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Stable recovery of a non-compactly supported coefficient of a Schrödinger equation on an infinite waveguide
Convergence rates of Tikhonov regularization for recovering growth rates in a Lotka-Volterra competition model with diffusion
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China |
In this paper, we shall study the convergence rates of Tikhonov regularizations for the recovery of the growth rates in a Lotka-Volterra competition model with diffusion. The ill-posed inverse problem is transformed into a nonlinear minimization system by an appropriately selected version of Tikhonov regularization. The existence of the minimizers to the minimization system is demonstrated. We shall propose a new variational source condition, which will be rigorously verified under a Hölder type stability estimate. We will also derive the reasonable convergence rates under the new variational source condition.
References:
[1] |
H. Amann,
Compact embeddings of vector valued Sobolev and Besov spaces, Glasnik Matematički, 35 (2000), 161-177.
|
[2] |
S. W. Anzengruber, B. Hofmann and R. Ramlau, On the interplay of basis smoothness and specific range conditions occurring in sparsity regularization, Inverse Problems, 29 (2013), 125002, 21 pp.
doi: 10.1088/0266-5611/29/12/125002. |
[3] |
R. I. Bot and B. Hofmann,
An extension of the variational inequality approach for nonlinear ill-posed problems, Journal of Integral Equations and Applications, 22 (2010), 369-392.
doi: 10.1216/JIE-2010-22-3-369. |
[4] |
M. Burger, J. Flemming and B. Hofmann, Convergence rates in $\ell^1$-regularization if the sparsity assumption fails, Inverse Problems, 29 (2013), 025013, 16 pp.
doi: 10.1088/0266-5611/29/2/025013. |
[5] |
D.-H. Chen, B. Hofmann and J. Zou, Elastic-net regularization versus $\ell^1$-regularization for linear inverse problems with quasi-sparse solutions, Inverse Problem, 33 (2017), 015004, 17 pp.
doi: 10.1088/1361-6420/33/1/015004. |
[6] |
D.-H. Chen, D. Jiang and J. Zou, Convergence rates of Tikhonov regularizations for elliptic and parabolic inverse radiativity problems, Inverse Problem, 36 (2020), no. 7, 075001, 21 pp.
doi: 10.1088/1361-6420/ab8449. |
[7] |
D.-H. Chen and I. Yousept, Variational source condition for ill-posed backward nonlinear Maxwell's equations, Inverse Problem, 35 (2019), 025001, 25 pp.
doi: 10.1088/1361-6420/aaeebe. |
[8] |
I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, AKTA, Kharkiv, 1999. |
[9] |
E. C. M. Crooks, E. N. Dancer, D. Hilhorst, M. Mimura and H. Ninomiya,
Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions, Nonlinear Analysis: Real World Applications, 5 (2004), 645-665.
doi: 10.1016/j.nonrwa.2004.01.004. |
[10] |
E. N. Dancer and Y. H. Du,
Positive solutions for three-species competition system with diffusion I, General existence results, Nonlinear Anal., 24 (1995), 337-357.
doi: 10.1016/0362-546X(94)E0063-M. |
[11] |
E. N. Dancer and Y. H. Du,
Positive solutions for three-species competition system with diffusion II, The case of equal birth rates, Nonlinear Anal., 24 (1995), 359-373.
doi: 10.1016/0362-546X(94)E0064-N. |
[12] |
E. N. Dancer, K. Wang and Z. Zhang,
Dynamics of strongly competing systems with many species, Transactions of the American Mathematical Society, 364 (2012), 961-1005.
doi: 10.1090/S0002-9947-2011-05488-7. |
[13] |
E. N. Dancer and Z. Zhang,
Dynamics of Lotka-Volterra competition systems with large interaction, J. Differ. Equ., 182 (2002), 470-489.
doi: 10.1006/jdeq.2001.4102. |
[14] |
H. W. Engl, K. Kunisch and A. Neubauer,
Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Problems, 5 (1989), 523-540.
doi: 10.1088/0266-5611/5/4/007. |
[15] |
H. W. Engl and J. Zou,
A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction, Inverse Problems, 16 (2000), 1907-1923.
doi: 10.1088/0266-5611/16/6/319. |
[16] |
J. Flemming,
Theory and examples of variational regularization with non-metric fitting functionals, J. Inverse Ill-Posed Probl., 18 (2010), 677-699.
