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 & 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.  Google Scholar

[2]

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

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

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[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.   Google Scholar
[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

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

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[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.   Google Scholar
[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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