American Institute of Mathematical Sciences

doi: 10.3934/ipi.2021023

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

* Corresponding author: Daijun Jiang

Received  May 2020 Revised  November 2020 Published  March 2021

Fund Project: The first author is supported by National Natural Science Foundation of China (Nos. 11701205 and 11871240). The second author is supported by National Natural Science Foundation of China (No. 11871240) and self-determined research funds of CCNU from the colleges' basic research and operation of MOE (No. CCNU20TS003)

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.

Citation: De-han Chen, Daijun jiang. Convergence rates of Tikhonov regularization for recovering growth rates in a Lotka-Volterra competition model with diffusion. Inverse Problems & Imaging, doi: 10.3934/ipi.2021023
References:
 [1] H. Amann, Compact embeddings of vector valued Sobolev and Besov spaces, Glasnik Matematički, 35 (2000), 161-177.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, AKTA, Kharkiv, 1999.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, Cambridge, 1988.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer Science & Business Media, 2012. Google Scholar [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.  Google Scholar [32] A. E. Lotka, Elements of Mathematical Biology, Dover, New York, 1956. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] V. Volterra, Lecions sur la Theorie Mathematique de la Lutte Pour la Vie, Gauthier Villars, Paris, 1931.  Google Scholar [37] L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity, Mathematische Annalen, 319 (2001), 735-758.  doi: 10.1007/PL00004457.  Google Scholar [38] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.  Google Scholar [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.  Google Scholar [40] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013. doi: 10.1088/0266-5611/25/12/123013.  Google Scholar [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.  Google Scholar

show all references

References:
 [1] H. Amann, Compact embeddings of vector valued Sobolev and Besov spaces, Glasnik Matematički, 35 (2000), 161-177.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, AKTA, Kharkiv, 1999.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, Cambridge, 1988.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer Science & Business Media, 2012. Google Scholar [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.  Google Scholar [32] A. E. Lotka, Elements of Mathematical Biology, Dover, New York, 1956. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] V. Volterra, Lecions sur la Theorie Mathematique de la Lutte Pour la Vie, Gauthier Villars, Paris, 1931.  Google Scholar [37] L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity, Mathematische Annalen, 319 (2001), 735-758.  doi: 10.1007/PL00004457.  Google Scholar [38] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.  Google Scholar [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.  Google Scholar [40] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013. doi: 10.1088/0266-5611/25/12/123013.  Google Scholar [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.  Google Scholar
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