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2013, 7(4): 1157-1182. doi: 10.3934/ipi.2013.7.1157

Identification of nonlinearities in transport-diffusion models of crowded motion

1. 

Department for Computational and Applied Mathematics, University of Münster, Einsteinstr. 62, 48149 Münster, Germany, Germany

2. 

DAMTP, Center for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

Received  March 2012 Revised  May 2013 Published  November 2013

The aim of this paper is to formulate a class of inverse problems of particular relevance in crowded motion, namely the simultaneous identification of entropies and mobilities. We study a model case of this class, which is the identification from flux-based measurements in a stationary setup. This leads to an inverse problem for a nonlinear transport-diffusion model, where boundary values and possibly an external potential can be varied. In specific settings we provide a detailed theory for the forward map and an adjoint problem useful in the analysis and numerical solution. We further verify the simultaneous identifiability of the nonlinearities and present several numerical tests yielding further insight into the way variations in boundary values and external potential affect the quality of reconstructions.
Citation: Martin Burger, Jan-Frederik Pietschmann, Marie-Therese Wolfram. Identification of nonlinearities in transport-diffusion models of crowded motion. Inverse Problems & Imaging, 2013, 7 (4) : 1157-1182. doi: 10.3934/ipi.2013.7.1157
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A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow,, SIAM Journal on Applied Mathematics, 60 (2000), 916. doi: 10.1137/S0036139997332099.

[2]

H. Berry and H. Cható, Anomalous subdiffusion due to obstacles : A critical survey, preprint,, 2011., ().

[3]

M. Bodnar and J. Velazquez, An integro-differential equation arising as a limit of individual cell-based models,, Journal of Differential Equations, 222 (2006), 341. doi: 10.1016/j.jde.2005.07.025.

[4]

S. Boi, V. Capasso and D. Morale, Modeling the aggregative behavior of ants of the species polyergus rufescens,, Nonlinear Analysis: Real World Applications, 1 (2000), 163. doi: 10.1016/S0362-546X(99)00399-5.

[5]

L. Boltzmann, Vorlesungen Über Gastheorie,, 2 vols. 1896, (1896).

[6]

A. Bruhn, J. Weickert and C. Schnörr, Combining the advantages of local and global optic flow methods,, in Proceedings of the 24th DAGM Symposium on Pattern Recognition, (2002), 454. doi: 10.1007/3-540-45783-6_55.

[7]

M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions,, Nonlinear Analysis: Real World Applications, 8 (2007), 939. doi: 10.1016/j.nonrwa.2006.04.002.

[8]

M. Burger, M. Di Francesco and Y. Dolak-Struss, The Keller-Segel model for chemotaxis with prevention of overcrowding: linear vs. nonlinear diffusion,, SIAM J. Math. Anal., 38 (2006), 1288. doi: 10.1137/050637923.

[9]

M. Burger, P. A. Markowich and J.-F. Pietschmann, Continuous limit of a crowd motion and herding model: Analysis and numerical simulations,, Kinetic and Related Models, 4 (2011), 1025. doi: 10.3934/krm.2011.4.1025.

[10]

M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries,, Nonlinearity, 25 (2012), 961. doi: 10.1088/0951-7715/25/4/961.

[11]

C. Cercignani, The Boltzmann Equation and Its Applications,, Springer-Verlag, (1988). doi: 10.1007/978-1-4612-1039-9.

[12]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Springer, (1994).

[13]

T. J. Connolly and D. J. N. Wall, On Frechet differentiability of some nonlinear operators occurring in inverse problems: An implicit function theorem approach,, Inverse Problems, 6 (1990), 949. doi: 10.1088/0266-5611/6/6/006.

[14]

O. Debeir, P. V. Ham, R. Kiss and C. Decaestecker, Tracking of migrating cells under phase-contrast video microscopy with combined mean-shift processes,, IEEE Trans. Med. Imaging, (2005), 697. doi: 10.1109/TMI.2005.846851.

[15]

M. Di Francesco and J. Rosado, Fully parabolic Keller-Segel model for chemotaxis with prevention of overcrowding,, Nonlinearity, 21 (2008), 2715. doi: 10.1088/0951-7715/21/11/012.

