Ben Leimkuhler's Research Page

My research group is based in the School of Mathematics at the University of Edinburgh. We combine expertise in dynamics, classical, quantum and statistical mechanics with advanced scientific computing techniques to develop forefront simulation methodology for applications such as molecular dynamics.
On these pages you will find links to preprints and articles in journals, as well as some general information about the group. People interested in possibly joining the group, whether for upper division undergraduate, MSc, doctoral, or postdoctoral study are encouraged to use the email address below to find out about opportunities.

Contact Info:

email: b dot leimkuhler at-sign ed.ac.uk

mail: Ben Leimkuhler, School of Mathematics, James Clerk Maxwell Building, King's Buildings, University of Edinburgh, Edinburgh EH9 3JZ

** Benedict Leimkuhler**

** Studies**

PhD 1988, MS 1986, University of Illinois

BS 1983, Purdue University

** Career History**

2006 -- Chair of Applied Mathematics, University of Edinburgh

2000 -- 2006 Professor of Applied Mathematics, University of Leicester

1996 -- 1999 Associate Professor of Mathematics, University of Kansas

1990 -- 1996 Assistant Professor of Mathematics, University of Kansas

1988 -- 1990 Researcher, Helsinki University of Technology

1986 Researcher, Lawrence Livermore National Laboratory

1983 -- 1988 Research Assistant, University of Illinois

**Research Interests **

My research is in the broad area of the computational modelling of dynamical systems, such systems ranging in scale from computer simulation of the motion of atoms and molecules to the modelling of celestial mechanics. This involves the development of appropriate numerical methods to solve the system of equations driving the dynamics. It is desirable to develop approximation schemes that preserve important qualitative features, so-called geometric integrators, and this has been the focus of much of my work. Of particular importance in this regard are the symplectic integrators for Hamiltonian systems which preserve the symplectic structure of phase space and have superior stability properties, particularly for long time computations.

While many of my articles are related to Hamiltonian systems and the development of geometry-preserving integration methods, I have lately been more focussed on stochastic differential equations. Results in this direction have been obtained on both formulation and numerical solution of SDE models for thermodynamic modelling, including proving the ergodicity of degenerate diffusion techniques and studying the perturbation of dynamics by stochastic methods. Most recently, I have focused on the design of Langevin dynamics integration strategies, including the construction of superconvergent Langevin dynamics methods for invariant measures relevant for molecular dynamics.

From 2009 to present I have been heavily involved with the EPSRC (Science and Innovation) Centre for Numerical Algorithms and Intelligent Software (NAIS) which links Edinburgh, Heriot-Watt and Strathclyde Universities.

In addition to NAIS, I have an ERC-funded collaboration in Biological Modelling with V. Danos in Informatics, and I am part of the ExTASY Project (Extensible Toolkit for Advanced Sampling and analYsis) which is cofunded by the UK’s EPSRC and the US NSF to develop advanced methods for the study of biological molecular energy surfaces.

** Associations, Positions, Roles**

2014-- Fellow of the Royal Society of Edinburgh (FRSE)

2012-- Fellow of the Institute of Mathematics

2012-- Co-Director, Maxwell Institute for Mathematical Sciences

2012 Senior Research Fellow, Netherlands Science Foundation

2011 JT Oden Fellowship, University of Texas

2009 - Member, Steering Committee of the Centre for Numerical Algorithms (NAIS)

2009 -11 Director, NAIS

2008 -11 Deputy Director, Maxwell Institute for Mathematical Sciences

2007-- Member, Board of the International Centre for Mathematical Sciences (ICMS)

2007-- Member, Programme Committee, ICMS

2004-5 Leverhulme Trust Research Fellow

2004-5 Visiting Researcher, Institute for Mathematics and Its Applications, Minneapolis

1998 Member, Mathematical Sciences Research Insitute, Berkeley

1996-7 Visiting Researcher, Cambridge University

** Editorial Boards**

2014-- Proceedings of the Royal Society A

2012-- Journal of Computational Dynamics (AIMS)

2009-13 Nonlinearity

2009-13 SIAM Journal on Numerical Analysis

2008-- IMA Journal on Numerical Analysis

2001-7 SIAM Journal on Scientific Computing

**Grants and Projects **

2013 -- 2016 NSF-EPSRC SI2-CHE:ExTASY Extensible Tools for Advanced Sampling and analYsis

2013 -- 2018 ERC Advanced Grant in Ruled-Based Modelling for Biology

2009 -- 2014 EPSRC (S&I)/SFC Numerical Algorithms and Intelligent Software for the Evolving HPC Platform

2008 -- 2010 EPSRC Network (Bath, Bristol, Edinburgh, Warwick) “Mathematical Challenges of Molecular Dynamics”

2004 -- 2007 Australian RC Geometric Integration

2004 US NIH Algorithms for Macromolecular Modelling

2004 -- 2005 EPSRC Algorithms for Macromolecular Modelling

2004 -- 2007 EPSRC Developing an Efficient Method for Locating Periodic Orbits

2003 -- 2005 SRIF Advanced Computing Facility (HPC)

