# American Institute of Mathematical Sciences

March  2020, 15(1): 1-28. doi: 10.3934/nhm.2020001

## Matrix valued inverse problems on graphs with application to mass-spring-damper systems

 Mathematics Department, University of Utah, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, USA

* Corresponding author: F. Guevara Vasquez

Received  June 2018 Revised  August 2019 Published  December 2019

Fund Project: This work was supported by the National Science Foundation grants DMS-1411577 and DMS-1439786.

We consider the inverse problem of finding matrix valued edge or node quantities in a graph from measurements made at a few boundary nodes. This is a generalization of the problem of finding resistors in a resistor network from voltage and current measurements at a few nodes, but where the voltages and currents are vector valued. The measurements come from solving a series of Dirichlet problems, i.e. finding vector valued voltages at some interior nodes from voltages prescribed at the boundary nodes. We give conditions under which the Dirichlet problem admits a unique solution and study the degenerate case where the edge weights are rank deficient. Under mild conditions, the map that associates the matrix valued parameters to boundary data is analytic. This has practical consequences to iterative methods for solving the inverse problem numerically and to local uniqueness of the inverse problem. Our results allow for complex valued weights and give also explicit formulas for the Jacobian of the parameter to data map in terms of certain products of Dirichlet problem solutions. An application to inverse problems arising in networks of springs, masses and dampers is presented.

Citation: Travis G. Draper, Fernando Guevara Vasquez, Justin Cheuk-Lum Tse, Toren E. Wallengren, Kenneth Zheng. Matrix valued inverse problems on graphs with application to mass-spring-damper systems. Networks and Heterogeneous Media, 2020, 15 (1) : 1-28. doi: 10.3934/nhm.2020001
##### References:
 [1] G. Alessandrini, Remark on a paper by H. Bellout and A. Friedman: "Identification problems in potential theory" [Arch. Rational Mech. Anal., 101 (1988), 143–160; MR0921936 (89c: 31003)], Boll. Un. Mat. Ital. A (7), 3 (1989), 243–249. [2] C. Araúz, A. Carmona and A. Encinas, Dirichlet-to-Robin matrix on networks, Electronic Notes in Discrete Mathematics, 46 (2014), 65-72.  doi: 10.1016/j.endm.2014.08.010. [3] C. Araúz, A. Carmona and A. Encinas, Dirichlet-to-Robin maps on finite networks, Applicable Analysis and Discrete Mathematics, 9 (2015), 85-102.  doi: 10.2298/AADM150207004A. [4] C. Araúz, A. Carmona and A. Encinas, Overdetermined partial boundary value problems on finite networks, Journal of Mathematical Analysis and Applications, 423 (2015), 191-207.  doi: 10.1016/j.jmaa.2014.09.025. [5] L. Borcea, F. Guevara Vasquez and A. V. Mamonov, A discrete Liouville identity for numerical reconstruction of Schrödinger potentials, Inverse Probl. Imaging, 11 (2017), 623-641.  doi: 10.3934/ipi.2017029. [6] J. Boyer, J. J. Garzella and F. Guevara Vasquez, On the solvability of the discrete conductivity and Schrödinger inverse problems, SIAM J. Applied Math., 76 (2016), 1053-1075.  doi: 10.1137/15M1043479. [7] M. Camar-Eddine and P. Seppecher, Determination of the closure of the set of elasticity functionals, Arch. Ration. Mech. Anal., 170 (2003), 211-245.  doi: 10.1007/s00205-003-0272-7. [8] F. R. K. Chung, Spectral Graph Theory, vol. 92 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997. [9] S.-Y. Chung, Identification of resistors in electrical networks, J. Korean Math. Soc., 47 (2010), 1223-1238. [10] Y. Colin de Verdière, Réseaux électriques planaires. Ⅰ, Comment. Math. Helv., 69 (1994), 351-374. [11] Y. Colin de Verdière, Spectres de Graphes, vol. 4 of Cours Spécialisés [Specialized Courses], Société Mathématique de France, Paris, 1998. [12] Y. Colin de Verdière, I. Gitler and D. Vertigan, Réseaux électriques planaires. Ⅱ, Comment. Math. Helv., 71 (1996), 144-167.  doi: 10.1007/BF02566413. [13] E. Curtis, E. Mooers and J. Morrow, Finding the conductors in circular networks from boundary measurements, RAIRO Modél. Math. Anal. Numér., 28 (1994), 781-814.  doi: 10.1051/m2an/1994280707811. [14] E. B. Curtis, D. Ingerman and J. A. Morrow, Circular planar graphs and resistor networks, Linear Algebra Appl., 283 (1998), 115-150. [15] V. Druskin, personal communication, 2015. [16] P. Erdős and A. Rényi, On random graphs. I, Publ. Math. Debrecen, 6 (1959), 290-297. [17] A. Gondolo and F. Guevara Vasquez, Characterization and synthesis of Rayleigh damped elastodynamic networks, Networks and Heterogeneous Media, 9 (2014), 299-314.  doi: 10.3934/nhm.2014.9.299. [18] F. Guevara Vasquez, G. W. Milton and D. Onofrei, Complete characterization and synthesis of the response function of elastodynamic networks, J. Elasticity, 102 (2011), 31-54.  doi: 10.1007/s10659-010-9260-y. [19] R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. [20] R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd edition, Cambridge University Press, Cambridge, 2013. [21] T. Lam and P. Pylyavskyy, Inverse problem in cylindrical electrical networks, SIAM J. Appl. Math., 72 (2012), 767-788.  doi: 10.1137/110846476. [22] J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994, Corrected reprint of the 1983 original. [23] J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. [24] W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976, International Series in Pure and Applied Mathematics. [25] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153–169. doi: 10.2307/1971291.

