American Institute of Mathematical Sciences

Advanced
August  2017, 11(4): 623-641. doi: 10.3934/ipi.2017029

A discrete Liouville identity for numerical reconstruction of Schrödinger potentials

Liliana Borcea 1, , Fernando Guevara Vasquez 2,, and  Alexander V. Mamonov 3,

1. 

Mathematics, University of Michigan, 2074 E Hall, 530 Church St, Ann Arbor, MI 48109-1043, USA
2. 

Mathematics, University of Utah, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, USA
3. 

Mathematics, University of Houston, 4800 Calhoun Rd., Houston, TX, 77004, USA

* Corresponding author: F. Guevara Vasquez

Received  January 2016 Revised  April 2017 Published  June 2017

Fund Project: LB acknowledges partial support from ONR Grant N00014-14-1-0077 and NSF grant DMS-1510429. The work of FGV was partially supported by the National Science Foundation grant DMS-1411577. The work of AVM was partially supported by the National Science Foundation grant DMS-1619821

We propose a discrete approach for solving an inverse problem for the two-dimensional Schrödinger equation, where the unknown potential is to be determined from the Dirichlet to Neumann map. In the continuum, the problem for absorptive potentials can be transformed with the Liouville identity into a conductivity inverse problem. Its discrete analogue is to find a resistor network matching the measurements, and is well understood. Here we use a discrete Liouville identity to transform its solution to that of Schrödinger's problem. The discrete Schrödinger potential given by the discrete Liouville identity can be used to reconstruct the potential in the continuum in two ways. First, we can obtain a direct but coarse reconstruction by interpreting the values of the discrete Schrödinger potential as averages of the continuum Schrödinger potential on a special sensitivity grid. Second, the discrete Schrödinger potential may be used to reformulate the conventional nonlinear output least squares formulation of the inverse Schrödinger problem. Instead of minimizing the boundary measurement misfit, we minimize the misfit between discrete Schrödinger potentials. This results in a better behaved optimization problem converging in a single Gauss-Newton iteration, and gives good quality reconstructions of the potential, as illustrated by the numerical results.

Keywords: Schrödinger equation, Liouville transform, Dirichlet to Neumann map, absorptive potentials, nonlinear preconditioner.
Mathematics Subject Classification: Primary: 05C22, 35R30, 35J10; Secondary: 05C50, 78A46.
Citation: Liliana Borcea, Fernando Guevara Vasquez, Alexander V. Mamonov. A discrete Liouville identity for numerical reconstruction of Schrödinger potentials. Inverse Problems & Imaging, 2017, 11 (4) : 623-641. doi: 10.3934/ipi.2017029
References:
[1]

C. Araúz, A. Carmona and A. Encinas, Dirichlet-to-Robin matrix on networks, Electronic Notes in Discrete Mathematics, 46 (2014), 65–72, Jornadas de Matemática Discreta y Algorítmica. doi: 10.1016/j.endm.2014.08.010.  Google Scholar

[2]

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.  Google Scholar

[3]

G. Bal, Optical tomography for small volume absorbing inclusions, Inverse Problems, 19 (2003), 371-386.  doi: 10.1088/0266-5611/19/2/308.  Google Scholar

[4]

P. Benner, S. Gugercin and K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Review, 57 (2015), 483-531.  doi: 10.1137/130932715.  Google Scholar

[5]

L. Borcea, V. Druskin, A. Mamonov and F. Guevara Vasquez, Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements Inverse Problems, 26 (2010), 105009, 36pp. doi: 10.1088/0266-5611/26/10/105009.  Google Scholar

[6]

L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136, Topical Review.  doi: 10.1088/0266-5611/18/6/201.  Google Scholar

[7]

L. Borcea, V. Druskin and F. Guevara Vasquez, Electrical impedance tomography with resistor networks Inverse Problems, 24 (2008), 035013, 31pp. doi: 10.1088/0266-5611/24/3/035013.  Google Scholar

[8]

L. Borcea, V. Druskin, F. Guevara Vasquez and A. V. Mamonov, Resistor network approaches to electrical impedance tomography, Inside Out Ⅱ (ed. G. Uhlmann), vol. 60, MSRI Publications, 2012. Google Scholar

[9]

L. Borcea, V. Druskin and L. Knizhnerman, On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids, Communications on Pure and Applied Mathematics, 58 (2005), 1231-1279.  doi: 10.1002/cpa.20073.  Google Scholar

[10]

L. Borcea, V. Druskin, A. V. Mamonov and M. Zaslavsky, A model reduction approach to numerical inversion for a parabolic partial differential equation Inverse Problems, 30 (2014), 125011, 30pp. doi: 10.1088/0266-5611/30/12/125011.  Google Scholar