[16]

P. Duchateau, Monotonicity and invertibility of coefficient-to-data mappings for parabolic inverse problems,, SIAM Journal on Mathematical Analysis, 26 (1995), 1473. doi: 10.1137/S0036141093259257.

[17]

S. Dümmel and M. Pfaffe, Identifikation eines Koeffizienten in der eindimensionalen Wärmeleitungsgleichung,, Wiss. Z. Tech. Univ. Chemnitz, 34 (1992), 45.

[18]

H. Egger and J. Schöberl, A hybrid mixed discontinuous Galerkin finite-element method for convection diffusion problems,, IMA Journal of Numerical Analysis, 30 (2010), 1206. doi: 10.1093/imanum/drn083.

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A. Eriksson, M. Nilsson Jacobi, J. Nystrm and K. Tunstrm, Determining interaction rules in animal swarms,, Behavioral Ecology, 21 (2010), 1106. doi: 10.1093/beheco/arq118.

[21]

L. C. Evans, Partial Differential Equations,, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, (2010).

[22]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,, Springer-Verlag, (1977).

[23]

D. Gillespie, W. Nonner and R. S. Eisenberg, Density functional theory of charged, hard-sphere fluids,, Phys. Rev. E., 68 (2003). doi: 10.1103/PhysRevE.68.031503.

[24]

D. Hall and M. Hoshino, Effects of macromolecular crowding on intracellular diffusion from a single particle perspective,, Biophysical Reviews, 2 (2010), 39. doi: 10.1007/s12551-010-0029-0.

[25]

S. Handrock-Meyer, Identifiability of distributed parameters for a class of quasilinear differential equations,, Journal of Inverse and Ill-posed Problems, 5 (1997). doi: 10.1515/jiip.1997.5.1.19.

[26]

A. Hasanov, Identification of unknown diffusion and convection coefficients in ion transport problems from flux data: An analytical approach,, J. Math. Chem., 48 (2010), 413. doi: 10.1007/s10910-010-9683-5.

[27]

A. Hasanov and A. Erdem, Determination of unknown coefficient in a non-linear elliptic problem related to the elastoplastic torsion of a bar,, IMA J. Appl. Math., 73 (2008), 579. doi: 10.1093/imamat/hxm056.

[28]

T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding,, Advances in Applied Mathematics, 26 (2001), 280. doi: 10.1006/aama.2001.0721.

[29]

S. P. Hoogendoorn, W. Daamen and P. H. L. Bovy, Extracting microscopic pedestrian characteristics from video data,, in TRB 2004 Annual Meeting. CD-Rom, (2004).

[30]

B. K. P. Horn and B. G. Schunck, Determining optical flow: A Retrospective,, Artif. Intell., 59 (1993), 81. doi: 10.1016/0004-3702(93)90173-9.

[31]

R. L. Hughes, A continuum theory for the flow of pedestrians,, Transportation Research Part B: Methodological, 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7.

[32]

T. L. Jackson and H. M. Byrne, A mechanical model of tumor encapsulation and transcapsular spread,, Mathematical Biosciences, 180 (2002), 307. doi: 10.1016/S0025-5564(02)00118-9.

[33]

F. James and M. Postel, Numerical gradient methods for flux identification in a system of conservation laws,, Journal of Engineering Mathematics, 60 (2008), 293. doi: 10.1007/s10665-007-9165-3.

[34]

F. James and M. Sepúlveda, Convergence results for the flux identification in a scalar conservation law,, SIAM Journal on Control and Optimization, 37 (1999), 869. doi: 10.1137/S0363012996272722.

[35]

A. Jüngel and I. V. Stelzer, Entropy structure of a cross-diffusion tumor-growth model,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512500091.

[36]

S. Kaczmarz, Approximate solution of systems of linear equations,, Internat. J. Control, 57 (1993), 1269. doi: 10.1080/00207179308934446.

[37]

T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics. Springer-Verlag, (1995).

[38]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5.

