Mark Embree

Professor, Department of Mathematics
Luther and Alice Hamlett Professor, College of Science
Leader, Computational Modeling and Data Analytics (CMDA) program

Email:   embree@vt.edu
Office:   414 Data & Decision Sciences (VT maps)
Mail:  225 Stanger Street 0123, Virginia Tech, Blacksburg, Virginia 24061

research interests

• Numerical analysis, especially matrix computations
• Dynamics and perturbation theory for non-self-adjoint operators
• Spectral theory for Schrödinger operators

Virginia Tech's Computational Modeling and Data Analytics (CMDA) major

In Spring 2015, Virginia Tech launched this new undergraduate major, a collaboration between the departments of Computer Science, Mathematics, and Statistics. Faculty have developed ten new courses for the major. Other departments have designed specialized options for students seeking to exercise CMDA skills in particular applications (biology; cryptography and cybersecurity; economics; geosciences; physics). The first class graduated in May 2017. As of July 2024, CMDA has graduated over 800 alumni (primary and secondary majors) and has roughly 500 current students pursuing the major.

• Flowcharts for the standard option, Biological Sciences option, Cryptography & Cybersecurity option, Economics option, Geosciences option, Physics option.
• CMDA is affiliated with the Calhoun Discovery Program in the Honors College.
  
• For further details, see the CMDA webpage.
• CMDA 3606, Math Modeling: Methods and Tools: course notes (June 2023).
• CMDA includes a Capstone Project program: opportunities for project sponsors are available.

Fall 2024

• Office Hours: TBD in D&DS 414 - or by appointment (email embree@vt.edu)
  

slides for lectures / short courses

Stability and Transient Dynamics for Linearized Reduced Order Models
Nonlinear Model Reduction for Control conference, May 2023.

Contour Integral Methods for Nonlinear Eigenvalue Problems: A Systems Theory Perspective
E-NLA Online Seminar Series on Numerical Linear Algebra (updated slides), June 2020.

Convergence Theory for Iterative Eigensolvers
Simons Institute workshop on Randomized Numerical Linear Algebra and Applications, September 2018, UC Berkeley.

Nonlinear Eigenvalue Problems: Interpolatory Algorithms and Transient Dyanmics
SIAM Conference on Applied Linear Algebra, May 2018, Hong Kong Baptist University
Version with pseudospectra movie is available here.

CUR Matrix Factorizations: Algorithms, Analysis, Applications

Pseudospectra and Nonnormal Dynamical Systems (4th Elgersburg School, March 2012)

• Slides for lecture 1, lecture 2, lecture 3, lecture 4, lecture 5

relics from past classes

• Lecture notes for CMDA 3606 (an applied linear algebra course at Virginia Tech)
• Lecture notes on Numerical Analysis (for MATH 5466 at Virginia Tech)
• Physical Laboratory manual to accompany undergraduate Matrix Analysis
  (with S. J. Cox, J. M. Hokanson, and others)
  
• Data from vibrating beaded strings for solving the inverse eigenvalue problem described in this article.
• MATH 5524: Matrix Theory (Spring 2017)

ph.d. students, past and present

• Jonathan Baker
  Lyapunov equations, convex optimization, nonlinear eigenvalue problems (PhD 2023)
  Dissertation: Vibrations of Mechanical Structures: Source Localization and Nonlinear Eigenvalue Problems for Mode Calculation
  
  
• Russell Carden
  Arnoldi convergence, Ritz values (1, 2), inverse field of values problems (PhD 2011)
  Thesis: Ritz Values and Arnoldi Convergence for Non-Hermitian Matrices
  
  
• Jeffrey Hokanson
  exponential data fitting, inverse eigenvalue problems (PhD 2013)
  Thesis: Numerically Stable and Statistically Efficient Algorithms for Large Scale Exponential Fitting
  
  
• Nilton Alan Garcia Hilares
  Power system dyanmics, transient growth, Loewner singular value decay (PhD 2023)
  Dissertation: Mathematical Modeling and Dynamic Recovery of Power Systems
  
  
• Sean Reiter
  Model reduction for power systems (in progress)
  
  
• John Sabino
  Thesis: Solution of Large-Scale Lyapunov Equations via the Block Modified Smith Method (PhD 2006)
  
  
• Josef Sifuentes
  Thesis: Preconditioned Iterative Methods for Inhomogeneous Acoustic Scattering Applications (PhD 2010)

Spectra and Pseudospectra

Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators Lloyd N. Trefethen and Mark Embree Princeton University Press, 2005. Princeton; amazon.com; Barnes & Noble; AddAll search.

EigTool

EigTool, Tom Wright's excellent MATLAB code for computing pseudospectra, can be downloaded via GitHub.

Papers, technical reports, preprints, etc.