[11]

L. Borcea, V. Druskin and A. Mamonov, Circular resistor networks for electrical impedance tomography with partial boundary measurements, Inverse Problems, 26 (2010), 045010, 30pp. doi: 10.1088/0266-5611/26/4/045010.  Google Scholar

[12]

K. Chadan, D. Colton, L. Päivärinta and W. Rundell, An Introduction to Inverse Scattering and Inverse Spectral Problems, SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997, With a foreword by Margaret Cheney. doi: 10.1137/1.9780898719710.  Google Scholar

[13]

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.  Google Scholar

[14]

Y. Colin de Verdiére, Réseaux électriques planaires. Ⅰ, Comment. Math. Helv., 69 (1994), 351-374.  doi: 10.1007/BF02564493.  Google Scholar

[15]

Y. Colin de Verdiére, Spectres de Graphes vol. 4 of Cours Spécialisés [Specialized Courses], Société Mathématique de France, Paris, 1998.  Google Scholar

[16]

Y. Colin de Verdiére, I. Gitler and D. Vertigan, Réseaux électriques planaires. Ⅱ, Comment. Math. Helv., 71 (1996), 144-167.  doi: 10.1007/BF02566413.  Google Scholar

[17]

E. Curtis, T. Edens and J. Morrow, Calculating resistors in a network, Engineering in Medicine and Biology Society, 1989. Images of the Twenty-First Century. Proceedings of the Annual International Conference of the IEEE Engineering in, 2 (1989), 451-452.  doi: 10.1109/IEMBS.1989.95813.  Google Scholar

[18]

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.  Google Scholar

[19]

E. B. Curtis, D. Ingerman and J. A. Morrow, Circular planar graphs and resistor networks, Linear Algebra Appl., 283 (1998), 115-150.  doi: 10.1016/S0024-3795(98)10087-3.  Google Scholar

[20]

E. B. Curtis and J. A. Morrow, Determining the resistors in a network, SIAM J. Appl. Math., 50 (1990), 918-930.  doi: 10.1137/0150055.  Google Scholar

[21]

E. B. Curtis and J. A. Morrow, The Dirichlet to Neumann map for a resistor network, SIAM J. Appl. Math., 51 (1991), 1011-1029.  doi: 10.1137/0151051.  Google Scholar

[22]

E. B. Curtis and J. A. Morrow, Inverse Problems for Electrical Networks, vol. 13 of Series on Applied Mathematics, World Scientific, 2000. Google Scholar

[23]

V. Druskin, V. Simoncini and M. Zaslavsky, Solution of the time-domain inverse resistivity problem in the model reduction framework part Ⅰ. one-dimensional problem with SISO data, SIAM Journal on Scientific Computing, 35 (2013), A1621-A1640.  doi: 10.1137/110852607.  Google Scholar

[24]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[25]

D. Gisser, D. Isaacson and J. Newell, Electric current computed tomography and eigenvalues, SIAM Journal on Applied Mathematics, 50 (1990), 1623-1634.  doi: 10.1137/0150096.  Google Scholar

[26]

G. H. Golub and C. F. Van Loan, Matrix Computations, 4th edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013.  Google Scholar

[27]

R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd edition, Cambridge University Press, Cambridge, 2013.  Google Scholar

[28]

D. Ingerman, Schrodinger equation, 2012, http://en.wikibooks.org/wiki/User:Daviddaved/Schrodinger_equation, Retrieved November 2013. Google Scholar

[29]

D. V. Ingerman, Discrete and continuous Dirichlet-to-Neumann maps in the layered case, SIAM Journal on Mathematical Analysis, 31 (2000), 1214-1234.  doi: 10.1137/S0036141097326581.  Google Scholar

[30]

A. V. Mamonov, V. Druskin and M. Zaslavsky, Nonlinear seismic imaging via reduced order model backprojection, SEG Technical Program Expanded Abstracts, (2015), 4375–4379, arXiv: 1504.00094. doi: 10.1190/segam2015-5830429.1.  Google Scholar

[31]

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.  Google Scholar

show all references

References:
[1]

C. Araúz, A. Carmona and A. Encinas, Dirichlet-to-Robin matrix on networks, Electronic Notes in Discrete Mathematics, 46 (2014), 65–72, Jornadas de Matemática Discreta y Algorítmica. doi: 10.1016/j.endm.2014.08.010.  Google Scholar

[2]

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.  Google Scholar

[3]

