Uniqueness from pointwise observations in a multi-parameter inverse problem

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January 2012, 11(1): 173-188. doi: 10.3934/cpaa.2012.11.173

Uniqueness from pointwise observations in a multi-parameter inverse problem

Michel Cristofol ^1, , Jimmy Garnier ^2, , François Hamel ^3, and Lionel Roques ^4,

1.	Aix-Marseille Université, LATP, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20, France
2.	UR 546 Biostatistique et Processus Spatiaux, INRA, F-84000 Avignon, France, and Aix-Marseille Université, LATP, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20, France
3.	Aix-Marseille Université & Institut Universitaire de France, LATP, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20
4.	UR 546 Biostatistique et Processus Spatiaux, INRA, F-84000 Avignon, France

Received March 2010 Revised November 2010 Published September 2011

Abstract
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In this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coefficients of one-dimensional reaction-diffusion equations. Such reaction-diffusion equations include the classical model of Kolmogorov, Petrovsky and Piskunov as well as more sophisticated models from biology. When the reaction term contains an unknown polynomial part of degree $N,$ with non-constant coefficients $\mu_k(x),$ our result gives a sufficient condition for the uniqueness of the determination of this polynomial part. This sufficient condition only involves pointwise measurements of the solution $u$ of the reaction-diffusion equation and of its spatial derivative $\partial u / \partial x$ at a single point $x_0,$ during a time interval $(0,\varepsilon).$ In addition to this uniqueness result, we give several counter-examples to uniqueness, which emphasize the optimality of our assumptions. Finally, in the particular cases $N=2$ and $N=3,$ we show that such pointwise measurements can allow an efficient numerical determination of the unknown polynomial reaction term.

Keywords: multi-parameter, heterogeneous media, inverse problem, uniqueness., Reaction-diffusion.

Mathematics Subject Classification: 65M32, 35K20, 35K55, 35K5.

Citation: Michel Cristofol, Jimmy Garnier, François Hamel, Lionel Roques. Uniqueness from pointwise observations in a multi-parameter inverse problem. Communications on Pure and Applied Analysis, 2012, 11 (1) : 173-188. doi: 10.3934/cpaa.2012.11.173

References:

