×

Local regularization methods for inverse Volterra equations applicable to the structure of solid surfaces. (English) Zbl 1321.65191

Author’s abstract: Deconvolution of appearance potential spectra is an old strategy commonly used to investigate electronic properties of solids in a surface region. Recently, this strategy was found to be effective in the study of nanostructures. In this context, the density of unoccupied states in the surface region of a solid is recovered from the measured AP-spectrum data from the governing equation \(k* x* x= g\), where \(k\) is a Lorentzian type function, \(g\) is a measured APS-signal and \(x\) is the density function to be recovered. As an important step in solving for \(x\), it’s necessary to solve the autoconvolution problem \(x* x= f\), which is a nonlinear ill-posed Volterra problem.
In this paper, we first improve upon the existing local regularization theory developed in [Z. Dai and P. K. Lamm, SIAM J. Numer. Anal. 46, No. 2, 832–868 (2008; Zbl 1205.65333)] for solving the autoconvolution problem, allowing for \(L_p\) data, where \(1\leq p\leq\infty\). It’s proven that the solutions of the regularized equation \(x^\delta_\alpha\in L_\infty(0, 1)\) (smoother than \(x^\delta_\alpha\in L_2(0, 1)\) as in [loc. cit.]) converge to the true solution \(\overline x\) of the autoconvolution equation in \(L_\infty\) norm (stronger than \(L_2\) norm as in [loc. cit.]) when the noise level in the data shrinks to \(0\). It is worth noting that the improved convergence theory is obtained while imposing less restrictions on the true solution \(\overline x\); namely \(\overline x\in C^1(0,1)\) in contrast to \(\overline x\in W^{2,\infty}(0,1)\).
Further, for the stable deconvolution of appearance potential spectra, we apply the local regularization methods to solve a combination of two ill-posed Volterra problems: the linear problem of determining \(f\) from \(f* k= g\) and then the nonlinear autoconvolution problem of determining \(x\) from \(x* x= f\). The results include a convergence theory and a fast sequential numerical method which essentially preserves the causal nature of the combined deconvolution problem. Numerical examples are included to show the effectiveness and efficiency of the methods.

MSC:

65R20 Numerical methods for integral equations
45D05 Volterra integral equations
65R32 Numerical methods for inverse problems for integral equations

Citations:

