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October  2021, 15(5): 951-974. 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  October 2021 Early access  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 and Imaging, 2021, 15 (5) : 951-974. doi: 10.3934/ipi.2021023
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. CrooksE. N. DancerD. HilhorstM. 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. DancerK. 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. EnglK. 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. GrammaticosJ. Moulin-OllagnierA. RamaniJ.-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. HofmannB. KaltenbacherC. 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. CrooksE. N. DancerD. HilhorstM. 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. DancerK. 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. EnglK. 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. GrammaticosJ. Moulin-OllagnierA. RamaniJ.-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. HofmannB. KaltenbacherC. 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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