# Mark J. Friedman

## Professor

### Personal Information

Mathematical Sciences Department
University of Alabama in Huntsville
201N SC
Huntsville, AL 35899

Tel: (256) 824-6470

Fax: (256) 824-6173

friedman@math.uah.edu

### Teaching Schedule for Fall 2012

• M,W: 3:55 - 5:15 PM, SC 219 MA 515
• Tu,Th: 5:30 -7:30 PM, SC 105
• Office Hours: M,W: 3:25 - 3:40, 5:15 - 6:50; Tu,Th: 4:30 - 5:30, 7:30 - 7:40 or by appointment.

### Courses I often teach

• Dynamical Systems 1 (MA 524), Numerical Methods (MA 415) and Analysis (MA 515),
• Applied Linear Algebra (MA 508), Numerical Linear Algebra (MA 614), Numerical Methods for Partial Differential Equations I (MA 615).

### Brief Career History

• B.S. (physics/math) Moscow Physical Technical Institute (USSR), 1971.
• 1971-1978 Unskilled laborer, pending exit visa (USSR).
• 4/1979 - 8/1979 Staff Scientist 2, Lawrence Berkeley Laboratory, University of California; research in the area of magnetic field problems.
• M.S. (math) Cornell University, 1981.
• Ph.D. (math) Cornell University, 1982. Thesis title Numerical Analysis of the Nonlinear Magnetostatic Problem, advisor James H. Bramble.
• 1982-87 Assistant Professor, Iowa State University.
• 1987- 3/2001 Associate Professor, University of Alabama in Huntsville.
• 3/2001- Professor, University of Alabama in Huntsville.

### Recent visiting positions

• 2002 (Fall) - Visiting scholar, University of California at Berkeley.
• 2002 (June/August ) - NASA Summer faculty fellow, Marshall Space Flight Center.
• 2000 (July) - Visiting Professor, Philpps-Universitaet, Marburg, Germany.
• 1999 (June/August ) - NASA Summer faculty fellow, Marshall Space Flight Center.
• 1999 (May/June) - Visiting Professor, Philpps-Universitaet, Marburg, Germany.
• 1998 (June/August ) - NASA Summer faculty fellow, Marshall Space Flight Center.
• 1988 (Winter/Spring) - Visiting Associate Professor, Georgia Institute of Technology
• 1997 (Fall) - Visiting scholar, University of California at Berkeley.
• 1996 (June) - Visiting Scientist, Los Alamos National. Laboratory.
• 1992 (June) - Visiting scholar, Cornell Theory Center, Cornell University.

## Research

My research interests include:

(a) Numerical linear algebra.
(b) Numerical aspects of bifurcation theory.
(c) Scientific computing and applied dynamical systems. Understanding complicated dynamics
(e.g. chaos) in physical systems governed by ordinary or partial differential equations in
terms of bifurcation theory. Global bifurcations (homoclinic and heteroclinic orbits). Applications
to science and engineering.
(d) Numerical solution of elliptic PDEs by the Multiquadric Method, a recent meshless collocation
method.

My ongoing research is focused on the following topics:

• Numerical linear algebra. Development of efficient algorithms and software for continuation of invariant subspaces of large parameter dependent matrices.
• Numerical bifurcation theory. Development of algorithms for bifurcation analysis in large systems via subspace reduction and their implementation in MATLAB.
• Applications. Study of stability and bifurcations in MEMS (MicroElectroMechanical) devices.
• A.C. Monteiro, Algorithms for computing heteroclinic orbits, MA Thesis, 1992.
• X. Z. Shi, Numerical investigation of the stable nocturnal boundary layer, Ph.D Thesis, 1997.

### Recent Research Grants

• 2002 - 2005 Bifurcation Analysis for Large Problems: Algorithms, Software, Applications to MicroElectroMechanical Systems (MEMS), NSF, $199,308, DMS-0209536. • 1994 - 1997 Computational methods for global analysis of connecting orbits: development of algorithms and applications, NSF,$60,000, DMS-9404912.
• 1994 - 1997 Analytical and computational studies of oscillations in age structured population models, U.S. Bulgaria cooperative research, NSF (Co-PI, J. Li. PI), $10,000, INT-9412284. • 1994 (June 26-July 9) Academy of Consciousness Studies, Princeton University. • 1992 - 1995 Nonlinear Dynamical Analysis of Time Dependent Nocturnal Boundary Layers, NSF (R. T. McNider, PI),$44,023, ATM- 912-0321.
• 1992 - 1994 Computational methods for global analysis of homoclinic and heteroclinic orbits, theoretical analysis and applications, NSF, \$45,000, DMS-910-7705.