Zbl 1205.65333
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] J. Baumeister, Deconvolution of appearance potential spectra , in Direct and inverse boundary value problems , R. Kleinman, R. Kress, and E. Martensen, eds., Lang, Frankfurt am Main, Germany, 1991. · Zbl 0738.65095
[2] J.V. Beck, B. Blackwell and C.R. St. Clair, Jr., Inverse heat conduction , Wiley-Interscience, New York, 1985. · Zbl 0633.73120
[3] C.D. Brooks, A discrepancy principle for parameter selection in the local regularization of linear Volterra inverse problems , Ph.D. thesis, Department of Mathematics, Michigan State University, East Lansing, MI, 2007.
[4] C.D. Brooks and P.K. Lamm, A discrepancy principle for parameter selection in the local regularization of linear Volterra inverse problems , preprint, 2007.
[5] A.C. Cinzori, Continuous future polynomial regularization of \(1\)-smoothing Volterra problems , Inverse Prob. 20 (2004), 1791-1806. · Zbl 1077.45002 · doi:10.1088/0266-5611/20/6/006
[6] A.C. Cinzori and P.K. Lamm, Future polynomial regularization of ill-posed Volterra equations , SIAM J. Numer. Anal. 37 (2000), 949-979. · Zbl 0947.65138 · doi:10.1137/S0036142998347358
[7] C. Cui, Local regularization methods for \(n\)-Dimensional first-kind integral equations , Ph.D. thesis, Michigan State University, East Lansing, MI, 2005.
[8] Z. Dai, Local Regularization For The Autoconvolution Problem , Ph.D. thesis, Department of Mathematics, Michigan State University, East Lansing, MI, 2005.
[9] Z. Dai and P.K. Lamm, Local regularization for the nonlinear inverse autoconvolution problem , SIAM J. Numer. Anal. 46 (2008), 832-868. · Zbl 1205.65333 · doi:10.1137/070679247
[10] V. Dose and Th. Dose, Deconvolution of appearance potential spectra , Appl. Phys. 19 (1979), 19-23
[11] V. Dose and H. Scheidt, Deconvolution of appearance potential spectra II, Appl. Phys. 20 (1979), 299-303.
[12] H.W. Engl, M. Hanke and A. Neubauer, Regularization of inverse problems , Kluwer Academic Publishers, Dordrecht, Netherlands, 1996. · Zbl 0859.65054
[13] H.W. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems , Inverse Prob. 5 (1989), 524-540. · Zbl 0695.65037 · doi:10.1088/0266-5611/5/4/007
[14] G. Fleischer, R. Gorenflo and B. Hofmann, On the autoconvolution equation and total variation constraints , ZAMM Z. Angew. Math. Mech. 79 (1999), 149-159. · Zbl 0923.65098 · doi:10.1002/(SICI)1521-4001(199903)79:3<149::AID-ZAMM149>3.0.CO;2-N
[15] G. Fleischer and B. Hofmann, On inversion rates for the autoconvolution equation , Inverse Prob. 12 (1996), 419-435. · Zbl 0862.45007 · doi:10.1088/0266-5611/12/4/006
[16] —, The local degree of ill-posedness and the autoconvolution equation , Nonlinear Anal. 30 (1997), 3323-3332. · Zbl 0894.47048 · doi:10.1016/S0362-546X(96)00334-3
[17] Y. Fukuda, Appearance potential spectroscopy (APS): Old Method, but applicable to study of nano-structures , Anal. Sci. 26 (2010), 187-197.
[18] R. Gorenflo and B. Hofmann, On autoconvolution and regularization , Inverse Prob. 10 (1994), 353-373. · Zbl 0804.45003 · doi:10.1088/0266-5611/10/2/011
[19] G. Gripenberg, S.O. Londen and O. Saffens, Volterra integral and functional equations , Cambridge University Press, Cambridge, 1990.
[20] H. Hagstrum, Ion-neutralization spectroscopy of solids and solid surfaces , Phys. Rev. 150 (1966), 495-515.
[21] T. Hein and B. Hofmann, Approximate source conditions for nonlinear ill-posed problems-chances and limitations , Inverse Prob. 25 (2009), 035003. · Zbl 1233.47054
[22] J. Janno, On a regularization method for the autoconvolution equation , Z. Angew. Math. Mech. 77 (1997), 393-394. · Zbl 0879.65094 · doi:10.1002/zamm.19970770521
[23] —, Lavrent’ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation , Inverse Prob. 16 (2000), 333-348. · Zbl 1014.65038 · doi:10.1088/0266-5611/16/2/305
[24] P.K. Lamm, Approximation of ill-posed Volterra problems via predictor-corrector regularization methods , SIAM J. Appl. Math. 56 (1996), 524-541. · Zbl 0852.65127 · doi:10.1137/S0036139994274496
[25] —, Future-sequential regularization methods for ill-posed Volterra equations : Applications to the inverse heat conduction problem , J. Math. Anal. Appl. 195 (1995), 469-494. · Zbl 0851.65094 · doi:10.1006/jmaa.1995.1368
[26] —, Regularized inversion of finitely smoothing Volterra operators : Predictor-corrector regularization methods , Inverse Prob. 13 (1997), 375-402. · Zbl 0878.65120 · doi:10.1088/0266-5611/13/2/012
[27] —, Variable-smoothing regularization methods for inverse problems , in Theory and practice of control and systems , A. Conte and A.M. Perdon, eds., World Scientific, Singapore, 1999.
[28] —, A survey of regularization methods for first-kind Volterra equations , in Surveys on solution methods for inverse problems , D. Colton, H.W. Engl, A. Louis, J.R. McLaughlin and W. Rundell, eds., Springer, Vienna, 2000. · doi:10.1007/978-3-7091-6296-5_4
[29] —, Variable-smoothing local regularization methods for first-kind integral equations , Inverse Prob. 19 (2003), 195-216. · Zbl 1037.65135 · doi:10.1088/0266-5611/19/1/311
[30] —, Full convergence of sequential local regularization methods for Volterra inverse problems , Inverse Prob. 21 (2005), 785-803. · Zbl 1085.65123 · doi:10.1088/0266-5611/21/3/001
[31] P.K. Lamm and Z. Dai, On local regularization methods for linear Volterra problems and nonlinear equations of Hammerstein type , Inverse Prob. 21 (2005), 1773-1790. · Zbl 1081.65126 · doi:10.1088/0266-5611/21/5/016
[32] P.K. Lamm and L. Eldén, Numerical solution of first-kind Volterra equations by sequential Tikhonov regularization , SIAM J. Numer. Anal. 34 (1997), 1432-1450. · Zbl 0891.65135 · doi:10.1137/S003614299528081X
[33] P.K. Lamm and X. Luo, Local regularization methods for nonlinear Hammerstein equations , · Zbl 1220.65184
[34] P.K. Lamm and T. Scofield, Sequential predictor-corrector methods for the variable regularization of Volterra inverse problems , Inverse Prob. 16 (2000), 373-399. · Zbl 0972.65121 · doi:10.1088/0266-5611/16/2/308
[35] —, Local regularization methods for the stabilization for ill-posed Volterra problems , Numer. Funct. Anal. Optim. 23 (2001), 913-940. · Zbl 1003.65148
[36] R. Miller, Nonlinear Volterra integral equations , W.A. Benjamin, New York, 1971. · Zbl 0448.45004
[37] R.L. Park, Recent developments in appearance potential spectroscopy , Surface Sci. 48 (1975), 80-98.
[38] R.L. Park and J.E. Houston, The electronic structure of solid surfaces : Core level excitation techniques , J. Vac. Sci. Tech. 11 (1974), 176-182.
[39] —, Soft x-ray appearance potential spectroscopy , J. Vac. Sci. Tech. 10 (1973), 1-18.
[40] R. Ramlau, Morozov’s discrepancy principle for Tikhonov regularization of nonlinear operators , Numer. Funct. Anal. Optim. 23 (2002), 147-172. · Zbl 1002.65064 · doi:10.1081/NFA-120003676
[41] W. Ring and J. Prix, Sequential predictor-corrector regularization methods and their limitations , Inverse Prob. 16 (2000), 619-634. · Zbl 0968.65112 · doi:10.1088/0266-5611/16/3/306
[42] O. Scherzer, The use of Morozov’s discrepancy principle for Tikhkonov regularization for solving nonlinear ill-posed problems , Computing 51 (1993), 45-60. · Zbl 0801.65053 · doi:10.1007/BF02243828
[43] S.W. Schultz, K.Th. Schleicher, D.M. Ruck and H.U. Chun, Derivation of the density of unoccupied states in polycrystalline ferromagnetic Fe, Co , and Ni from highly resolved appearance potential spectra, J. Vac. Sci. Tech.: Vacuum, Surfaces, and Films (1984), 822-825.
[44] S. Schwabik, Generalized ordinary differential equations , World Scientific Publishing Co., New Jersey, 1992. · Zbl 0781.34003
[45] D. Willett, A linear generalization of Gronwall’s inequality , Proc. Amer. Math. Soc. 16 (1965), 774-778. \noindentstyle · Zbl 0128.27604 · doi:10.2307/2033920
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.