• Optimal Algorithms for Quantifying Spectral Size with Applications to Quasicrystals
  M. Colbrook, M. Embree, and J. Fillman
  Preprint: arXiv:2407.20353
  
  
• Extending Elman's Bound for GMRES
  Preprint: arXiv:2312.15022
  
  
• On the balanced trncation error bound and sign parameters from arrowhead realizations
  S. Reiter, T. Damm, M. Embree, and S. Gugercin
  Adv. Comp. Math. 50 (2024) article 10 (29 pages), open access
  
  
• Gap labels for zeros of the partition function of the 1D Ising model via the Schwartzman homomorphism
  D. Damanik, M. Embree, and J. Fillman
  To appear, Indagationes Mathematicae
  Preprint: arxiv:2211.01558
  
  
• Analysis of GMRES for low-rank and small-norm perturbations of the identity matrix
  A. K. Carr, E. de Sturler, and M. Embree
  Proc. Appl. Math. Mech. (PAMM) 22:1 (2022) 6pp.
  Preprint: arxiv:2210.12053
  
  
• Discontinuities of the integrated density of states for Laplacians associated with Penrose and Ammann-Beenker tilings
  D. Damanik, M. Embree, J. Fillman, and M. Mei
  Experimental Mathematics 33 (2024) 588-610.
  Preprint: arxiv:2209.01443
  
  
• Contour integral methods for nonlinear eigenvalue problems: a system theoretic approach
  M. Brennan, M. Embree, S. Gugercin
  SIAM Review 65 (2023) 439-470.
  Preprint: arXiv:2012.14979
  
  
• Mathematics of magic angles in a model of twisted bilayer graphene
  S. Becker, M. Embree, J. Wittsten, and M. Zworski
  Prob. Math. Phys. 3 (2022) 69-103.
  Preprint: arXiv:2008.08489
  
  
• Spectral characterization of magic angles in twisted bilayer graphene
  S. Becker, M. Embree, J. Wittsten, and M. Zworski
  Phys. Rev. B. 103 (2021) 164113.
  Preprint: arXiv:2010.05279
  
  
• Pseudospectra of Loewner matrix pencils
  M. Embree, C. Ionita
  published version (revised preprint; original version: arXiv:1910.12153)
  In Realization and Model Reduction of Dynamical Systems: A Festschrift in Honor of the 70th Birthday of Thanos Antoulas
  Beattie, Benner, Embree, Lefteriu, Gugercin, eds., Springer, Cham, 2022.
  
  
• Polynomial preconditioned Arnoldi with stability control
  M. Embree, J. A. Loe, and R. B. Morgan
  SIAM J. Sci. Comp. 43 (2021) A1-A25.
  Preprint (earlier version): arXiv:1806.08020
  Presentation at 2021 SIAM Applied Linear Algebra Conference: slides
  
  
• Unstable modes in projection-based reduced-order models: How many can there be, and what do they tell you?
  M. Embree
  Systems Control Lett. 124 (2019) 49-59.
  Preprint: arXiv:1801.00234
  
  
• Spectra of discrete two-dimensional periodic Schrödinger operators with small potentials
  M. Embree and J. Fillman
  J. Spectral Theory 9 (2019) 1063-1087.
  Preprint: arXiv:1701.00863 [math.SP]
  
  
• Weighted inner products for GMRES and Arnoldi iterations
  M. Embree, R. B. Morgan, and H. V. Nguyen
  SIAM J. Sci. Comp. 39 (2017) S610--S632. (Copyright SIAM, 2017)
  Preprint: arXiv:1607.00255 [math.NA]
  
  
• Pseudospectra of matrix pencils for transient analysis of differential-algebraic equations
  M. Embree and B. Keeler
  SIAM J. Matrix Anal. Appl. 38 (2017) 1028-1054. (Copyright SIAM, 2017)
  Preprint: arXiv:1601.00044 [math.NA]
  
  
• Fast singular value decay for Lyapunov solutions with nonnormal coefficients
  J. Baker, M. Embree, and J. Sabino
  SIAM J. Matrix Anal. Appl. 36 (2015) 656-668. (Copyright SIAM, 2015)
  Preprint: arXiv:1410.8741 [math.NA]
  
  
• Spectral approximation for quasiperiodic Jacobi operators
  C. Puelz, M. Embree, and J. Fillman
  Integral Equations Operator Theory 82 (2015) 533-554.
  Preprint: arXiv:1408.0370 [math.SP]
  
  
• A DEIM induced CUR factorization
  D. C. Sorensen and M. Embree
  SIAM J. Sci. Comp. 38 (2016) A1454-A1482. (Copyright SIAM, 2016)
  Preprint: arXiv:1407.5516 [math.NA]
  
  
• Spectral properties of Schrödinger operators arising in the study of quasicrystals
  D. Damanik, M. Embree, A. Gorodetski
  In Mathematics of Aperiodic Order, pages 307-370; Kellendonk, Lenz, Savinien, eds., Springer, 2015
  Preprint: arXiv:1210.5753 [math-ph]
  
  
• The stability of GMRES convergence, with application to approximate deflation preconditioning
  J. A. Sifuentes, M. Embree, and R. B. Morgan
  SIAM J. Matrix Anal. Appl. 34 (2013) 1066-1088. (Copyright SIAM, 2013)
  