[39]

K. Keren, P. T. Yam, A. Kinkhabwala, A. Mogilner and J. A. Theriot, Intracellular fluid flow in rapidly moving cells,, Nature Cell Biology, 11 (2009), 1219. doi: 10.1038/ncb1965.

[40]

J. Kerridge, S. Keller, T. Chamberlain and N. Sumpter, Collecting pedestrian trajectory data in real-time,, in Pedestrian and Evacuation Dynamics 2005 (editors, (2005), 27. doi: 10.1007/978-3-540-47064-9_3.

[41]

P. Knabner and B. Igler, Structural identification of nonlinear coefficient functions in transport processes through porous media,, in Lectures on Applied Mathematics (Munich, (1999), 157.

[42]

R. Kowar and O. Scherzer, Convergence analysis of a Landweber-Kaczmarz method for solving nonlinear ill-posed problems,, in Ill-Posed and Inverse Problems, (2002), 253.

[43]

P. Kügler, Identification of a temperature dependent heat conductivity from single boundary measurements,, SIAM J. Numer. Anal., 41 (2003), 1543. doi: 10.1137/S0036142902415900.

[44]

B. D. Lucas and T. Kanade, An Iterative Image Registration Technique with an Application to Stereo Vision,, in IJCAI81, (1981), 674.

[45]

R. Lukeman, Y.-X. Li and L. Edelstein-Keshet, Inferring individual rules from collective behavior,, Proceedings of the National Academy of Sciences, 107 (2010), 12576. doi: 10.1073/pnas.1001763107.

[46]

M. Moeller, M. Burger, P. Dieterich and A. Schwab, A Framework for Automated Cell Tracking in Phase Contrast Microscopic Videos Based on Normal Velocities,, Technical Report, (2010).

[47]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm,, Journal of Mathematical Biology, 38 (1999), 534. doi: 10.1007/s002850050158.

[48]

D. Morale, V. Capasso and K. Oelschlger, An interacting particle system modelling aggregation behavior: from individuals to populations,, Journal of Mathematical Biology, 50 (2005), 49. doi: 10.1007/s00285-004-0279-1.

[49]

S. Olla and S. R. S. Varadhan, Scaling limit for interacting Ornstein-Uhlenbeck processes,, Comm. Math. Phys., 135 (1991), 355. doi: 10.1007/BF02098047.

[50]

S. Olla, S. R. S. Varadhan and H.-T. Yau, Hydrodynamical limit for a Hamiltonian system with weak noise,, Comm. Math. Phys., 155 (1993), 523. doi: 10.1007/BF02096727.

[51]

Y. H. Ou, A. Hasanov and Z. H. Liu, Inverse coefficient problems for nonlinear parabolic differential equations,, Acta. Math. Sin. (Engl. Ser.), 24 (2008), 1617. doi: 10.1007/s10114-008-6384-0.

[52]

K. Painter and T. Hillen, Volume-filling and quorum sensing in models for chemosensitive movement,, Canadian Applied Mathematics Quaterly, 10 (2003), 280.

[53]

N. Papenberg, A. Bruhn, T. Brox, S. Didas and J. Weickert, Highly accurate optic flow computation with theoretically justified warping,, International Journal of Computer Vision, 67 (2006), 141. doi: 10.1007/s11263-005-3960-y.

[54]

I. Sbalzarini and P. Koumoutsakos, Feature point tracking and trajectory analysis for video imaging in cell biology,, Journal of Structural Biology, 151 (2005), 182. doi: 10.1016/j.jsb.2005.06.002.

[55]

J. Schauder, Der Fixpunktsatz in Funktionalräumen,, Studia Math., 2 (1930), 171.

[56]

M. J. Simpson, B. D. Hughes and K. A. Landman, Diffusion populations: Ghosts or folks,, Australasian Journal of Engineering Education, 15 (2009), 59.

[57]

M. J. Simpson, K. A. Landman and B. D. Hughes, Multi-species simple exclusion process,, Physica A, 388 (2009), 399. doi: 10.1016/j.physa.2008.10.038.

[58]

C. Topaz, A. Bertozzi and M. Lewis, A nonlocal continuum model for biological aggregation,, Bulletin of Mathematical Biology, 68 (2006), 1601. doi: 10.1007/s11538-006-9088-6.