G. Bal, Optical tomography for small volume absorbing inclusions, Inverse Problems, 19 (2003), 371-386.  doi: 10.1088/0266-5611/19/2/308.  Google Scholar

[4]

P. Benner, S. Gugercin and K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Review, 57 (2015), 483-531.  doi: 10.1137/130932715.  Google Scholar

[5]

L. Borcea, V. Druskin, A. Mamonov and F. Guevara Vasquez, Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements Inverse Problems, 26 (2010), 105009, 36pp. doi: 10.1088/0266-5611/26/10/105009.  Google Scholar

[6]

L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136, Topical Review.  doi: 10.1088/0266-5611/18/6/201.  Google Scholar

[7]

L. Borcea, V. Druskin and F. Guevara Vasquez, Electrical impedance tomography with resistor networks Inverse Problems, 24 (2008), 035013, 31pp. doi: 10.1088/0266-5611/24/3/035013.  Google Scholar

[8]

L. Borcea, V. Druskin, F. Guevara Vasquez and A. V. Mamonov, Resistor network approaches to electrical impedance tomography, Inside Out Ⅱ (ed. G. Uhlmann), vol. 60, MSRI Publications, 2012. Google Scholar

[9]

L. Borcea, V. Druskin and L. Knizhnerman, On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids, Communications on Pure and Applied Mathematics, 58 (2005), 1231-1279.  doi: 10.1002/cpa.20073.  Google Scholar

[10]

L. Borcea, V. Druskin, A. V. Mamonov and M. Zaslavsky, A model reduction approach to numerical inversion for a parabolic partial differential equation Inverse Problems, 30 (2014), 125011, 30pp. doi: 10.1088/0266-5611/30/12/125011.  Google Scholar

[11]

L. Borcea, V. Druskin and A. Mamonov, Circular resistor networks for electrical impedance tomography with partial boundary measurements, Inverse Problems, 26 (2010), 045010, 30pp. doi: 10.1088/0266-5611/26/4/045010.  Google Scholar

[12]

K. Chadan, D. Colton, L. Päivärinta and W. Rundell, An Introduction to Inverse Scattering and Inverse Spectral Problems, SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997, With a foreword by Margaret Cheney. doi: 10.1137/1.9780898719710.  Google Scholar

[13]

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.  Google Scholar

[14]

Y. Colin de Verdiére, Réseaux électriques planaires. Ⅰ, Comment. Math. Helv., 69 (1994), 351-374.  doi: 10.1007/BF02564493.  Google Scholar

[15]

Y. Colin de Verdiére, Spectres de Graphes vol. 4 of Cours Spécialisés [Specialized Courses], Société Mathématique de France, Paris, 1998.  Google Scholar

[16]

Y. Colin de Verdiére, I. Gitler and D. Vertigan, Réseaux électriques planaires. Ⅱ, Comment. Math. Helv., 71 (1996), 144-167.  doi: 10.1007/BF02566413.  Google Scholar

[17]

E. Curtis, T. Edens and J. Morrow, Calculating resistors in a network, Engineering in Medicine and Biology Society, 1989. Images of the Twenty-First Century. Proceedings of the Annual International Conference of the IEEE Engineering in, 2 (1989), 451-452.  doi: 10.1109/IEMBS.1989.95813.  Google Scholar

[18]

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.  Google Scholar

[19]

E. B. Curtis, D. Ingerman and J. A. Morrow, Circular planar graphs and resistor networks, Linear Algebra Appl., 283 (1998), 115-150.  doi: 10.1016/S0024-3795(98)10087-3.  Google Scholar

[20]

E. B. Curtis and J. A. Morrow, Determining the resistors in a network, SIAM J. Appl. Math., 50 (1990), 918-930.  doi: 10.1137/0150055.  Google Scholar

[21]

E. B. Curtis and J. A. Morrow, The Dirichlet to Neumann map for a resistor network, SIAM J. Appl. Math., 51 (1991), 1011-1029.  doi: 10.1137/0151051.  Google Scholar

[22]

E. B. Curtis and J. A. Morrow, Inverse Problems for Electrical Networks, vol. 13 of Series on Applied Mathematics, World Scientific, 2000. Google Scholar

[23]

V. Druskin, V. Simoncini and M. Zaslavsky, Solution of the time-domain inverse resistivity problem in the model reduction framework part Ⅰ. one-dimensional problem with SISO data, SIAM Journal on Scientific Computing, 35 (2013), A1621-A1640.  doi: 10.1137/110852607.  Google Scholar

[24]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[25]

D. Gisser, D. Isaacson and J. Newell, Electric current computed tomography and eigenvalues, SIAM Journal on Applied Mathematics, 50 (1990), 1623-1634.  doi: 10.1137/0150096.  Google Scholar