[1]	W. C. Allee, "The Social Life of Animals," Norton, New York, 1938.
[2]	M. Bellassoued and M. Yamamoto, Inverse source problem for a transmission problem for a parabolic equation, J. Inverse Ill-Posed Probl., 14 (2006), 47-56. doi: 10.1163/156939406776237456.
[3]	A. Benabdallah, M. Cristofol, P. Gaitan and M. Yamamoto, Inverse problem for a parabolic system with two components by measurements of one component, Appl. Anal., 88 (2009), 683-710. doi: 10.1080/00036810802555490.
[4]	H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: I - Species persistence, J. Math. Biol., 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3.
[5]	A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Soviet Math. Doklady, 24 (1981), 244-247.
[6]	R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," John Wiley & Sons Ltd, Chichester, UK, 2003. doi: 10.1002/0470871296.
[7]	M. Choulli, E. M. Ouhabaz and M. Yamamoto, Stable determination of a semilinear term in a parabolic equation, Commun. Pure Appl. Anal., 5 (2006), 447-462. doi: 10.3934/cpaa.2006.5.447.
[8]	M. Cristofol, P. Gaitan and H. Ramoul, Inverse problems for a $2\times 2$ reaction-diffusion system using a carleman estimate with one observation, Inverse Problems, 22 (2006), 1561-1573. doi: 10.1088/0266-5611/22/5/003.
[9]	M. Cristofol and L. Roques, Biological invasions: deriving the regions at risk from partial measurements, Math. Biosci., 215 (2008), 158-166. doi: 10.1016/j.mbs.2008.07.004.
[10]	B. Dennis, Allee effects: population growth, critical density, and the chance of extinction, Natur. Resource Modeling, 3 (1989), 481-538.
[11]	P. DuChateau and W. Rundell, Unicity in an inverse problem for an unknown reaction term in a reaction-diffusion equation, J. Differential Equations, 59 (1985), 155-164.
[12]	H. Egger, H. W. Engl and M. V. Klibanov, Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation, Inverse Problems, 21 (2005), 271-290. doi: 10.1088/0266-5611/21/1/017.
[13]	M. El Smaily, F. Hamel and L. Roques, Homogenization and influence of fragmentation in a biological invasion model, Discrete Contin. Dyn. Syst. - A, 25 (2009), 321-342. doi: 10.3934/dcds.2009.25.321.
[14]	R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 335-369.
[15]	A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Englewood Cliffs, NJ, 1964.
[16]	F. Hamel, J. Fayard and L. Roques, Spreading speeds in slowly oscillating environments, Bull. Math. Biol., 72 (2010), 1166-1191. doi: 10.1007/s11538-009-9486-7.
[17]	O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245. doi: 10.1088/0266-5611/14/5/009.
[18]	T. H. Keitt, M. A. Lewis and R. D. Holt, Allee effects, invasion pinning, and species' borders, American Naturalist, 157 (2001), 203-216. doi: 10.1086/318633.
[19]	M. V. Klibanov and A. Timonov, "Carleman Estimates for Coefficient Inverse Problems and Numerical Applications," Inverse And Ill-Posed Series, VSP, Utrecht, 2004.
[20]	A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. État Moscou, Sér. Internationale A, 1 (1937), 1-26.
[21]	M. A. Lewis and P. Kareiva, Allee dynamics and the speed of invading organisms, Theor. Population Biol., 43 (1993), 141-158. doi: 10.1006/tpbi.1993.1007.
[22]	A. Lorenzi, An inverse problem for a semilinear parabolic equation, Ann. Mat. Pura Appl., 131 (1982), 145-166. doi: 10.1007/BF01765150.
[23]	H. Matano, K.-I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit, Netw. Heterog. Media, 1 (2006), 537-568.
[24]	J. D. Murray, "Mathematical Biology," 3rd edition, Interdisciplinary Applied Mathematics 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868.
[25]	S.-I. Nakamura, A note on uniqueness in an inverse problem for a semilinear parabolic equation, Nihonkai Math. J., 12 (2001), 71-73.
[26]	C. V. Pao, "Nonlinear Parabolic and Elliptic Equations," Plenum Press, New York, 1992.
[27]	M. S. Pilant and W. Rundell,, An inverse problem for a nonlinear parabolic equation, Comm. Partial Differential Equations, 11 (1986), 445-457. doi: 10.1080/03605308608820430.
[28]	L. Roques and M. D. Chekroun, On population resilience to external perturbations, SIAM J. Appl. Math., 68 (2007), 133-153. doi: 10.1137/060676994.
[29]	L. Roques and M. Cristofol, On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation, Nonlinearity, 23 (2010), 675-686. doi: 10.1088/0951-7715/23/3/014.