doi: 10.1515/JIIP.2010.031. |
[17] |
B. Grammaticos, J. Moulin-Ollagnier, A. Ramani, J.-M. Strelcyn and S. Wojciechowski,
Integrals of quadratic ordinary differential equations in ${{\mathbb R}}^3$: The Lotka-Volterra system, Phys. A, 163 (1990), 683-722.
doi: 10.1016/0378-4371(90)90152-I. |
[18] |
M. Grasmair, Generalized Bregman distances and convergence rates for non-convex regularization methods, Inverse Problems, 26 (2010), 115014, 16 pp.
doi: 10.1088/0266-5611/26/11/115014. |
[19] |
D. N. Hào and T. N. Quyen, Convergence rates for Tikhonov regularization of coefficient identification problems in Laplace-type equations, Inverse Problems, 26 (2010), 125014, 23 pp.
doi: 10.1088/0266-5611/26/12/125014. |
[20] |
J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, Cambridge, 1988.
![]() ![]() |
[21] |
B. Hofmann, B. Kaltenbacher, C. Pöschl and O. Scherzer,
A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.
doi: 10.1088/0266-5611/23/3/009. |
[22] |
B. Hofmann and P. Mathé, Parameter choice in Banach space regularization under variational inequalities, Inverse Problems, 28 (2012), 104006, 17 pp.
doi: 10.1088/0266-5611/28/10/104006. |
[23] |
T. Hohage and F. Weilding,
Characerizations of variational source conditions, converse results, and maxisets of spectral regularization methods, SIAM J. Numer. Anal., 55 (2017), 598-620.
doi: 10.1137/16M1067445. |
[24] |
T. Hohage and F. Weilding,
Variational source condition and stability estimates for inverse electromagnetic medium scattering problems, Inverse Problems Imaging, 11 (2017), 203-220.
doi: 10.3934/ipi.2017010. |
[25] |
T. Hohage and F. Weilding, Verification of a variational source condition for acoustic inverse medium scattering problems, Inverse Problem, 31 (2015), 075006 14 pp.
doi: 10.1088/0266-5611/31/7/075006. |
[26] |
G. Israel and A. M. Gasca, The Biology of Numbers, Science Networks. Historical Studies, 26. Birkhäuser, Basel, 2002.
doi: 10.1007/978-3-0348-8123-4. |
[27] |
D. Jiang, H. Feng and J. Zou, Convergence rates of Tikhonov regularizations for parameter identification in a parabolic-elliptic system, Inverse Problem, 28 (2012), 104002, 20pp.
doi: 10.1088/0266-5611/28/10/104002. |
[28] |
Y. Kan-on,
Bifurcation structure of positive stationary solutions for a Lotka-Volterra competition model with diffusion, II, Global structure, Discrete Contin. Dyn. Syst., 14 (2006), 135-148.
doi: 10.3934/dcds.2006.14.135. |
[29] |
Y. Kan-on,
Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model, Discrete Contin. Dyn. Syst., 8 (2002), 147-162.
doi: 10.3934/dcds.2002.8.147. |
[30] |
J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer Science & Business Media, 2012. |
[31] |
A. Lorz, J.-F. Pietschmann and M. Schlottbom, Parameter identification in a structured population model, Inverse Problems, 35 (2019), 095008.
doi: 10.1088/1361-6420/ab1af4. |
[32] |
A. E. Lotka, Elements of Mathematical Biology, Dover, New York, 1956. |
[33] |
K. Nakashima and T. Wakasa,
Generation of interfaces for Lotka-Volterra competition-diffusion system with large interaction rates, J. Differ. Equ., 235 (2007), 586-608.
doi: 10.1016/j.jde.2007.01.002. |
[34] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.
doi: 10.1007/978-94-010-0732-0.![]() ![]() ![]() |
[35] |
L. Roques and M. Cristofol, The inverse problem of determining several coefficients in a nonlinear Lotka-Volterra system, Inverse Problems, 28 (2012), 075007.
doi: 10.1088/0266-5611/28/7/075007. |
[36] |
V. Volterra, Lecions sur la Theorie Mathematique de la Lutte Pour la Vie, Gauthier Villars, Paris, 1931. |
[37] |
L. Weis,
Operator-valued Fourier multiplier theorems and maximal $ L_p $-regularity, Mathematische Annalen, 319 (2001), 735-758.
doi: 10.1007/PL00004457. |
[38] |
J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.