  
• Ritz value localization for non-Hermitian matrices
  R. Carden and M. Embree
  SIAM J. Matrix Anal. Appl. 33 (2012) 1320-1338. (Copyright SIAM, 2012)
  
  
• Short-term recurrence Krylov subspace methods for nearly Hermitian matrices
  M. Embree , J. A. Sifuentes, K. M. Soodhalter, D. B. Szyld, and F. Xue
  SIAM J. Matrix Anal. Appl. 33 (2012) 480-500. (Copyright SIAM, 2012)
  
  
• Reconstructing an even damping from a single spectrum
  S. J. Cox and M. Embree
  Inverse Problems 27 (2011) 035012 (18pp).
  
  
• One can hear the composition of a string: experiments with an inverse eigenvalue problem
  S. J. Cox, M. Embree, and J. Hokanson
  SIAM Review 54 (2012) 157-178. (Copyright SIAM, 2012)
  Additional data sets are available at www.caam.rice.edu/~beads.
  
  
• Dynamical systems and non-Hermitian iterative eigensolvers
  M. Embree and R. B. Lehoucq
  SIAM J. Num. Anal. 47 (2009) 1445-1473. (Copyright SIAM, 2009)
  
  
• The fractal dimension of the spectrum of the Fibonacci Hamiltonian
  D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev
  Commun. Math. Phys. 280 (2008) 499-516.
  
  
• The Arnoldi eigenvalue iteration with exact shifts can fail
  SIAM J. Matrix Anal. Appl. 31 (2009) 1-10. (Copyright SIAM, 2009)
  
  
• Parallel solution of large-scale free surface viscoelastic flows via sparse approximate inverse preconditioning
  Z. Castillo, X. Xie, D. C. Sorensen, M. Embree, and M. Pasquali
  J. Non-Newtonian Fluid Mech. 157 (2009) 44-54.
  
  
• The role of the penalty in the local discontinuous Galerkin method for Maxwell's eigenvalue problem
  T. Warburton and Mark Embree
  Comp. Methods Appl. Mech. Eng. 195 (2006) 3205-3323.
  
  
• Convergence of polynomial restart Krylov methods for eigenvalue computation
  Christopher A. Beattie, Mark Embree, D. C. Sorensen
  SIAM Review 47 (2005) 492-515. (Copyright SIAM, 2005)
  
  
• Convergence of restarted Krylov subspaces to invariant subspaces
  Christopher Beattie, Mark Embree, John Rossi
  SIAM J. Matrix Anal. Appl. 25 (2004) 1074-1109. (Copyright SIAM, 2004)
  
  
• The tortoise and the hare restart GMRES
  SIAM Review 45 (2003) 259-266. (Copyright SIAM, 2003)
  
  
• The spectra of large Toeplitz band matrices with a randomly perturbed entry
  Albrecht Böttcher, Mark Embree, V. I. Sokolov
  Math. Comp. 72 (2003), 1329-1348.
  
  
• On large Toeplitz band matrices with an uncertain block
  Albrecht Böttcher, Mark Embree, V. I. Sokolov
  Linear Algebra Appl. 366 (2003), 87-97.
  
  
• Spectral approximation of banded Laurent matrices with localized random perturbations
  Albrecht Böttcher, Mark Embree, Marko Lindner
  Integral Equations Operator Theory 42 (2002), 142-165. (Copyright Birkhäuser, 2002)
  
  
• Piecewise continuous Toeplitz matrices and operators: slow approach to infinity
  Albrecht Böttcher, Mark Embree, Lloyd N. Trefethen
  SIAM J. Matrix Anal. Appl. 24 (2002) 484-489. (Copyright SIAM, 2002)
  
  
• Infinite Toeplitz and Laurent matrices with localized impurities
  Albrecht Böttcher, Mark Embree, V. I. Sokolov
  Linear Algebra Appl. 343-344 (2002), 101-118.
  
  
• Generalizing eigenvalue theorems to pseudospectra theorems
  Mark Embree and Lloyd N. Trefethen
  SIAM J. Sci. Comp. 23 (2001) 583-590. (Copyright SIAM, 2001)
  
  
• Spectra, pseudospectra, and localization for random bidiagonal matrices
  Lloyd N. Trefethen, Marco Contedini, Mark Embree
  Comm. Pure Applied Math. 54 (2001) 595-623.
  
  
• How descriptive are GMRES convergence bounds?
  Oxford University Computing Laboratory Numerical Analysis Report 99/08, June 1999.
  Revised, corrected, and extended version (arxiv:2209.01231), September 2022
  
  
• Green's functions for multiply connected domains via conformal mapping
  Mark Embree and Lloyd N. Trefethen
  SIAM Review 41 (1999) 745-761.
  
  
• Growth and decay of random Fibonacci sequences
  Mark Embree and Lloyd N. Trefethen
  Proc. Roy. Soc. London Series A 455 (1999) 2471-2485.