### Selected Publications

• A finite element method for the solution of a potential theory integral equation, Math. Meth. in Appl. Sci. 1 (1979), 581-587.
• Mathematical Study of the nonlinear singular integral magnetic field equation, 1. SIAM J. Appl. Math. 39 , No. 1 (1980), 14-20.
• Mathematical Study of the nonlinear singular integral magnetic field equation, 2. SIAM J. Numer. Anal. 18 , No. 1 (1981), 644-653.
• Mathematical Study of the nonlinear singular integral magnetic field equation, 3. SIAM J. Math. Anal. 12 , (1981) 536-540.
• Finite element formulation of the general magnetostatic problem in the space of solenoidal vector functions, Math. Comp. 43 , No. 168 (1984), 415-431.
• Spectral properties for the magnetization integral operator, (with J. E. Pasciak), Math. Comp. 43 , No. 168 (1984), 447-453.
• A new finite element-boundary integral procedure for the solution of the magnetostatic problem, COMPEL 4 , No. 3 (1985), 167-174.
• Finite Element Approximation of a Reaction-Diffusion Equation. Part I: Application of Topological Techniques to the Analysis of Asymptotic Behavior of the Semidiscrete Solutions, Quarterly of Appl. Math., XLIV (1986), 275-286.
• Application of topological techniques to the analysis of asymptotic behavior of numerical solutions for a reaction-diffusion equation, SIAM J. Math. Anal. 18, No. 1 (1987).
• Numerical computation of heteroclinic orbits (with E. J. Doedel), J. Comp. and Appl. Math. 26 (1989), 155-170.
• Numerical computation and continuation of invariant manifolds connecting fixed points with application to computation of combustion fronts (with E. J. Doedel), in: T.J. Chung and Gerald R. Karr, Ed., Finite Element Analysis in Fluids, Proc. 7th Int. Conf. on Finite Element Methods in flow problems, April 1989. (UAH Press, Huntsville, AL, 1989), 277-282.
• A dynamical systems approach to modeling meridians and Ki (with S. Birch and W. A. Tiller) in: Energy Fields in Medicine. A Study of Device Technology Based on acupuncture meridians and Chi Energy, Proc. Int. Roundtable, The John E. Fetzer Foundation, May 1989, 218-229.
• Towards the development of a mathematical model for acupuncture meridians (with S. Birch and W. A. Tiller), Acupuncture and Electro-Ther. Res. Int. J. 14, Nos. 3/4 (1989), 217-226.
• Mathematical Model development for the Law of five elements in acupuncture. (with S. Birch) J. Amer. Acupuncture 18 (1989) 361-366.
• Numerical computation and continuation of invariant manifolds, connecting fixed points (with E. J. Doedel), SIAM J. Numer. Anal. 28 (1991) 789-808.
• Development of efficient computational methods for global analysis of homoclinic and heteroclinic orbits: a case study (with E. J. Doedel), J. of Dynamics and Dif. Equations 5 , No. 1 (1993), 37-58.
• Numerical analysis and accurate computation of heteroclinic orbits in the case of center manifolds, J. of Dynamics and Dif. Equations 5 , No. 1 (1993), 59-87.
• Dynamical systems modeling as a research tool in traditional acupuncture (with S. Birch), Proc. First Symp. Soc. For Acupuncture Research, Rockville, MD, January 1993.
• On locating connecting orbits, (with E. J. Doedel and A.C. Monteiro)), Applied Math. And Comp. 65 , Nos. 1-3 (1994) 231-239.
• On computing connecting orbits: general algorithm and application to the Sine-Gordon and Hodgkin-Huxley equations (with E. J. Doedel and J. Guckenheimer), Special Section on Nonlinear Theory and Its Applications, The IEICE Trans. Fundamentals E77 A, No. 11 (1994) Japan.
• On the predictability of the stable atmospheric boundary layer (with R. T. McNider, D. England and X. Shi), J. of Atmospheric Sciences 52, No 10 (1995), 1602-1614.
• Successive continuation for locating connecting orbits (with E. J. Doedel and B. I. Kunin), Numer. Algorithms 14, (1997) 103-124.
• Mathematical Modeling as a tool for Basic Research in Acupuncture, (with S. Birch and W.A. Tiller), J. of Alternative and Complimentary Medicine 3, (1997), S89-S100.
• Grown-in point defects and microscopic defect formation in CZ silicon, Part I: the one- dimensional, steady state approximation (with W.A. Tiller, R. Shaw, N. Cuendet, and T.Halicioglu), Int. J. Crystal Growth. 186 (1998), 113 - 127.
• Solitary waves in a coupled Massive Thirring Model with self-phase modulation and dispersion (with A.R. Champneys and B. Malomed), Phys. Rev. Lett. 80 (1998), 4169-4173.
• On two simple models for competition between age classes, Mathematical Biosciences, 157 (1999), 65 - 89 (with T. Kostova and J. Li).
• Heteroclinic Loop Bifurcations with Nongeneric parameters (with S. N. Chow and B. Deng), SIAM, J. Appl. Math . 59 (1999), 1303-1321.
• Continuation for Nonlinear Elliptic Partial Differential Equations Discretized by the Multiquadric Method (with A.I. Fedoseyev and E.J. Kansa), Int. J. Bifur. Chaos, Int. J. Bifur. & Chaos, 10, No. 2 (2000), 481 - 492. [pdf]
• Computing connecting orbits via an improved algorithm for continuing invariant subspaces (with J. Demmel and L. Dieci), SIAM J. Sci. Comp., 22, No. 1 (2001), 81 - 94. [pdf]
• Continuation of Invariant Subspaces (with L. Dieci), Numer. Lin. Alg. Appl., 8 (2001), 317 - 327. [pdf]
• An improved detection of bifurcations in large nonlinear systems via the Continuation of Invariant Subspaces algorithm, Int. J. Bifur. & Chaos, 11 , No. 8 (2001), 2277-2285. [pdf]
• An improved Multiquadric method for nonlinear elliptic partial differential equations via PDE collocation on the boundary (with A.I. Fedoseyev and E.J. Kansa), Comput. Math. Applic., the Special Issue on Radial Basis Functions, 43, Nos (3-5) (2002), 439-455. [pdf]
• Continuation of Invariant Subspaces for Large Bifurcation Problems (with D. Bindel and J. Demmel), in: on-line PROCEEDINGS, SIAM 2003 Conference on Applied Linear Algebra. [pdf]
• D. Bindel, J. Demmel, M. Friedman, W. Govaerts, and Yu.A. Kuznetsov, "Bifurcation analysis of large equilibrium systems in Matlab". In: V.S. Sunderam et al. (eds.) "Proceedings of the International Conference on Computational Science ICCS 2005, Atlanta, GA, USA, May 22-25, 2005, Part I". Springer Verlag Lecture Notes in Computer Science 3514 (2005), 50-57. [pdf]
• B. Sautois, W. Govaerts, M. Friedman, and Yu.A. Kuznetsov, "Continuation of Homoclinic Orbits in Matlab". In: V.S. Sunderam et al. (eds.) "Proceedings of the International Conference on Computational Science ICCS 2005, Atlanta, GA, USA, May 22-25, 2005, Part I". Springer Verlag Lecture Notes in Computer Science 3514 (2005), 263-270. [pdf]
• D. Bindel, J. Demmel, M. Friedman, "Continuation of Invariant Subspaces for Large Bifurcation Problems", Tech. Report, University of California at Berkeley, 2006, [link]
• D. Bindel, J. Demmel, M. Friedman, "Continuation of Invariant Subspaces in Large Scale Bifurcation Problems", under review, SIAM J. Sci. Comp. [pdf]
• D. Bindel, J. Demmel, M. Friedman, "Sufficient conditions for continuously defined Invariant Subspaces", under review, Linear Algebra and Applications. [pdf]
• M. Friedman and W. Qiu, "On the location and continuation of Hopf bifurcations in large-scale problems", under review, Int. J. Bifur. & Chaos. [pdf]
• J. Hughes and M. Friedman, "A bisection-like algorithm for branch switching at a simple branch point", under review, Journal of Scientific Computing. [pdf]

### Software

• AUTO97a, a modification of AUTO97 to include new routines for numerical analysis of connecting orbits, an experimental version, 1997.
• SUBCON, a collection of fortran subroutines for continuing invariant subspaces of a parameter dependent matrix, an experimental version, 1998-1999.
• MQ-PDEs, A collection of fortran subroutines for discretizing 1D and 2D nonlinear elliptic partial differential equations by the Multiquadric Method, an experimental version (with A.I. Fedoseyev and E.J. Kansa), 1998-1999.
• D. Bindel, A. Dhooge, M. Friedman, W. Govaerts, J. Hughes, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet, and W. Qiu , "Cl_matcontL, Continuation Toolbox in MATLAB" University of Alabama in Huntsville, February 2007. [zip]

### Recent Invited Presentations

• Towards the development of a mathematical model for acupuncture meridians (with S. Birch and W. A. Tiller), 5th Annual International Symposium on Acupuncture and Electrotherapeutics, Columbia University, New York, New York, October 1989.
• Numerical analysis and efficient computation of heteroclinic orbits, SIAM Conference on Dynamical Systems, Orlando, Florida, May 1990.
• Computational methods for global analysis of homoclinic and heteroclinic orbits, Tokyo University, Tokyo, Japan, December 1990.
• Computational methods for global analysis of homoclinic and heteroclinic orbits, Kyoto Sangyo University, Kyoto, Japan, January 1991.
• Global analysis of homoclinic and heteroclinic orbits, First International Congress on Nonlinear Analysis, Tampa, Florida, August 1992.
• Dynamical systems modeling as a research tool in traditional acupuncture, First Symp. Soc. For Acupuncture Research, Rockville, MD, January 1993.
• The successive continuation method for obtaining homoclinic and heteroclinic orbits, SIAM Annual Meeting, Philadelphia, PA, July 1993.
• On global analysis of connecting orbits: numerical analysis, algorithms, applications, GA Tech., Atlanta, GA, November 1993.
• On global analysis of connecting orbits: numerical analysis, algorithms, applications, Kyoto University, Kyoto, Japan, February 1994.
• On global analysis of connecting orbits: numerical analysis, algorithms, applications, Hiroshima University, Hiroshima, Japan, February 1994.
• On global analysis of connecting orbits: numerical analysis, algorithms, applications, Waseda University, Tokyo, Japan, February 1994.
• On global analysis of connecting orbits: numerical analysis, algorithms, applications, The Weizmann Institute of Science, Rehovot, Israel, January 1995.
• On global analysis of connecting orbits: numerical analysis, algorithms, applications, Bulgarian Academy of Science, Sofia, Bulgaria, January 1995.
• On efficient computation of connecting orbits, International Conference on Scientific Computation and Differential Equations, Stanford, California, March 1995.
• Successive continuation for locating connecting orbits, The Tel Aviv University, Tel Aviv, Israel, March 1996.
• Successive continuation for locating connecting orbits, Los Alamos Nat. Lab., Los Alamos, New Mexico, July 1996. Mathematical Modeling as a tool for Basic Research in Acupuncture, First Int. Symposium; The Physiology of Acupuncture, Washington DC, November 1996
• An efficient algorithm for computing connecting orbits, Workshop on Numerical Methods for Bifurcation Problems, IMA, University of Minnesota, Minneapolis, MN, September 1997.
• Efficient algorithms for computing and continuing connecting orbits; applications, Georgia Inst. Technology, Atlanta, Georgia, March 1998.
• Employing numerical continuation for studying MHD Kelvin-Helmholts fluid instabilities in a solar physics problem. NASA/Marshall Space Flight Center, August 1998.
• Continuation of invariant subspaces algorithm. Applications to bifurcation analysis of large systems, Philpps-Universitaet Marburg, Germany, June 1999.
• Continuation of solutions to 1D and 2D Nonlinear Elliptic PDEs discretized by the Multiquadric Method, Philpps-Universitaet Marburg, Germany, June 1999.
• Continuation of invariant subspaces algorithm. Applications to bifurcation analysis of large systems. Continuation of solutions to 1D and 2D Nonlinear Elliptic PDEs discretized by the Multiquadric Method, Workshop on Bifurcations, Analysis, Numerical Methods, Software, University of Gent, Belgium, June 1999.
• Analysis of MHD Kelvin-Helmholts fluid instabilities in a solar physics problem, NASA/Marshall Space Flight Center, August 1999.
• An improved detection of bifurcations in large nonlinear systems via the Continuation of Invariant Subspaces algorithm, Workshop on Bifurcations, Analysis, Numerical Methods, Software, University of Gent, Belgium, June 2000.
• Continuation of Invariant Subspaces for large and sparse bifurcations problems, Workshop on Bifurcations, Analysis, Numerical Methods, Software, Utrecht University, The Netherlands, June 2001.
• Practical continuation of invariant subspaces for bifurcations problems, Workshop on Numerical Methods for Nonlinear Dynamics and Bifurcations, the University of Bristol, UK, July 2002.
• An improved RLV stability analysis via a continuation approach, NASA/Marshall Space Flight Center, August 2002.
• Practical continuation of invariant subspaces for bifurcations problems, University of California at Berkeley, November 2002.