American Institute of Mathematical Sciences

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September  2013, 12(5): 2277-2296. doi: 10.3934/cpaa.2013.12.2277

On behavior of signs for the heat equation and a diffusion method for data separation

Mi-Ho Giga 1, , Yoshikazu Giga 2, , Takeshi Ohtsuka 3, and  Noriaki Umeda 4,

1. 

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
2. 

Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914
3. 

Division of Mathematical Sciences, Graduate School of Engineering, Gunma University, 4-2 Aramaki-cho, Maebashi, 371-8510
4. 

Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo, 153-8914

Received  February 2012 Revised  October 2012 Published  January 2013

Consider the solution $u(x,t)$ of the heat equation with initial data $u_0$. The diffusive sign $S_D[u_0](x)$ is defined by the limit of sign of $u(x,t)$ as $t\to 0$. A sufficient condition for $x\in R^d$ and $u_0$ such that $S_D[u_0](x)$ is well-defined is given. A few examples of $u_0$ violating and fulfilling this condition are given. It turns out that this diffusive sign is also related to variational problem whose energy is the Dirichlet energy with a fidelity term. If initial data is a difference of characteristic functions of two disjoint sets, it turns out that the boundary of the set $S_D[u_0](x) = 1$ (or $-1$) is roughly an equi-distance hypersurface from $A$ and $B$ and this gives a separation of two data sets.
Keywords: equi-distance hypersurface, Diffusive sign, heat equations., sign-changing.
Mathematics Subject Classification: Primary: 35K05; Secondary: 35B4.
Citation: Mi-Ho Giga, Yoshikazu Giga, Takeshi Ohtsuka, Noriaki Umeda. On behavior of signs for the heat equation and a diffusion method for data separation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2277-2296. doi: 10.3934/cpaa.2013.12.2277
References:
[1]

F. Andreu-Vaillo, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,'', Progress in Mathematics, 223 (2004).  doi: 10.1007/978-3-0348-7928-6.  Google Scholar

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A. L. Bertozzi and A. Flenner, Diffuse interface models on graphs for classification of high dimensional data,, Multiscale Modeling and Simulation, 10 (2012), 1090.  doi: 10.1137/11083109X.  Google Scholar

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A. Chambolle, An algorithm for mean curvature motion,, Interfaces and free boundaries, 6 (2004), 195.  doi: 10.4171/IFB/97.  Google Scholar

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X. Chen, Generation and propagation of interfaces for reaction-diffusion equations,, J. Differential Equations, 96 (1992), 116.  doi: 10.1016/0022-0396(92)90146-E.  Google Scholar

[8]

X.-Y. Chen, A strong unique continuation theorem for parabolic equations,, Math. Ann., 311 (1998), 603.  doi: 10.1007/s002080050202.  Google Scholar

[9]

X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations,, J. Differential Equations, 78 (1989), 160.  doi: 10.1016/0022-0396(89)90081-8.  Google Scholar

[10]

R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, B. Nadler, F. Warner and S. W. Zucker, Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps,, Proc. Natl. Acad. Sci., 102 (2005), 7426.  doi: 10.1073/pnas.0500334102.  Google Scholar

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N. Cristianini and J. Shawe-Taylor, "An Introduction to Support Vector Machines and Other Kernel-based Learning Methods,'', Cambridge University Press, (2000).   Google Scholar

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M.-H. Giga and Y. Giga, Very singular diffusion equations: second and fourth order problems,, Japan J. Indust. Appl. Math., 27 (2010), 323.  doi: 10.1007/s13160-010-0020-y.  Google Scholar

[13]

M.-H. Giga, Y. Giga and J. Saal, "Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-similar Solutions,'', Progress in Nonlinear Differential Equations and Their Applications, 79 (2010).  doi: 10.1007/978-0-8176-4651-6.  Google Scholar

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K. Kielak, P. B. Mucha and P. Rybka, Almost classical solutions to the total variation flow., (to appear in Journal of Evolution Equations), ().  doi: 10.1007/s00028-012-0167-x.  Google Scholar

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H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation,, J. Fac. Sci. Univ. Tokyo Sect. IA Math. \textbf{29} (1982), 29 (1982), 401.   Google Scholar

[17]

L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza: il Teorema di Modica-Mortola,, Boll. Un. Mat. Ital., (B5) (1977), 285.   Google Scholar

[18]

K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleighungen,, J. Reine Angew. Math., 211 (1962), 78.  doi: 10.1515/crll.1962.211.78.  Google Scholar

[19]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'', Englewood Cliffs, (1967).   Google Scholar

[20]

W. Ring, Structural properties of solutions to total variation regularization problems,, M2AN Math. Model. Numer. Anal., 34 (2000), 799.  doi: 10.1051/m2an:2000104.  Google Scholar