doi: 10.1017/CBO9781139171755.![]() ![]() ![]() |
[39] |
A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04631-5. |
[40] |
M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013.
doi: 10.1088/0266-5611/25/12/123013. |
[41] |
M. Yamamoto and J. Zou,
Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Problems, 17 (2001), 1181-1202.
doi: 10.1088/0266-5611/17/4/340. |
show all references
References:
[1] |
H. Amann,
Compact embeddings of vector valued Sobolev and Besov spaces, Glasnik Matematički, 35 (2000), 161-177.
|
[2] |
S. W. Anzengruber, B. Hofmann and R. Ramlau, On the interplay of basis smoothness and specific range conditions occurring in sparsity regularization, Inverse Problems, 29 (2013), 125002, 21 pp.
doi: 10.1088/0266-5611/29/12/125002. |
[3] |
R. I. Bot and B. Hofmann,
An extension of the variational inequality approach for nonlinear ill-posed problems, Journal of Integral Equations and Applications, 22 (2010), 369-392.
doi: 10.1216/JIE-2010-22-3-369. |
[4] |
M. Burger, J. Flemming and B. Hofmann, Convergence rates in $\ell^1$-regularization if the sparsity assumption fails, Inverse Problems, 29 (2013), 025013, 16 pp.
doi: 10.1088/0266-5611/29/2/025013. |
[5] |
D.-H. Chen, B. Hofmann and J. Zou, Elastic-net regularization versus $\ell^1$-regularization for linear inverse problems with quasi-sparse solutions, Inverse Problem, 33 (2017), 015004, 17 pp.
doi: 10.1088/1361-6420/33/1/015004. |
[6] |
D.-H. Chen, D. Jiang and J. Zou, Convergence rates of Tikhonov regularizations for elliptic and parabolic inverse radiativity problems, Inverse Problem, 36 (2020), no. 7, 075001, 21 pp.
doi: 10.1088/1361-6420/ab8449. |
[7] |
D.-H. Chen and I. Yousept, Variational source condition for ill-posed backward nonlinear Maxwell's equations, Inverse Problem, 35 (2019), 025001, 25 pp.
doi: 10.1088/1361-6420/aaeebe. |
[8] |
I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, AKTA, Kharkiv, 1999. |
[9] |
E. C. M. Crooks, E. N. Dancer, D. Hilhorst, M. Mimura and H. Ninomiya,
Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions, Nonlinear Analysis: Real World Applications, 5 (2004), 645-665.
doi: 10.1016/j.nonrwa.2004.01.004. |
[10] |
E. N. Dancer and Y. H. Du,
Positive solutions for three-species competition system with diffusion I, General existence results, Nonlinear Anal., 24 (1995), 337-357.
doi: 10.1016/0362-546X(94)E0063-M. |
[11] |
E. N. Dancer and Y. H. Du,
Positive solutions for three-species competition system with diffusion II, The case of equal birth rates, Nonlinear Anal., 24 (1995), 359-373.
doi: 10.1016/0362-546X(94)E0064-N. |
[12] |
E. N. Dancer, K. Wang and Z. Zhang,
Dynamics of strongly competing systems with many species, Transactions of the American Mathematical Society, 364 (2012), 961-1005.
doi: 10.1090/S0002-9947-2011-05488-7. |
[13] |
E. N. Dancer and Z. Zhang,
Dynamics of Lotka-Volterra competition systems with large interaction, J. Differ. Equ., 182 (2002), 470-489.
doi: 10.1006/jdeq.2001.4102. |
[14] |
H. W. Engl, K. Kunisch and A. Neubauer,
Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Problems, 5 (1989), 523-540.
doi: 10.1088/0266-5611/5/4/007. |
[15] |
H. W. Engl and J. Zou,
A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction, Inverse Problems, 16 (2000), 1907-1923.
doi: 10.1088/0266-5611/16/6/319. |
[16] |
J. Flemming,
Theory and examples of variational regularization with non-metric fitting functionals, J. Inverse Ill-Posed Probl., 18 (2010), 677-699.
doi: 10.1515/JIIP.2010.031. |
[17] |
B. Grammaticos, J. Moulin-Ollagnier, A. Ramani, J.-M. Strelcyn and S. Wojciechowski,
Integrals of quadratic ordinary differential equations in ${{\mathbb R}}^3$: The Lotka-Volterra system, Phys. A, 163 (1990), 683-722.
doi: 10.1016/0378-4371(90)90152-I. |
[18] |
M. Grasmair, Generalized Bregman distances and convergence rates for non-convex regularization methods, Inverse Problems, 26 (2010), 115014, 16 pp.
doi: 10.1088/0266-5611/26/11/115014. |
[19] |
D. N. Hào and T. N. Quyen, Convergence rates for Tikhonov regularization of coefficient identification problems in Laplace-type equations, Inverse Problems, 26 (2010), 125014, 23 pp.
doi: 10.1088/0266-5611/26/12/125014. |
[20] |
J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, Cambridge, 1988.