2002 -- 2004 Australian RC Geometric Numerical Integration

2001 -- 2004 EPSRC A Mixed Atomistic and Continuum Model for Crossing Multiple Length and Time Scales

2001 -- 2004 EPSRC Geometric Integrators for Switched and Multiple Time-scale Dynamics

2000 -- 2004 EU Research Training Network MASIE (Mechanics and Symmetry in Europe)

1994 -- 99 Multiple grants awarded whilst at the University of Kansas, mostly by US National Science Foundation

Conferences and Workshops* [examples]*

Principal Organiser of the inaugural conference in 1994 of the series on Algorithms for Macromolecular Modelling (AM3) in Lawrence, Kansas, followed by service on the Organising Committee for subsequent meetings in the series (Berlin, 1997; New York City, 2000; Leicester 2004 as Principal Organiser; Austin, 2009)

Organising Committee, LMS Durham Symposium, 2000

Co-Organiser, Advanced Integration Methods for Molecular Dynamics, CECAM, Lyon, 2000

Scientific Committee, Prestissimo/DFG Conference on Molecular Simulation, Inst. Henri Poincaré, Paris, 2004

Co-Organiser, Workshop on Astrophysical N-body Problems, Inst. for Pure and Applied Mathematics, UCLA, 2005

Co-Organiser, The Interplay between Mathematical Theory and Applications, Newton Institute, 2007

Co-Organiser, NSF-NAIS Workshop on Intelligent Software, Edinburgh 2009

Co-Organiser, Capstone Conference, EPSRC Conference on Challenges in Scientific Computing, Warwick 2009

Co-organiser, Complex Molecular Systems, Lorentz Center, Leiden, 2012

** Books**

* Simulating Hamiltonian Dynamics * (Cambridge University Press, 2005), co-authored with S. Reich (Potsdam) is an introduction to the subject of Geometric Integration for undergraduate and graduate students in mathematics and cognate disciplines. * Molecular Dynamics* (Springer, in press) one of the first mathematical books on the subject.

** Selected Articles**

- Symplectic numerical integrators in contrained Hamiltonian systems (with R.D. Skeel), J. Comput. Phys. 112 (1994), 117-125.
- Integration methods for molecular dynamics (with S. Reich and R.D. Skeel), IMA Vol. Math. Appl. 82 (1996), 161-185.
- Symplectic methods for conservative multibody systems (with E. Barth), Fields Inst. Comm. 10 (1996), 25-43.
- The adaptive Verlet method (with W. Huang), SIAM J. Sci. Comput. 18 (1997), 239-256.
- Time-transformations for reversible variable stepsize integration (with S.D. Bond), Numer. Algorithms 19 (1998), 55-71.
- Comparison of geometric integrators for rigid body simulation, Lect. Notes Comput. Sci. Eng. 4 (1999), 349-362.
- The Nosé-Poincaré method for constant temperature molecular dynamics (with S.D.Bond and B.B. Laird), J. Comput. Phys 151 (1999), 114-134.
- Asymptotic error analysis of the adaptive Verlet method (with S. Cirilli and E. Hairer), BIT 39 (1999), 25-33.
- Explicit variable step-size and time-reversible integration (with T. Holder and S. Reich), Appl. Numer. Math 39 (2001), 367-377
- An efficient multiple time-scale reversible integrator for the gravitational N-body problem, Appl. Numer. Math. 43 (2002), 175-190.
- On the approximation of the Feynman-Kac path intergral (with B.B. Laird), J. Comput. Phys. 185 (2003), 472-483.
- The canonical ensemble via symplectic integration using Nosé and Nosé-Poincaré chains (with C. Sweet), J. Chem. Phys. 121 (2004), 108-116.
- Molecular dynamics and the accuracy of numerically computed averages (with. S.D. Bond), Acta Numer. 16 (2007), 1-65.
- Molecular simulation in the canonical ensemble and beyond (with Z. Jia), Math. Model. Numer. Anal. 41 (2007), 333-350.
- Stabilized integration of Hamiltonian systems with hard-sphere inequality constraints (with S.D. Bond), SIAM J. Sci. Comput. 30 (2007/2008), 134-147.
- Simplified modelling of a thermal bath, with applications to a fluid vortex system (with S. Dubinskina and J. Frank), Multiscale Model Simul. 8 (2010), 1182-1902.
- Comparing the efficiencies of stochastic isothermal molecular dynamics methods (with E. Noorizadeh and O. Penrose), J. Stat. Phys. 143 (2011), 921-942.1572-9613.
- Dimensional reductions for the computation of time-dependent quantum expectations (with G. Mazzi), SIAM J. Sci. Comput. 33 (2011), 2024-2038.
- Robust and efficient configurational molecular sampling via Langevin Dynamics (with C. Matthews), J. Chem. Phys., published online 1 May 2013 (Vol.138, Issue 17).
- Rational construction of stochastic numerical methods for molecular sampling (with C. Matthews), Applied Mathematics Research Express, 2013, (2013), 34-56.