[21]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Phys. D, 60 (1992), 259.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[22]

G. Steidl, Supervised learning by support vector machines,, "Handbook of Mathematical Methods in Imaging'' (O. Scherzer ed.),, 3 (2011), 959.   Google Scholar

[23]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems,, Arch. Rational Mech. Anal., 101 (1988), 209.  doi: 10.1007/BF00253122.  Google Scholar

[24]

D. G. Widder, "The Heat Equations,'', Academic Press, (1975).   Google Scholar

show all references

References:
[1]

F. Andreu-Vaillo, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,'', Progress in Mathematics, 223 (2004).  doi: 10.1007/978-3-0348-7928-6.  Google Scholar

[2]

S. Angenent, The zero set of a solution of a parabolic equation,, J. Reine Angew. Math., 390 (1988), 79.  doi: 10.1515/crll.1988.390.79.  Google Scholar

[3]

A. L. Bertozzi and A. Flenner, Diffuse interface models on graphs for classification of high dimensional data,, Multiscale Modeling and Simulation, 10 (2012), 1090.  doi: 10.1137/11083109X.  Google Scholar

[4]

M. Bonforte and A. Figalli, Total variation flow and sign fast diffusion in one dimension,, J. Differential Equations, 252 (2012), 4455.  doi: 10.1016/j.jde.2012.01.003.  Google Scholar

[5]

A. Briani, A. Chambolle, M. Novaga and G. Orlandi, On the gradient flow of a one-homogeneous functional,, Confluentes Mathematici, 3 (2011), 617.  doi: 10.1142/S1793744211000461.  Google Scholar

[6]

A. Chambolle, An algorithm for mean curvature motion,, Interfaces and free boundaries, 6 (2004), 195.  doi: 10.4171/IFB/97.  Google Scholar

[7]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations,, J. Differential Equations, 96 (1992), 116.  doi: 10.1016/0022-0396(92)90146-E.  Google Scholar

[8]

X.-Y. Chen, A strong unique continuation theorem for parabolic equations,, Math. Ann., 311 (1998), 603.  doi: 10.1007/s002080050202.  Google Scholar

[9]

X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations,, J. Differential Equations, 78 (1989), 160.  doi: 10.1016/0022-0396(89)90081-8.  Google Scholar

[10]

R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, B. Nadler, F. Warner and S. W. Zucker, Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps,, Proc. Natl. Acad. Sci., 102 (2005), 7426.  doi: 10.1073/pnas.0500334102.  Google Scholar

[11]

N. Cristianini and J. Shawe-Taylor, "An Introduction to Support Vector Machines and Other Kernel-based Learning Methods,'', Cambridge University Press, (2000).   Google Scholar

[12]

M.-H. Giga and Y. Giga, Very singular diffusion equations: second and fourth order problems,, Japan J. Indust. Appl. Math., 27 (2010), 323.  doi: 10.1007/s13160-010-0020-y.  Google Scholar

[13]

M.-H. Giga, Y. Giga and J. Saal, "Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-similar Solutions,'', Progress in Nonlinear Differential Equations and Their Applications, 79 (2010).  doi: 10.1007/978-0-8176-4651-6.  Google Scholar

[14]

D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations,, J. Differential Equations, 598 (1985), 165.  doi: 10.1016/0022-0396(85)90153-6.  Google Scholar

[15]

K. Kielak, P. B. Mucha and P. Rybka, Almost classical solutions to the total variation flow., (to appear in Journal of Evolution Equations), ().  doi: 10.1007/s00028-012-0167-x.  Google Scholar

[16]

H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation,, J. Fac. Sci. Univ. Tokyo Sect. IA Math. \textbf{29} (1982), 29 (1982), 401.   Google Scholar

[17]

L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza: il Teorema di Modica-Mortola,, Boll. Un. Mat. Ital., (B5) (1977), 285.   Google Scholar

[18]

K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleighungen,, J. Reine Angew. Math., 211 (1962), 78.  doi: 10.1515/crll.1962.211.78.  Google Scholar

[19]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'', Englewood Cliffs, (1967).   Google Scholar

[20]

W. Ring, Structural properties of solutions to total variation regularization problems,, M2AN Math. Model. Numer. Anal., 34 (2000), 799.  doi: 10.1051/m2an:2000104.  Google Scholar

[21]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Phys. D, 60 (1992), 259.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[22]

G. Steidl, Supervised learning by support vector machines,, "Handbook of Mathematical Methods in Imaging'' (O. Scherzer ed.),, 3 (2011), 959.   Google Scholar

[23]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems,, Arch. Rational Mech. Anal., 101 (1988), 209.  doi: 10.1007/BF00253122.  Google Scholar

[24]

D. G. Widder, "The Heat Equations,'', Academic Press, (1975).   Google Scholar

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