![]() ![]() |
[21] |
B. Hofmann, B. Kaltenbacher, C. Pöschl and O. Scherzer,
A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.
doi: 10.1088/0266-5611/23/3/009. |
[22] |
B. Hofmann and P. Mathé, Parameter choice in Banach space regularization under variational inequalities, Inverse Problems, 28 (2012), 104006, 17 pp.
doi: 10.1088/0266-5611/28/10/104006. |
[23] |
T. Hohage and F. Weilding,
Characerizations of variational source conditions, converse results, and maxisets of spectral regularization methods, SIAM J. Numer. Anal., 55 (2017), 598-620.
doi: 10.1137/16M1067445. |
[24] |
T. Hohage and F. Weilding,
Variational source condition and stability estimates for inverse electromagnetic medium scattering problems, Inverse Problems Imaging, 11 (2017), 203-220.
doi: 10.3934/ipi.2017010. |
[25] |
T. Hohage and F. Weilding, Verification of a variational source condition for acoustic inverse medium scattering problems, Inverse Problem, 31 (2015), 075006 14 pp.
doi: 10.1088/0266-5611/31/7/075006. |
[26] |
G. Israel and A. M. Gasca, The Biology of Numbers, Science Networks. Historical Studies, 26. Birkhäuser, Basel, 2002.
doi: 10.1007/978-3-0348-8123-4. |
[27] |
D. Jiang, H. Feng and J. Zou, Convergence rates of Tikhonov regularizations for parameter identification in a parabolic-elliptic system, Inverse Problem, 28 (2012), 104002, 20pp.
doi: 10.1088/0266-5611/28/10/104002. |
[28] |
Y. Kan-on,
Bifurcation structure of positive stationary solutions for a Lotka-Volterra competition model with diffusion, II, Global structure, Discrete Contin. Dyn. Syst., 14 (2006), 135-148.
doi: 10.3934/dcds.2006.14.135. |
[29] |
Y. Kan-on,
Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model, Discrete Contin. Dyn. Syst., 8 (2002), 147-162.
doi: 10.3934/dcds.2002.8.147. |
[30] |
J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer Science & Business Media, 2012. |
[31] |
A. Lorz, J.-F. Pietschmann and M. Schlottbom, Parameter identification in a structured population model, Inverse Problems, 35 (2019), 095008.
doi: 10.1088/1361-6420/ab1af4. |
[32] |
A. E. Lotka, Elements of Mathematical Biology, Dover, New York, 1956. |
[33] |
K. Nakashima and T. Wakasa,
Generation of interfaces for Lotka-Volterra competition-diffusion system with large interaction rates, J. Differ. Equ., 235 (2007), 586-608.
doi: 10.1016/j.jde.2007.01.002. |
[34] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.
doi: 10.1007/978-94-010-0732-0.![]() ![]() ![]() |
[35] |
L. Roques and M. Cristofol, The inverse problem of determining several coefficients in a nonlinear Lotka-Volterra system, Inverse Problems, 28 (2012), 075007.
doi: 10.1088/0266-5611/28/7/075007. |
[36] |
V. Volterra, Lecions sur la Theorie Mathematique de la Lutte Pour la Vie, Gauthier Villars, Paris, 1931. |
[37] |
L. Weis,
Operator-valued Fourier multiplier theorems and maximal $ L_p $-regularity, Mathematische Annalen, 319 (2001), 735-758.
doi: 10.1007/PL00004457. |
[38] |
J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.
doi: 10.1017/CBO9781139171755.![]() ![]() ![]() |
[39] |
A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04631-5. |
[40] |
M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013.
doi: 10.1088/0266-5611/25/12/123013. |
[41] |
M. Yamamoto and J. Zou,
Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Problems, 17 (2001), 1181-1202.
doi: 10.1088/0266-5611/17/4/340. |
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