[30]	L. Roques and F. Hamel, Mathematical analysis of the optimal habitat configurations for species persistence, Math. Biosci., 210 (2007), 34-59. doi: 10.1016/j.mbs.2007.05.007.
[31]	L. Roques, A. Roques, H. Berestycki and A. Kretzschmar, A population facing climate change: joint influences of Allee effects and environmental boundary geometry, Population Ecology, 50 (2008), 215-225. doi: 10.1007/s10144-007-0073-1.
[32]	L. Roques and R. S. Stoica, Species persistence decreases with habitat fragmentation: an analysis in periodic stochastic environments, J. Math. Biol., 55 (2007), 189-205. doi: 10.1007/s00285-007-0076-8.
[33]	N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice," Oxford Series in Ecology and Evolution, Oxford: Oxford University Press, 1997.
[34]	N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic-waves in heterogeneous environments, Theoret. Population Biol., 30 (1986), 143-160. doi: 10.1016/0040-5809(86)90029-8.
[35]	J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.
[36]	S. Soubeyrand, L. Held, M. Hohle and I. Sache, Modelling the spread in space and time of an airborne plant disease, J. Roy. Statist. Soc. Ser. C, 57 (2008), 253-272. doi: 10.1111/j.1467-9876.2007.00612.x.
[37]	S. Soubeyrand, S. Neuvonen and A. Penttinen, Mechanical-statistical modeling in ecology: from outbreak detections to pest dynamics, Bull. Math. Biol., 71 (2009), 318-338. doi: 10.1007/s11538-008-9363-9.
[38]	P. Turchin, "Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants," Sinauer Associates, Sunderland, MA, 1998.
[39]	A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Royal Soc. London - B, 237 (1952), 37-72.
[40]	R. R. Veit and M. A. Lewis, Dispersal, population growth, and the Allee effect: dynamics of the house finch invasion of eastern North America, American Naturalist, 148 (1996), 255-274. doi: 10.1086/285924.
[41]	C. K. Wikle, Hierarchical models in environmental science, Intern. Stat. Rev., 71 (2003), 181-199. doi: 10.1111/j.1751-5823.2003.tb00192.x.
[42]	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]	W. C. Allee, "The Social Life of Animals," Norton, New York, 1938.
[2]	M. Bellassoued and M. Yamamoto, Inverse source problem for a transmission problem for a parabolic equation, J. Inverse Ill-Posed Probl., 14 (2006), 47-56. doi: 10.1163/156939406776237456.
[3]	A. Benabdallah, M. Cristofol, P. Gaitan and M. Yamamoto, Inverse problem for a parabolic system with two components by measurements of one component, Appl. Anal., 88 (2009), 683-710. doi: 10.1080/00036810802555490.
[4]	H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: I - Species persistence, J. Math. Biol., 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3.
[5]	A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Soviet Math. Doklady, 24 (1981), 244-247.
[6]	R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," John Wiley & Sons Ltd, Chichester, UK, 2003. doi: 10.1002/0470871296.
[7]	M. Choulli, E. M. Ouhabaz and M. Yamamoto, Stable determination of a semilinear term in a parabolic equation, Commun. Pure Appl. Anal., 5 (2006), 447-462. doi: 10.3934/cpaa.2006.5.447.
[8]	M. Cristofol, P. Gaitan and H. Ramoul, Inverse problems for a $2\times 2$ reaction-diffusion system using a carleman estimate with one observation, Inverse Problems, 22 (2006), 1561-1573. doi: 10.1088/0266-5611/22/5/003.
[9]	M. Cristofol and L. Roques, Biological invasions: deriving the regions at risk from partial measurements, Math. Biosci., 215 (2008), 158-166. doi: 10.1016/j.mbs.2008.07.004.
[10]	B. Dennis, Allee effects: population growth, critical density, and the chance of extinction, Natur. Resource Modeling, 3 (1989), 481-538.
[11]	P. DuChateau and W. Rundell, Unicity in an inverse problem for an unknown reaction term in a reaction-diffusion equation, J. Differential Equations, 59 (1985), 155-164.
[12]	H. Egger, H. W. Engl and M. V. Klibanov, Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation, Inverse Problems, 21 (2005), 271-290. doi: 10.1088/0266-5611/21/1/017.
[13]	M. El Smaily, F. Hamel and L. Roques, Homogenization and influence of fragmentation in a biological invasion model, Discrete Contin. Dyn. Syst. - A, 25 (2009), 321-342. doi: 10.3934/dcds.2009.25.321.
[14]	R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 335-369.
[15]	A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Englewood Cliffs, NJ, 1964.
[16]	F. Hamel, J. Fayard and L. Roques, Spreading speeds in slowly oscillating environments, Bull. Math. Biol., 72 (2010), 1166-1191. doi: 10.1007/s11538-009-9486-7.
[17]	O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245. doi: 10.1088/0266-5611/14/5/009.
[18]	T. H. Keitt, M. A. Lewis and R. D. Holt, Allee effects, invasion pinning, and species' borders, American Naturalist, 157 (2001), 203-216. doi: 10.1086/318633.
[19]	M. V. Klibanov and A. Timonov, "Carleman Estimates for Coefficient Inverse Problems and Numerical Applications," Inverse And Ill-Posed Series, VSP, Utrecht, 2004.
[20]	A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. État Moscou, Sér. Internationale A, 1 (1937), 1-26.
[21]	M. A. Lewis and P. Kareiva, Allee dynamics and the speed of invading organisms, Theor. Population Biol., 43 (1993), 141-158. doi: 10.1006/tpbi.1993.1007.
[22]	A. Lorenzi, An inverse problem for a semilinear parabolic equation, Ann. Mat. Pura Appl., 131 (1982), 145-166. doi: 10.1007/BF01765150.
[23]	H. Matano, K.-I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit, Netw. Heterog. Media, 1 (2006), 537-568.
[24]	J. D. Murray, "Mathematical Biology," 3rd edition, Interdisciplinary Applied Mathematics 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868.
[25]	S.-I. Nakamura, A note on uniqueness in an inverse problem for a semilinear parabolic equation, Nihonkai Math. J., 12 (2001), 71-73.
[26]	C. V. Pao, "Nonlinear Parabolic and Elliptic Equations," Plenum Press, New York, 1992.
[27]	M. S. Pilant and W. Rundell,, An inverse problem for a nonlinear parabolic equation, Comm. Partial Differential Equations, 11 (1986), 445-457. doi: 10.1080/03605308608820430.
[28]	L. Roques and M. D. Chekroun, On population resilience to external perturbations, SIAM J. Appl. Math., 68 (2007), 133-153. doi: 10.1137/060676994.
[29]	L. Roques and M. Cristofol, On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation, Nonlinearity, 23 (2010), 675-686. doi: 10.1088/0951-7715/23/3/014.
[30]	L. Roques and F. Hamel, Mathematical analysis of the optimal habitat configurations for species persistence, Math. Biosci., 210 (2007), 34-59. doi: 10.1016/j.mbs.2007.05.007.
[31]	L. Roques, A. Roques, H. Berestycki and A. Kretzschmar, A population facing climate change: joint influences of Allee effects and environmental boundary geometry, Population Ecology, 50 (2008), 215-225. doi: 10.1007/s10144-007-0073-1.
[32]	L. Roques and R. S. Stoica, Species persistence decreases with habitat fragmentation: an analysis in periodic stochastic environments, J. Math. Biol., 55 (2007), 189-205. doi: 10.1007/s00285-007-0076-8.
[33]	N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice," Oxford Series in Ecology and Evolution, Oxford: Oxford University Press, 1997.
[34]	N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic-waves in heterogeneous environments, Theoret. Population Biol., 30 (1986), 143-160. doi: 10.1016/0040-5809(86)90029-8.
[35]	J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.
[36]	S. Soubeyrand, L. Held, M. Hohle and I. Sache, Modelling the spread in space and time of an airborne plant disease, J. Roy. Statist. Soc. Ser. C, 57 (2008), 253-272. doi: 10.1111/j.1467-9876.2007.00612.x.
[37]	S. Soubeyrand, S. Neuvonen and A. Penttinen, Mechanical-statistical modeling in ecology: from outbreak detections to pest dynamics, Bull. Math. Biol., 71 (2009), 318-338. doi: 10.1007/s11538-008-9363-9.
[38]	P. Turchin, "Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants," Sinauer Associates, Sunderland, MA, 1998.
[39]	A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Royal Soc. London - B, 237 (1952), 37-72.
[40]	R. R. Veit and M. A. Lewis, Dispersal, population growth, and the Allee effect: dynamics of the house finch invasion of eastern North America, American Naturalist, 148 (1996), 255-274. doi: 10.1086/285924.
[41]	C. K. Wikle, Hierarchical models in environmental science, Intern. Stat. Rev., 71 (2003), 181-199. doi: 10.1111/j.1751-5823.2003.tb00192.x.
[42]	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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