[59]

U. Weidmann, Transporttechnik der Fussgänger - Transporttechnische Eigenschaften des Fussgängerverkehrs (Literaturstudie),, Literature Research 90, (1993).

[60]

C. Zimmer, B. Zhang, A. Dufour, A. Thebaud, S. Berlemont, V. Meas-Yedid and J.-C. Marin, On the digital trail of mobile cells,, Signal Processing Magazine, 23 (2006), 54. doi: 10.1109/MSP.2006.1628878.

show all references

References:
[1]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow,, SIAM Journal on Applied Mathematics, 60 (2000), 916. doi: 10.1137/S0036139997332099.

[2]

H. Berry and H. Cható, Anomalous subdiffusion due to obstacles : A critical survey, preprint,, 2011., ().

[3]

M. Bodnar and J. Velazquez, An integro-differential equation arising as a limit of individual cell-based models,, Journal of Differential Equations, 222 (2006), 341. doi: 10.1016/j.jde.2005.07.025.

[4]

S. Boi, V. Capasso and D. Morale, Modeling the aggregative behavior of ants of the species polyergus rufescens,, Nonlinear Analysis: Real World Applications, 1 (2000), 163. doi: 10.1016/S0362-546X(99)00399-5.

[5]

L. Boltzmann, Vorlesungen Über Gastheorie,, 2 vols. 1896, (1896).

[6]

A. Bruhn, J. Weickert and C. Schnörr, Combining the advantages of local and global optic flow methods,, in Proceedings of the 24th DAGM Symposium on Pattern Recognition, (2002), 454. doi: 10.1007/3-540-45783-6_55.

[7]

M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions,, Nonlinear Analysis: Real World Applications, 8 (2007), 939. doi: 10.1016/j.nonrwa.2006.04.002.

[8]

M. Burger, M. Di Francesco and Y. Dolak-Struss, The Keller-Segel model for chemotaxis with prevention of overcrowding: linear vs. nonlinear diffusion,, SIAM J. Math. Anal., 38 (2006), 1288. doi: 10.1137/050637923.

[9]

M. Burger, P. A. Markowich and J.-F. Pietschmann, Continuous limit of a crowd motion and herding model: Analysis and numerical simulations,, Kinetic and Related Models, 4 (2011), 1025. doi: 10.3934/krm.2011.4.1025.

[10]

M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries,, Nonlinearity, 25 (2012), 961. doi: 10.1088/0951-7715/25/4/961.

[11]

C. Cercignani, The Boltzmann Equation and Its Applications,, Springer-Verlag, (1988). doi: 10.1007/978-1-4612-1039-9.

[12]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Springer, (1994).

[13]

T. J. Connolly and D. J. N. Wall, On Frechet differentiability of some nonlinear operators occurring in inverse problems: An implicit function theorem approach,, Inverse Problems, 6 (1990), 949. doi: 10.1088/0266-5611/6/6/006.

[14]

O. Debeir, P. V. Ham, R. Kiss and C. Decaestecker, Tracking of migrating cells under phase-contrast video microscopy with combined mean-shift processes,, IEEE Trans. Med. Imaging, (2005), 697. doi: 10.1109/TMI.2005.846851.

[15]

M. Di Francesco and J. Rosado, Fully parabolic Keller-Segel model for chemotaxis with prevention of overcrowding,, Nonlinearity, 21 (2008), 2715. doi: 10.1088/0951-7715/21/11/012.

[16]

P. Duchateau, Monotonicity and invertibility of coefficient-to-data mappings for parabolic inverse problems,, SIAM Journal on Mathematical Analysis, 26 (1995), 1473. doi: 10.1137/S0036141093259257.

[17]

S. Dümmel and M. Pfaffe, Identifikation eines Koeffizienten in der eindimensionalen Wärmeleitungsgleichung,, Wiss. Z. Tech. Univ. Chemnitz, 34 (1992), 45.

[18]

H. Egger and J. Schöberl, A hybrid mixed discontinuous Galerkin finite-element method for convection diffusion problems,, IMA Journal of Numerical Analysis, 30 (2010), 1206. doi: 10.1093/imanum/drn083.

[19]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, volume 375 of Mathematics and its Applications. Kluwer Academic Publishers Group, (1996). doi: 10.1007/978-94-009-1740-8.

[20]

A. Eriksson, M. Nilsson Jacobi, J. Nystrm and K. Tunstrm, Determining interaction rules in animal swarms,, Behavioral Ecology, 21 (2010), 1106. doi: 10.1093/beheco/arq118.

[21]

L. C. Evans, Partial Differential Equations,, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, (2010).

[22]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,, Springer-Verlag, (1977).

[23]

D. Gillespie, W. Nonner and R. S. Eisenberg, Density functional theory of charged, hard-sphere fluids,, Phys. Rev. E., 68 (2003). doi: 10.1103/PhysRevE.68.031503.

[24]

D. Hall and M. Hoshino, Effects of macromolecular crowding on intracellular diffusion from a single particle perspective,, Biophysical Reviews, 2 (2010), 39. doi: 10.1007/s12551-010-0029-0.

[25]

S. Handrock-Meyer, Identifiability of distributed parameters for a class of quasilinear differential equations,, Journal of Inverse and Ill-posed Problems, 5 (1997). doi: 10.1515/jiip.1997.5.1.19.

[26]

A. Hasanov, Identification of unknown diffusion and convection coefficients in ion transport problems from flux data: An analytical approach,, J. Math. Chem., 48 (2010), 413. doi: 10.1007/s10910-010-9683-5.

[27]

A. Hasanov and A. Erdem, Determination of unknown coefficient in a non-linear elliptic problem related to the elastoplastic torsion of a bar,, IMA J. Appl. Math., 73 (2008), 579. doi: 10.1093/imamat/hxm056.

[28]

T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding,, Advances in Applied Mathematics, 26 (2001), 280. doi: 10.1006/aama.2001.0721.

[29]

S. P. Hoogendoorn, W. Daamen and P. H. L. Bovy, Extracting microscopic pedestrian characteristics from video data,, in TRB 2004 Annual Meeting. CD-Rom, (2004).

[30]

B. K. P. Horn and B. G. Schunck, Determining optical flow: A Retrospective,, Artif. Intell., 59 (1993), 81. doi: 10.1016/0004-3702(93)90173-9.

[31]

R. L. Hughes, A continuum theory for the flow of pedestrians,, Transportation Research Part B: Methodological, 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7.

[32]

T. L. Jackson and H. M. Byrne, A mechanical model of tumor encapsulation and transcapsular spread,, Mathematical Biosciences, 180 (2002), 307. doi: 10.1016/S0025-5564(02)00118-9.

[33]

F. James and M. Postel, Numerical gradient methods for flux identification in a system of conservation laws,, Journal of Engineering Mathematics, 60 (2008), 293. doi: 10.1007/s10665-007-9165-3.

[34]

F. James and M. Sepúlveda, Convergence results for the flux identification in a scalar conservation law,, SIAM Journal on Control and Optimization, 37 (1999), 869. doi: 10.1137/S0363012996272722.

[35]

A. Jüngel and I. V. Stelzer, Entropy structure of a cross-diffusion tumor-growth model,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512500091.

[36]

S. Kaczmarz, Approximate solution of systems of linear equations,, Internat. J. Control, 57 (1993), 1269. doi: 10.1080/00207179308934446.

[37]

T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics. Springer-Verlag, (1995).

[38]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5.

[39]

K. Keren, P. T. Yam, A. Kinkhabwala, A. Mogilner and J. A. Theriot, Intracellular fluid flow in rapidly moving cells,, Nature Cell Biology, 11 (2009), 1219. doi: 10.1038/ncb1965.

[40]

J. Kerridge, S. Keller, T. Chamberlain and N. Sumpter, Collecting pedestrian trajectory data in real-time,, in Pedestrian and Evacuation Dynamics 2005 (editors, (2005), 27. doi: 10.1007/978-3-540-47064-9_3.

[41]

P. Knabner and B. Igler, Structural identification of nonlinear coefficient functions in transport processes through porous media,, in Lectures on Applied Mathematics (Munich, (1999), 157.

[42]

R. Kowar and O. Scherzer, Convergence analysis of a Landweber-Kaczmarz method for solving nonlinear ill-posed problems,, in Ill-Posed and Inverse Problems, (2002), 253.

[43]

P. Kügler, Identification of a temperature dependent heat conductivity from single boundary measurements,, SIAM J. Numer. Anal., 41 (2003), 1543. doi: 10.1137/S0036142902415900.

[44]

B. D. Lucas and T. Kanade, An Iterative Image Registration Technique with an Application to Stereo Vision,, in IJCAI81, (1981), 674.

[45]

R. Lukeman, Y.-X. Li and L. Edelstein-Keshet, Inferring individual rules from collective behavior,, Proceedings of the National Academy of Sciences, 107 (2010), 12576. doi: 10.1073/pnas.1001763107.

[46]

M. Moeller, M. Burger, P. Dieterich and A. Schwab, A Framework for Automated Cell Tracking in Phase Contrast Microscopic Videos Based on Normal Velocities,, Technical Report, (2010).

[47]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm,, Journal of Mathematical Biology, 38 (1999), 534. doi: 10.1007/s002850050158.

[48]

D. Morale, V. Capasso and K. Oelschlger, An interacting particle system modelling aggregation behavior: from individuals to populations,, Journal of Mathematical Biology, 50 (2005), 49. doi: 10.1007/s00285-004-0279-1.

[49]

S. Olla and S. R. S. Varadhan, Scaling limit for interacting Ornstein-Uhlenbeck processes,, Comm. Math. Phys., 135 (1991), 355. doi: 10.1007/BF02098047.

[50]

S. Olla, S. R. S. Varadhan and H.-T. Yau, Hydrodynamical limit for a Hamiltonian system with weak noise,, Comm. Math. Phys., 155 (1993), 523. doi: 10.1007/BF02096727.

[51]

Y. H. Ou, A. Hasanov and Z. H. Liu, Inverse coefficient problems for nonlinear parabolic differential equations,, Acta. Math. Sin. (Engl. Ser.), 24 (2008), 1617. doi: 10.1007/s10114-008-6384-0.

[52]

K. Painter and T. Hillen, Volume-filling and quorum sensing in models for chemosensitive movement,, Canadian Applied Mathematics Quaterly, 10 (2003), 280.

[53]

N. Papenberg, A. Bruhn, T. Brox, S. Didas and J. Weickert, Highly accurate optic flow computation with theoretically justified warping,, International Journal of Computer Vision, 67 (2006), 141. doi: 10.1007/s11263-005-3960-y.

[54]

I. Sbalzarini and P. Koumoutsakos, Feature point tracking and trajectory analysis for video imaging in cell biology,, Journal of Structural Biology, 151 (2005), 182. doi: 10.1016/j.jsb.2005.06.002.

[55]

J. Schauder, Der Fixpunktsatz in Funktionalräumen,, Studia Math., 2 (1930), 171.

[56]

M. J. Simpson, B. D. Hughes and K. A. Landman, Diffusion populations: Ghosts or folks,, Australasian Journal of Engineering Education, 15 (2009), 59.

[57]

M. J. Simpson, K. A. Landman and B. D. Hughes, Multi-species simple exclusion process,, Physica A, 388 (2009), 399. doi: 10.1016/j.physa.2008.10.038.

[58]

C. Topaz, A. Bertozzi and M. Lewis, A nonlocal continuum model for biological aggregation,, Bulletin of Mathematical Biology, 68 (2006), 1601. doi: 10.1007/s11538-006-9088-6.

[59]

U. Weidmann, Transporttechnik der Fussgänger - Transporttechnische Eigenschaften des Fussgängerverkehrs (Literaturstudie),, Literature Research 90, (1993).

[60]

C. Zimmer, B. Zhang, A. Dufour, A. Thebaud, S. Berlemont, V. Meas-Yedid and J.-C. Marin, On the digital trail of mobile cells,, Signal Processing Magazine, 23 (2006), 54. doi: 10.1109/MSP.2006.1628878.

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