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##### References:
 [1] G. Alessandrini, Remark on a paper by H. Bellout and A. Friedman: "Identification problems in potential theory" [Arch. Rational Mech. Anal., 101 (1988), 143–160; MR0921936 (89c: 31003)], Boll. Un. Mat. Ital. A (7), 3 (1989), 243–249. [2] C. Araúz, A. Carmona and A. Encinas, Dirichlet-to-Robin matrix on networks, Electronic Notes in Discrete Mathematics, 46 (2014), 65-72.  doi: 10.1016/j.endm.2014.08.010. [3] C. Araúz, A. Carmona and A. Encinas, Dirichlet-to-Robin maps on finite networks, Applicable Analysis and Discrete Mathematics, 9 (2015), 85-102.  doi: 10.2298/AADM150207004A. [4] C. Araúz, A. Carmona and A. Encinas, Overdetermined partial boundary value problems on finite networks, Journal of Mathematical Analysis and Applications, 423 (2015), 191-207.  doi: 10.1016/j.jmaa.2014.09.025. [5] L. Borcea, F. Guevara Vasquez and A. V. Mamonov, A discrete Liouville identity for numerical reconstruction of Schrödinger potentials, Inverse Probl. Imaging, 11 (2017), 623-641.  doi: 10.3934/ipi.2017029. [6] J. Boyer, J. J. Garzella and F. Guevara Vasquez, On the solvability of the discrete conductivity and Schrödinger inverse problems, SIAM J. Applied Math., 76 (2016), 1053-1075.  doi: 10.1137/15M1043479. [7] M. Camar-Eddine and P. Seppecher, Determination of the closure of the set of elasticity functionals, Arch. Ration. Mech. Anal., 170 (2003), 211-245.  doi: 10.1007/s00205-003-0272-7. [8] F. R. K. Chung, Spectral Graph Theory, vol. 92 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997. [9] S.-Y. Chung, Identification of resistors in electrical networks, J. Korean Math. Soc., 47 (2010), 1223-1238. [10] Y. Colin de Verdière, Réseaux électriques planaires. Ⅰ, Comment. Math. Helv., 69 (1994), 351-374. [11] Y. Colin de Verdière, Spectres de Graphes, vol. 4 of Cours Spécialisés [Specialized Courses], Société Mathématique de France, Paris, 1998. [12] Y. Colin de Verdière, I. Gitler and D. Vertigan, Réseaux électriques planaires. Ⅱ, Comment. Math. Helv., 71 (1996), 144-167.  doi: 10.1007/BF02566413. [13] E. Curtis, E. Mooers and J. Morrow, Finding the conductors in circular networks from boundary measurements, RAIRO Modél. Math. Anal. Numér., 28 (1994), 781-814.  doi: 10.1051/m2an/1994280707811. [14] E. B. Curtis, D. Ingerman and J. A. Morrow, Circular planar graphs and resistor networks, Linear Algebra Appl., 283 (1998), 115-150. [15] V. Druskin, personal communication, 2015. [16] P. Erdős and A. Rényi, On random graphs. I, Publ. Math. Debrecen, 6 (1959), 290-297. [17] A. Gondolo and F. Guevara Vasquez, Characterization and synthesis of Rayleigh damped elastodynamic networks, Networks and Heterogeneous Media, 9 (2014), 299-314.  doi: 10.3934/nhm.2014.9.299. [18] F. Guevara Vasquez, G. W. Milton and D. Onofrei, Complete characterization and synthesis of the response function of elastodynamic networks, J. Elasticity, 102 (2011), 31-54.  doi: 10.1007/s10659-010-9260-y. [19] R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. [20] R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd edition, Cambridge University Press, Cambridge, 2013. [21] T. Lam and P. Pylyavskyy, Inverse problem in cylindrical electrical networks, SIAM J. Appl. Math., 72 (2012), 767-788.  doi: 10.1137/110846476. [22] J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994, Corrected reprint of the 1983 original. [23] J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. [24] W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976, International Series in Pure and Applied Mathematics. [25] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153–169. doi: 10.2307/1971291.
A simple spring, mass and damper network that we use in example 1. Here $V = \{0, 1, 2, 3\}$ and $E = \{\{0, 1\}, \{0, 2\}, \{0, 3\}\}$. The node positions at equilibrium are given in black and correspond to $x_0 = (0, 0)^T$, and $x_i = (\cos(2\pi i / 3), \sin(2\pi i / 3))^T$, $i = 1, \ldots, 3$. A displaced configuration is given in blue, where the new node positions are $x_i + u_i$, with displacements $u_i$ for $i = 0, \ldots, 3$
The Laplacian for a scalar conductivity on the cylindrical graph $C \equiv P_5 \times P_3$ can be seen as a matrix valued Schrödinger operator on the graph $P_5$, as explained in example 2. To fix ideas, $s^4 \in \mathbb{R}^2$ represents the conductivities of $C$ within the $4-$th group in red and defines the matrix valued Schrödinger potential $q(4)$. The conductivity $s^{2, 3} \in \mathbb{R}^3$ represents the conductivities of the 3 edges between the 2nd and 3rd group and is used to define the matrix valued conductivity $\sigma(\{2, 3\})$
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