October  2020, 14(5): 939-966. doi: 10.3934/ipi.2020043

On the simultaneous recovery of the conductivity and the nonlinear reaction term in a parabolic equation

1. 

Department of Mathematics, Alpen-Adria-Universität Klagenfurt, 9020 Klagenfurt, Austria

2. 

Department of Mathematics, Texas A & M University, College Station, Texas 77843, USA

* Corresponding author: William Rundell

Received  March 2020 Revised  May 2020 Published  July 2020

Fund Project: The first author is supported by FWF grant P30054; the second author is supported by NFS grant DMS-1620138

This paper considers the inverse problem of recovering both the unknown, spatially-dependent conductivity $ a(x) $ and the nonlinear reaction term $ f(u) $ in a reaction-diffusion equation from overposed data. These measurements can consist of: the value of two different solution measurements taken at a later time $ T $; time-trace profiles from two solutions; or both final time and time-trace measurements from a single forwards solve data run. We prove both uniqueness results and the convergence of iteration schemes designed to recover these coefficients. The last section of the paper shows numerical reconstructions based on these algorithms.

Citation: Barbara Kaltenbacher, William Rundell. On the simultaneous recovery of the conductivity and the nonlinear reaction term in a parabolic equation. Inverse Problems and Imaging, 2020, 14 (5) : 939-966. doi: 10.3934/ipi.2020043
References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Advances in Mathematics, 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.

[2]

R. A. Arrhenius, über die dissociationswärme und den einflußder temperatur auf den dissociationsgrad der elektrolyte, Z. Phys. Chem., 4 (1889), 96-116. 

[3] G. A. Baker and P. Graves-Morris, Padé Approximants, Cambridge University Press, Cambridge, second edition, 1996.  doi: 10.1017/CBO9780511530074.
[4]

G. Borg, Eine umkehrung der Sturm-Liouville eigenwertaufgabe, Acta Mathematica, 76 (1946), 1-96.  doi: 10.1007/BF02421600.

[5]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, The Journal of Chemical Physics, 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.

[6]

K. Chadan, D. Colton, L. Päivärinta and W. Rundell, An Introduction to Inverse Scattering and Inverse Spectral Problems, SIAM Monographs on Mathematical Modeling and Computation, SIAM, 1997. doi: 10.1137/1.9780898719710.

[7]

E. L. Cussler, Diffusion: Mass Transfer in Fluid Systems, Cambridge University Press., 1997.

[8]

H. EggerH. 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.

[9]

C. M. Elliott and Z. Songmu, On the cahn-hilliard equation, Arch. Rational Mech. Anal, 96 (1986), 339-357.  doi: 10.1007/BF00251803.

[10]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, 1964.

[11]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.

[12]

I. M. Gel'fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl., 1 (1951), 253-291.  doi: 10.1090/trans2/001/11.

[13] P. Grindrod, The Theory and Applications of Reaction-Diffusion Equations: Patterns and Waves, 2$^nd$ edition, Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, Oxford University Press, New York, 1996. 
[14]

V. Isakov, Inverse parabolic problems with the final overdetermination, Comm. Pure Appl. Math., 44 (1991), 185-209.  doi: 10.1002/cpa.3160440203.

[15]

V. Isakov, Inverse problems for partial differential equations, 2$^nd$ edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.

[16]

B. Kaltenbacher and W. Rundell, On an inverse potential problem for a fractional reaction-diffusion equation, Inverse Problems, 35 (2019), 065004. doi: 10.1088/1361-6420/ab109e.

[17]

B. Kaltenbacher and W. Rundell, On the identification of a nonlinear term in a reaction-diffusion equation, Inverse Problems, 35 (2019), 115007. doi: 10.1088/1361-6420/ab2aab.

[18]

B. Kaltenbacher and W. Rundell, Recovery of multiple coefficients in a reaction-diffusion equation, J. Math. Anal. Appl., 481 (2019), 123475. doi: 10.1016/j.jmaa.2019.123475.

[19]

B. Kaltenbacher and W. Rundell, The inverse problem of reconstructing reaction-diffusion systems, Inverse Problems, 36 (2020).

[20]

C. Kuttler, Reaction-diffusion equations and their application on bacterial communication, in Disease Modelling and Public Health, Handbook of Statistics, Elsevier Science, 2017.

[21]

H. A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32 (1990), 262-288.  doi: 10.1137/1032046.

[22]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Modern Birkhäuser Classics, Springer Basel, 1995.

[23]

E. A. Mason, Gas, State of Matter, Encyclopedia Britannica, 2020.

[24]

J. D. Murray, Mathematical Biology I, 3$^rd$ edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002.

[25]

A. Pierce, Unique identification of eigenvalues and coefficients in a parabolic problem, SIAM J. Control Optim., 17 (1979), 494-499.  doi: 10.1137/0317035.

[26]

M. Pilant and W. Rundell, Iteration schemes for unknown coefficient problems in parabolic equations, Numer. Methods for P.D.E., 3 (1987), 313–325, 1987. doi: 10.1002/num.1690030404.

[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]

J. Pöschel and E. Trubowitz, Inverse spectral theory, in Pure and Applied Mathematics, 130, Academic Press, Inc., Boston, MA, 1987.

[29]

W. Rundell and P. E. Sacks, The reconstruction of Sturm-Liouville operators, Inverse Problems, 8 (1992), 457-482.  doi: 10.1088/0266-5611/8/3/007.

[30]

W. Rundell and P. E. Sacks, Reconstruction techniques for classical inverse Sturm-Liouville problems, Math. Comp., 58 (1992), 161-183.  doi: 10.1090/S0025-5718-1992-1106979-0.

[31]

A. M. Savchuk and A. A. Shkalikov, On the eigenvalues of the Sturm-Liouville operator with potentials in Sobolev spaces, Mat. Zametki, 80 (2006), 864-884.  doi: 10.1007/s11006-006-0204-6.

[32]

Z. Zhang and Z. Zhou, Recovering the potential term in a fractional diffusion equation, IMA J. Appl. Math., 82 (2017), 579-600.  doi: 10.1093/imamat/hxx004.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Advances in Mathematics, 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.

[2]

R. A. Arrhenius, über die dissociationswärme und den einflußder temperatur auf den dissociationsgrad der elektrolyte, Z. Phys. Chem., 4 (1889), 96-116. 

[3] G. A. Baker and P. Graves-Morris, Padé Approximants, Cambridge University Press, Cambridge, second edition, 1996.  doi: 10.1017/CBO9780511530074.
[4]

G. Borg, Eine umkehrung der Sturm-Liouville eigenwertaufgabe, Acta Mathematica, 76 (1946), 1-96.  doi: 10.1007/BF02421600.

[5]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, The Journal of Chemical Physics, 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.

[6]

K. Chadan, D. Colton, L. Päivärinta and W. Rundell, An Introduction to Inverse Scattering and Inverse Spectral Problems, SIAM Monographs on Mathematical Modeling and Computation, SIAM, 1997. doi: 10.1137/1.9780898719710.

[7]

E. L. Cussler, Diffusion: Mass Transfer in Fluid Systems, Cambridge University Press., 1997.

[8]

H. EggerH. 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.

[9]

C. M. Elliott and Z. Songmu, On the cahn-hilliard equation, Arch. Rational Mech. Anal, 96 (1986), 339-357.  doi: 10.1007/BF00251803.

[10]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, 1964.

[11]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.

[12]

I. M. Gel'fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl., 1 (1951), 253-291.  doi: 10.1090/trans2/001/11.

[13] P. Grindrod, The Theory and Applications of Reaction-Diffusion Equations: Patterns and Waves, 2$^nd$ edition, Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, Oxford University Press, New York, 1996. 
[14]

V. Isakov, Inverse parabolic problems with the final overdetermination, Comm. Pure Appl. Math., 44 (1991), 185-209.  doi: 10.1002/cpa.3160440203.

[15]

V. Isakov, Inverse problems for partial differential equations, 2$^nd$ edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.

[16]

B. Kaltenbacher and W. Rundell, On an inverse potential problem for a fractional reaction-diffusion equation, Inverse Problems, 35 (2019), 065004. doi: 10.1088/1361-6420/ab109e.

[17]

B. Kaltenbacher and W. Rundell, On the identification of a nonlinear term in a reaction-diffusion equation, Inverse Problems, 35 (2019), 115007. doi: 10.1088/1361-6420/ab2aab.

[18]

B. Kaltenbacher and W. Rundell, Recovery of multiple coefficients in a reaction-diffusion equation, J. Math. Anal. Appl., 481 (2019), 123475. doi: 10.1016/j.jmaa.2019.123475.

[19]

B. Kaltenbacher and W. Rundell, The inverse problem of reconstructing reaction-diffusion systems, Inverse Problems, 36 (2020).

[20]

C. Kuttler, Reaction-diffusion equations and their application on bacterial communication, in Disease Modelling and Public Health, Handbook of Statistics, Elsevier Science, 2017.

[21]

H. A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32 (1990), 262-288.  doi: 10.1137/1032046.

[22]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Modern Birkhäuser Classics, Springer Basel, 1995.

[23]

E. A. Mason, Gas, State of Matter, Encyclopedia Britannica, 2020.

[24]

J. D. Murray, Mathematical Biology I, 3$^rd$ edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002.

[25]

A. Pierce, Unique identification of eigenvalues and coefficients in a parabolic problem, SIAM J. Control Optim., 17 (1979), 494-499.  doi: 10.1137/0317035.

[26]

M. Pilant and W. Rundell, Iteration schemes for unknown coefficient problems in parabolic equations, Numer. Methods for P.D.E., 3 (1987), 313–325, 1987. doi: 10.1002/num.1690030404.

[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]

J. Pöschel and E. Trubowitz, Inverse spectral theory, in Pure and Applied Mathematics, 130, Academic Press, Inc., Boston, MA, 1987.

[29]

W. Rundell and P. E. Sacks, The reconstruction of Sturm-Liouville operators, Inverse Problems, 8 (1992), 457-482.  doi: 10.1088/0266-5611/8/3/007.

[30]

W. Rundell and P. E. Sacks, Reconstruction techniques for classical inverse Sturm-Liouville problems, Math. Comp., 58 (1992), 161-183.  doi: 10.1090/S0025-5718-1992-1106979-0.

[31]

A. M. Savchuk and A. A. Shkalikov, On the eigenvalues of the Sturm-Liouville operator with potentials in Sobolev spaces, Mat. Zametki, 80 (2006), 864-884.  doi: 10.1007/s11006-006-0204-6.

[32]

Z. Zhang and Z. Zhou, Recovering the potential term in a fractional diffusion equation, IMA J. Appl. Math., 82 (2017), 579-600.  doi: 10.1093/imamat/hxx004.

Figure 1.  Logarithms of the singular values for $ a(x) $ and $ q(x) $ recovery from time trace data using $ F:\delta c \to u(1,t) $ with $ \{\delta c = \sin(n\pi x)\} $
Figure 2.  Recovery of $ a(x) $ and $ f(u) $ from final data
Figure 3.  Recovery of $ \,a(x), $ and $ \,f(u)\, $ from space-and-time data
Figure 4.  Recovery of a(x); and f(u) from space-and-time data: 1% noise
Figure 5.  Recovery of $ \,a(x)\, $ and $ \,f(u)\, $ from space-and-time data when $ u_0\not = 0 $
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