbH & Co. KG; 2008.
   Google Scholar
• Tautenhahn U. On the asymptotical regularization of nonlinear ill-posed problems. Inverse Probl. 1994;10:1405–1418. doi: 10.1088/0266-5611/10/6/014
   Web of Science ®Google Scholar
• Vainikko G, Veretennikov A. Iteration procedures in ill-posed problems. Moscow: Nauka; 1986. (In Russian)
   Google Scholar
• Rieder A. Runge–Kutta integrators yield optimal regularization schemes. Inverse Probl. 2005;21:453–471. doi: 10.1088/0266-5611/21/2/003
   Web of Science ®Google Scholar
• Neubauer A. On Landweber iteration for nonlinear ill-posed problems in Hilbert scales. Numer Math. 2000;85:309–328. doi: 10.1007/s002110050487
   Web of Science ®Google Scholar
• Jin Q. On a regularized Levenberg–Marquardt method for solving nonlinear inverse problems. Numer Math. 2010;115:229–259. doi: 10.1007/s00211-009-0275-x
   Web of Science ®Google Scholar
• Jin Q. On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems. Numer Math. 2009;111:509–558. doi: 10.1007/s00211-008-0198-y
   Web of Science ®Google Scholar
• Neubauer A. On Nesterov acceleration for Landweber iteration of linear ill-posed problems. J Inverse Ill-Pose Probl. 2017;25:381–390.
   Web of Science ®Google Scholar
• Hubmer S, Ramlau R. Convergence analysis of a two-point gradient method for nonlinear ill-posed problems. Inverse Probl. 2017;33:095004. doi: 10.1088/1361-6420/aa7ac7
   Web of Science ®Google Scholar
• Alvarez F. On the minimizing property of a second-order dissipative system in Hilbert spaces. SIAM J Control Optim. 2000;38:1102–1119. doi: 10.1137/S0363012998335802
   Web of Science ®Google Scholar
• Alvarez F, Attouch H, Bolte J, et al. A second-order gradient-like dissipative dynamical system with hessian-driven damping. Application to optimization and mechanics. J Math Pures Appl. 2002;81:747–779. doi: 10.1016/S0021-7824(01)01253-3
   Web of Science ®Google Scholar
• Attouch H, Goudou X, Redont P. The heavy ball with friction method. I. The continuous dynamical system. Commun Contemp Math. 2000;2:1–34.
   Web of Science ®Google Scholar
• Zhang Y, Gong R, Cheng X, et al. A dynamical regularization algorithm for solving inverse source problems of elliptic partial differential equations. Inverse Probl. 2018;34:065001.
   Web of Science ®Google Scholar
• Edvardsson S, Neuman M, Edström P, et al. Solving equations through particle dynamics. Comput Phys Commun. 2015;197:169–181. doi: 10.1016/j.cpc.2015.08.028
   Web of Science ®Google Scholar
• Edvardsson S, Gulliksson M, Persson J. The dynamical functional particle method: an approach for boundary value problems. J Appl Mech. 2012;79:021012. doi: 10.1115/1.4005563
   Web of Science ®Google Scholar
• Sandin P, Gulliksson M. Numerical solution of the stationary multicomponent nonlinear schrodinger equation with a constraint on the angular momentum. Phys Rev E. 2016;93:033301. doi: 10.1103/PhysRevE.93.033301
   PubMed Web of Science ®Google Scholar
• Hofmann B, Mathé P. Analysis of profile functions for general linear regularization methods. SIAM J Numer Anal. 2007;45:1122–1141. doi: 10.1137/060654530
   Web of Science ®Google Scholar
• Mathé P, Pereverzev SV. Geometry of linear ill-posed problems in variable Hilbert scales. Inverse Probl. 2003;19(3):789–803. doi: 10.1088/0266-5611/19/3/319
   Web of Science ®Google Scholar
• Mathé P. Saturation of regularization methods for linear ill-posed problems in Hilbert spaces. SIAM J Numer Anal. 2006;42:968–973. doi: 10.1137/S0036142903420947
   Web of Science ®Google Scholar
• Boţ R, Csetnek E. Second order forward–backward dynamical systems for monotone inclusion problems. SIAM J Control Optim. 2016;54:1423–1443. doi: 10.1137/15M1012657
   Web of Science ®Google Scholar
• Schock E. Approximate solution of ill-posed equations: arbitrarily slow convergence vs. superconvergence. In: Hämmerlin G, Hoffmann KH, editors. Constructive methods for the practical treatment of integral equations. Vol. 73. Basel: Birkhäuser; 1985 p. 234–243.
   Google Scholar
• Hein T, Hofmann B. Approximate source conditions for nonlinear ill-posed problems – chances and limitations. Inverse Probl. 2009;25:035003.
   Web of Science ®Google Scholar
• Hofmann B, Kaltenbacher B, Poschl C, et al. A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse Probl. 2007;23:987–1010. doi: 10.1088/0266-5611/23/3/009
   Web of Science ®Google Scholar
• Hochbruck M, Lubich C, Selhofer H. Exponential integrators for large systems of differential equations. SIAM J Sci Comput. 1998;19:1152–1174.
   Web of Science ®Google Scholar
• Hairer E, Wanner G, Lubich C. Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. 2nd ed. New York (NY): Springer; 2006.
   Google Scholar
• Tadmor E. A review of numerical methods for nonlinear partial differential equations. Bull Am Math Soc. 2012;49(4):507–554. doi: 10.1090/S0273-0979-2012-01379-4
   Web of Science ®Google Scholar
• Lions J, Magenes E. Non-homogeneous boundary value problems and applications. Vol. I. Berlin: Springer; 1972.
   Google Scholar
• Hanke M. Conjugate gradient type methods for ill-posed problems. New York (NY): Wiley; 1995.
   Google Scholar
• Nesterov Y. A method of solving a convex programming problem with convergence rate. Sov Math Dokl. 1983;27:372–376.
   Google Scholar
• Su W, Boyd S, Candes E. A differential equation for modeling Nesterov's accelerated gradient method: theory and insights. J Mach Learn Res. 2016;17:1–43.
   Web of Science ®Google Scholar