[26]

G. H. Golub and C. F. Van Loan, Matrix Computations, 4th edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013.  Google Scholar

[27]

R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd edition, Cambridge University Press, Cambridge, 2013.  Google Scholar

[28]

D. Ingerman, Schrodinger equation, 2012, http://en.wikibooks.org/wiki/User:Daviddaved/Schrodinger_equation, Retrieved November 2013. Google Scholar

[29]

D. V. Ingerman, Discrete and continuous Dirichlet-to-Neumann maps in the layered case, SIAM Journal on Mathematical Analysis, 31 (2000), 1214-1234.  doi: 10.1137/S0036141097326581.  Google Scholar

[30]

A. V. Mamonov, V. Druskin and M. Zaslavsky, Nonlinear seismic imaging via reduced order model backprojection, SEG Technical Program Expanded Abstracts, (2015), 4375–4379, arXiv: 1504.00094. doi: 10.1190/segam2015-5830429.1.  Google Scholar

[31]

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.  Google Scholar

Figure 1.  Graphs of class $C(m, n)$, where $n$ the number of boundary nodes and $m$ is equal to the number of edges from the center in a fixed radial direction, plus the number of concentric layers of $n$ edges each. The boundary nodes are shown as circles and the interior nodes as filled (black) circles. Both graphs are critical since $n = 2m+1$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  A graph $\mathcal{G}$ in black and its line graph $\widetilde{\mathcal{G}}$ in red
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Conductivities $\sigma^{(i)}$, corresponding to different Schrödinger potentials $q^{(i)}$, $i=1, 2$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Sensitivity functions for $n=17$, $q = 1$, i.e. the ''rows'' of $D\mathit{\boldsymbol{Q}}[\mathit{\boldsymbol{M}}(q)] D\mathit{\boldsymbol{M}}[q]$. The other sensitivity functions can be obtained by rotations of integer multiples of $2\pi/17$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Sensitivity grids. The ''x'' are for $q=1$ and the ''$\circ$'' for $q=3$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Gauss-Newton iterates for smooth $q$ (sensitivity grid)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Gauss-Newton iterates for piecewise constant $q$ (sensitivity grid)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  A typical convergence history for the preconditioned Gauss-Newton method. We show convergence in terms of the unpreconditioned residual (green), the preconditioned residual (red) and the projected preconditioned residual (blue)
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Jussi Behrndt, A. F. M. ter Elst. The Dirichlet-to-Neumann map for Schrödinger operators with complex potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 661-671. doi: 10.3934/dcdss.2017033
[2]

Victor Isakov, Jenn-Nan Wang. Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2014, 8 (4) : 1139-1150. doi: 10.3934/ipi.2014.8.1139
[3]

Victor Isakov. Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 631-640. doi: 10.3934/dcdss.2011.4.631
[4]

Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745
[5]

Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems & Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959
[6]

Yutian Lei. Liouville theorems and classification results for a nonlocal Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5351-5377. doi: 10.3934/dcds.2018236
[7]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[8]

Haidong Liu, Zhaoli Liu. Positive solutions of a nonlinear Schrödinger system with nonconstant potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1431-1464. doi: 10.3934/dcds.2016.36.1431
[9]

Xing Cheng, Ze Li, Lifeng Zhao. Scattering of solutions to the nonlinear Schrödinger equations with regular potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2999-3023. doi: 10.3934/dcds.2017129
[10]

Yongsheng Jiang, Huan-Song Zhou. A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1723-1730. doi: 10.3934/cpaa.2010.9.1723
[11]

Jing Yang. Segregated vector Solutions for nonlinear Schrödinger systems with electromagnetic potentials. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1785-1805. doi: 10.3934/cpaa.2017087
[12]

Rémi Carles. Global existence results for nonlinear Schrödinger equations with quadratic potentials. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 385-398. doi: 10.3934/dcds.2005.13.385
[13]

Zaihui Gan, Boling Guo, Jian Zhang. Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1303-1312. doi: 10.3934/cpaa.2009.8.1303
[14]

Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705
[15]

Ihsane Bikri, Ronald B. Guenther, Enrique A. Thomann. The Dirichlet to Neumann map - An application to the Stokes problem in half space. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 221-230. doi: 10.3934/dcdss.2010.3.221
[16]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807
[17]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[18]

Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063
[19]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[20]

Shouchuan Hu, Nikolaos S. Papageorgiou. Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2889-2922. doi: 10.3934/cpaa.2013.12.2889

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (15)
  • HTML views (16)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences