American Institute of Mathematical Sciences

Advanced
September  2021, 16(3): 493-511. doi: 10.3934/nhm.2021014

Irrigable measures for weighted irrigation plans

Qing Sun

Department of Mathematics, Penn State University, University Park, PA 16803, USA

Received  January 2020 Revised  April 2021 Published  September 2021 Early access  July 2021

A model of irrigation network, where lower branches must be thicker in order to support the weight of the higher ones, was recently introduced in [7]. This leads to a countable family of ODEs, describing the thickness of every branch, solved by backward induction. The present paper determines what kind of measures can be irrigated with a finite weighted cost. Indeed, the boundedness of the cost depends on the dimension of the support of the irrigated measure, and also on the asymptotic properties of the ODE which determines the thickness of branches.

Keywords: Optimization, ordinary differential equation, calculus of variation.
Mathematics Subject Classification: Primary: 34A05, 34A36; Secondary: 92B05.
Citation: Qing Sun. Irrigable measures for weighted irrigation plans. Networks and Heterogeneous Media, 2021, 16 (3) : 493-511. doi: 10.3934/nhm.2021014
References:
[1]

M. Bernot, V. Caselles and J. -M. Morel, Optimal Transportation Networks. Models and Theory, Lecture Notes in Mathematics, 1955, Springer, Berlin, 2009.

[2]

M. Bernot, V. Caselles and J.-M. Morel, Traffic plans, Publicacions Matemàtiques, 49 (2005), 417-451.  doi: 10.5565/PUBLMAT_49205_09.

[3]

A. Brancolini and S. Solimini, Fractal regularity results on optimal irrigation patterns, J. Math. Pures Appl., 102 (2014), 854-890.  doi: 10.1016/j.matpur.2014.02.008.

[4]

A. Bressan, M. Palladino and Q. Sun, Variational problems for tree roots and branches, Calc. Var. & Part. Diff. Equat., 59 (2020), Paper No. 7, 31 pp. doi: 10.1007/s00526-019-1666-1.

[5]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Matematica Italiana B, 2 (1988), 641-656. 

[6]

A. Bressan and Q. Sun, On the optimal shape of tree roots and branches, Math. Models & Methods Appl. Sci., 28 (2018), 2763-2801.  doi: 10.1142/S0218202518500604.

[7]

A. Bressan and Q. Sun, Weighted irrigation plans, submitted, arXiv: 1906.02232.

[8]

G. Devillanova and S. Solimini, On the dimension of an irrigable measure, Rend. Sem. Mat. Univ. Padova., 117 (2007), 1-49. 

[9]

G. Devillanova and S. Solimini, Elementary properties of optimal irrigation patterns, Calc. Var. & Part. Diff. Equat., 28 (2007), 317-349.  doi: 10.1007/s00526-006-0046-9.

[10]

G. Devillanova and S. Solimini, Some remarks on the fractal structure of irrigation balls, Advanced Nonlinear Studies, 19 (2019), 55-68.  doi: 10.1515/ans-2018-2035.

[11] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition. CRC Press, 2015. 
[12]

E. N. Gilbert, Minimum cost communication networks, Bell System Tech. J., 46 (1967), 2209-2227.  doi: 10.1002/j.1538-7305.1967.tb04250.x.

[13]

P. Hartman, Ordinary Differential Equations, , Second Edition, Birkhäuser, Boston, Mass., 1982.

[14]

F. Maddalena and S. Solimini, Synchronic and asynchronic descriptions of irrigation problems, Adv. Nonlinear Stud., 13 (2013), 583-623.  doi: 10.1515/ans-2013-0303.

[15]

F. Maddalena, G. Taglialatela and J.-M. Morel, A variational model of irrigation patterns, Interfaces Free Bound., 5 (2003), 391-415.  doi: 10.4171/IFB/85.

[16]

Q. Xia, Optimal paths related to transport problems, Comm. Contemp. Math., 5 (2003), 251-279.  doi: 10.1142/S021919970300094X.

[17]

Q. Xia, Motivations, ideas and applications of ramified optimal transportation, ESAIM Math. Model. Numer. Anal., 49 (2015), 1791-1832.  doi: 10.1051/m2an/2015028.

show all references

References:
[1]

M. Bernot, V. Caselles and J. -M. Morel, Optimal Transportation Networks. Models and Theory, Lecture Notes in Mathematics, 1955, Springer, Berlin, 2009.

[2]

M. Bernot, V. Caselles and J.-M. Morel, Traffic plans, Publicacions Matemàtiques, 49 (2005), 417-451.  doi: 10.5565/PUBLMAT_49205_09.

[3]

A. Brancolini and S. Solimini, Fractal regularity results on optimal irrigation patterns, J. Math. Pures Appl., 102 (2014), 854-890.  doi: 10.1016/j.matpur.2014.02.008.

[4]

A. Bressan, M. Palladino and Q. Sun, Variational problems for tree roots and branches, Calc. Var. & Part. Diff. Equat., 59 (2020), Paper No. 7, 31 pp. doi: 10.1007/s00526-019-1666-1.

[5]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Matematica Italiana B, 2 (1988), 641-656. 

[6]

A. Bressan and Q. Sun, On the optimal shape of tree roots and branches, Math. Models & Methods Appl. Sci., 28 (2018), 2763-2801.  doi: 10.1142/S0218202518500604.

[7]

A. Bressan and Q. Sun, Weighted irrigation plans, submitted, arXiv: 1906.02232.

[8]

G. Devillanova and S. Solimini, On the dimension of an irrigable measure, Rend. Sem. Mat. Univ. Padova., 117 (2007), 1-49. 

[9]

G. Devillanova and S. Solimini, Elementary properties of optimal irrigation patterns, Calc. Var. & Part. Diff. Equat., 28 (2007), 317-349.  doi: 10.1007/s00526-006-0046-9.

[10]

G. Devillanova and S. Solimini, Some remarks on the fractal structure of irrigation balls, Advanced Nonlinear Studies, 19 (2019), 55-68.  doi: 10.1515/ans-2018-2035.

[11] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition. CRC Press, 2015. 
[12]

E. N. Gilbert, Minimum cost communication networks, Bell System Tech. J., 46 (1967), 2209-2227.  doi: 10.1002/j.1538-7305.1967.tb04250.x.

[13]

P. Hartman, Ordinary Differential Equations, , Second Edition, Birkhäuser, Boston, Mass., 1982.

[14]

F. Maddalena and S. Solimini, Synchronic and asynchronic descriptions of irrigation problems, Adv. Nonlinear Stud., 13 (2013), 583-623.  doi: 10.1515/ans-2013-0303.

[15]

F. Maddalena, G. Taglialatela and J.-M. Morel, A variational model of irrigation patterns, Interfaces Free Bound., 5 (2003), 391-415.  doi: 10.4171/IFB/85.

[16]

Q. Xia, Optimal paths related to transport problems, Comm. Contemp. Math., 5 (2003), 251-279.  doi: 10.1142/S021919970300094X.

[17]

Q. Xia, Motivations, ideas and applications of ramified optimal transportation, ESAIM Math. Model. Numer. Anal., 49 (2015), 1791-1832.  doi: 10.1051/m2an/2015028.

Figure 1.  Left: A free standing tree with 5 branches. In this example, $ \mathcal{O}(1) = \{2,3 \},\mathcal{O}(3) = \{4,5\}, \mathcal{O}(2) = \mathcal{O}(4) = \mathcal{O}(5) = \emptyset $. Right: On each branch, the weight decreases as one moves from the lower portion to the tip
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Left: Two finite truncation plans, showing three maximal $ \varepsilon $-good paths (thick lines) and six maximal $ \varepsilon' $-good paths (thin lines), for $ 0<\varepsilon'<\varepsilon $. Right: The three maximal $ \varepsilon $-good paths can be partitioned into five elementary branches, by the Path Splitting Algorithm
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Left: The dyadic approxmiated measure $ \mu_1 $ is supported on the four centers $ x_1^1,\ldots,x^1_4 $ of small cubes. Right: Dyadic approximated measures corresponding to a family of partitions into dyadic cubes in $ \bf{R}^2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The dyadic irrigation plans in $ \bf{R}^2 $. Left: The dyadic irrigation plan $ \chi_1 $. The multiplicity on each branch equals to the mass on the terminal point. Right: The dyadic irrigation plan $ \chi_2 $. The particles are first transported to the 4 centers in $ \mathcal{P}_1 $, then on each center in $ \mathcal{P}_1 $, the particles are transported to the neighboring 4 centers in $ \mathcal{P}_2 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 321-337. doi: 10.3934/naco.2021008
[2]

Sriram Nagaraj. Optimization and learning with nonlocal calculus. Foundations of Data Science, 2022  doi: 10.3934/fods.2022009
[3]

Daniel G. Alfaro Vigo, Amaury C. Álvarez, Grigori Chapiro, Galina C. García, Carlos G. Moreira. Solving the inverse problem for an ordinary differential equation using conjugation. Journal of Computational Dynamics, 2020, 7 (2) : 183-208. doi: 10.3934/jcd.2020008
[4]

Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147
[5]

Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268
[6]

Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709
[7]

Michael Herty, Veronika Sachers. Adjoint calculus for optimization of gas networks. Networks and Heterogeneous Media, 2007, 2 (4) : 733-750. doi: 10.3934/nhm.2007.2.733
[8]

Ivar Ekeland. From Frank Ramsey to René Thom: A classical problem in the calculus of variations leading to an implicit differential equation. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1101-1119. doi: 10.3934/dcds.2010.28.1101
[9]

Lijuan Wang, Yashan Xu. Admissible controls and controllable sets for a linear time-varying ordinary differential equation. Mathematical Control and Related Fields, 2018, 8 (3&4) : 1001-1019. doi: 10.3934/mcrf.2018043
[10]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure and Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229
[11]

Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure and Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945
[12]

Jan-Hendrik Webert, Philip E. Gill, Sven-Joachim Kimmerle, Matthias Gerdts. A study of structure-exploiting SQP algorithms for an optimal control problem with coupled hyperbolic and ordinary differential equation constraints. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1259-1282. doi: 10.3934/dcdss.2018071
[13]

Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040
[14]

Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87
[15]

Janosch Rieger. The Euler scheme for state constrained ordinary differential inclusions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2729-2744. doi: 10.3934/dcdsb.2016070
[16]

Zvi Artstein. Averaging of ordinary differential equations with slowly varying averages. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 353-365. doi: 10.3934/dcdsb.2010.14.353
[17]

Tomasz Kapela, Piotr Zgliczyński. A Lohner-type algorithm for control systems and ordinary differential inclusions. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 365-385. doi: 10.3934/dcdsb.2009.11.365
[18]

Stefano Maset. Conditioning and relative error propagation in linear autonomous ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2879-2909. doi: 10.3934/dcdsb.2018165
[19]

W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209
[20]

Jacson Simsen, Mariza Stefanello Simsen, José Valero. Convergence of nonautonomous multivalued problems with large diffusion to ordinary differential inclusions. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2347-2368. doi: 10.3934/cpaa.2020102

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (86)
  • HTML views (190)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy