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

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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

Abstract
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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.
[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.
[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.
[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.
[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.
[6]	A. Chambolle, An algorithm for mean curvature motion,, Interfaces and free boundaries, 6 (2004), 195. doi: 10.4171/IFB/97.
[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.
[8]	X.-Y. Chen, A strong unique continuation theorem for parabolic equations,, Math. Ann., 311 (1998), 603. doi: 10.1007/s002080050202.
[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.
[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.
[11]	N. Cristianini and J. Shawe-Taylor, "An Introduction to Support Vector Machines and Other Kernel-based Learning Methods,'', Cambridge University Press, (2000).
[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.
[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.
[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.
[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.
[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.
[17]	L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza: il Teorema di Modica-Mortola,, Boll. Un. Mat. Ital., (B5) (1977), 285.
[18]	K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleighungen,, J. Reine Angew. Math., 211 (1962), 78. doi: 10.1515/crll.1962.211.78.
[19]	M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'', Englewood Cliffs, (1967).
[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.
[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.
[22]	G. Steidl, Supervised learning by support vector machines,, "Handbook of Mathematical Methods in Imaging'' (O. Scherzer ed.),, 3 (2011), 959.
[23]	P. Sternberg, The effect of a singular perturbation on nonconvex variational problems,, Arch. Rational Mech. Anal., 101 (1988), 209. doi: 10.1007/BF00253122.
[24]	D. G. Widder, "The Heat Equations,'', Academic Press, (1975).

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.
[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.
[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.
[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.
[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.
[6]	A. Chambolle, An algorithm for mean curvature motion,, Interfaces and free boundaries, 6 (2004), 195. doi: 10.4171/IFB/97.
[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.
[8]	X.-Y. Chen, A strong unique continuation theorem for parabolic equations,, Math. Ann., 311 (1998), 603. doi: 10.1007/s002080050202.
[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.
[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.
[11]	N. Cristianini and J. Shawe-Taylor, "An Introduction to Support Vector Machines and Other Kernel-based Learning Methods,'', Cambridge University Press, (2000).
[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.
[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.
[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.
[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.
[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.
[17]	L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza: il Teorema di Modica-Mortola,, Boll. Un. Mat. Ital., (B5) (1977), 285.
[18]	K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleighungen,, J. Reine Angew. Math., 211 (1962), 78. doi: 10.1515/crll.1962.211.78.
[19]	M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'', Englewood Cliffs, (1967).
[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.
[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.
[22]	G. Steidl, Supervised learning by support vector machines,, "Handbook of Mathematical Methods in Imaging'' (O. Scherzer ed.),, 3 (2011), 959.
[23]	P. Sternberg, The effect of a singular perturbation on nonconvex variational problems,, Arch. Rational Mech. Anal., 101 (1988), 209. doi: 10.1007/BF00253122.
[24]	D. G. Widder, "The Heat Equations,'', Academic Press, (1975).

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