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:* 575 McBryde Hall (VT maps)

*Mail:* 225 Stanger Street 0123, Virginia Tech, Blacksburg, Virginia 24061

### research interests

Luther and Alice Hamlett Professor, College of Science

Leader, Computational Modeling and Data Analytics (CMDA) program

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

*
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 September 2022, CMDA has graduated over 500 alumni and has roughly 600 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 (Fall 2022).
- CMDA includes a Capstone Project program: opportunities for project sponsors are available.

- Office Hours: by appointment (email embree@vt.edu)

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)

- 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)

- Jonathan Baker

*Lyapunov equations, convex optimization, data analytics for smart buildings*(in progress)

- Russell Carden

*Arnoldi convergence, Ritz values (1, 2), inverse field of values problems*(PhD 2011) - Jeffrey Hokanson

*exponential data fitting, inverse eigenvalue problems*(PhD 2013) - Alan Garcia

Nonlinear eigenvalue problems, model reduction (in progress)

- 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: The Behavior of Nonnormal Matrices and OperatorsLloyd N. Trefethen and Mark Embree Princeton University Press, 2005. Princeton; amazon.com; Barnes & Noble; AddAll search. |

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

- On the tightness of the balanced truncation error bound, with applications to arrowhead systems

S. Reiter, M. Embree, S. Gugercin

Preprint: arXiv:2011.07170

(Revision and extension of arXiv:2011.07170v1)

- Gap labels for zeros of the partition function of the 1D Ising model via the Schwartzman homomorphism

D. Damanik, M. Embree, and J. Fillman

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

Preprint: arxiv:2209.01443

- Contour integral methods for nonlinear eigenvalue problems: a system theoretic approach

M. Brennan, M. Embree, S. Gugercin

*SIAM Review*, to appear.

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.