My group's work is performed in collaboration with researchers in other areas of mathematics (numerical analysis, probability theory, analysis) and also with chemists, physicists, engineers and biomolecular modellers. This broad-based approach combines model development with simulation and rigorous numerical analysis. Our goal is to use the flexibility of our toolset to expand understanding of algorithms without much regard to whether or not the approach taken fits a particular formalism or mathematical school. We always appreciate the elegance that only a full mathematical theory can provide, but at the same time we are pragmatic: there are many problems which cannot be resolved in complete rigor using current theories. In these cases the boundaries of knowledge can usually still be explored using computational experiments on carefully chosen model problems. We use painstaking numerical testing to formulate conjectures which can later be subjects of in-depth mathematical study.

A few links to collaborators (former and future), ex-students/postdocs, other research colleagues and friends:

Assyr Abdulle (EPFL, Lausanne)

Teijo Arponen (Aalto, Finland)

Eric Barth (Kalamazoo)

Stephen Bond (Sandia, New Mexico)

Nawaf Bou-Rabee (Rutgers)

Chris Budd (Bath)

Giovanni Ciccott (Dublin/Rome)

Carl Dettmann (Bristol)

Ron Elber (Texas)

Jason Frank (CWI-Amsterdam)

Gero Friesecke (Munich)

Martin Hairer (Warwick)

Ernst Hairer (Geneva)

Arieh Iserles (Cambridge)

Markos Katsoulakis (Amherst)

Claude Lebris (ENPC, Paris)

Frederic Legoll (ENPC, Paris)

Tony Lelievre (ENPC, Paris)

Mitch Luskin (Minnesota)

Giacomo Mazzi (KU Leuven)

Houman Owhadi (Caltech)

Greg Pavliotis (Imperial)

Oliver Penrose (Heriot-Watt)

Petr Plechac (Delaware)

Sebastian Reich (Potsdam)

Christof Schuette (Berlin)

Robert Skeel (Purdue)

Gabriel Stoltz (ENPC, Paris)

Andrew Stuart (Warwick)

Chris Sweet (Notre Dame)

Florian Theil (Warwick)

Michael Tretyakov (Leicester)

Mark Tuckerman (NYU)

Eric Vanden-Eijnden (NYU)

Jonathan Weare (Chicago)

Jonannes Zimmer (Bath)

Teijo Arponen (Aalto, Finland)

Eric Barth (Kalamazoo)

Stephen Bond (Sandia, New Mexico)

Nawaf Bou-Rabee (Rutgers)

Chris Budd (Bath)

Giovanni Ciccott (Dublin/Rome)

Carl Dettmann (Bristol)

Ron Elber (Texas)

Jason Frank (CWI-Amsterdam)

Gero Friesecke (Munich)

Martin Hairer (Warwick)

Ernst Hairer (Geneva)

Arieh Iserles (Cambridge)

Markos Katsoulakis (Amherst)

Claude Lebris (ENPC, Paris)

Frederic Legoll (ENPC, Paris)

Tony Lelievre (ENPC, Paris)

Mitch Luskin (Minnesota)

Giacomo Mazzi (KU Leuven)

Houman Owhadi (Caltech)

Greg Pavliotis (Imperial)

Oliver Penrose (Heriot-Watt)

Petr Plechac (Delaware)

Sebastian Reich (Potsdam)

Christof Schuette (Berlin)

Robert Skeel (Purdue)

Gabriel Stoltz (ENPC, Paris)

Andrew Stuart (Warwick)

Chris Sweet (Notre Dame)

Florian Theil (Warwick)

Michael Tretyakov (Leicester)

Mark Tuckerman (NYU)

Eric Vanden-Eijnden (NYU)

Jonathan Weare (Chicago)

Jonannes Zimmer (Bath)

*Book:* Simulating Hamiltonian Dynamics, B. Leimkuhler and S. Reich, Cambridge University Press, 2005.

The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics

B. Leimkuhler, C. Matthews and G. Stoltz, , , 2013. PDF WEB

Robust and efficient configurational molecular sampling via Langevin Dynamics

B. Leimkuhler and C. Matthews, Journal of Chemical Physics, 138: , 2013. JOURNAL PDF WEB

Stochastic resonance-free multiple time-step algorithm for molecular dynamics with very large time steps

B. Leimkuhler, D. Margul and M. Tuckerman, Molecular Physics, , 2013. JOURNAL PDF WEB

Weakly coupled heat bath models for Gibbs-like invariant states in nonlinear wave equations

J. Bajars, J. Frank and B. Leimkuhler, Nonlinearity, 26: 1945-1973, 2013. JOURNAL PDF

Rational construction of stochastic numerical methods for molecular sampling

B. Leimkuhler and C. Matthews, Applied Mathematics Research Express, 2013: 34-56, 2013. JOURNAL PDF WEB

Comparing the efficiencies of stochastic isothermal molecular dynamics methods

B. Leimkuhler, E. Noorizadeh and O. Penrose, Journal of Statistical Physics, 143: 921-942, 2011. JOURNAL PDF

Adaptive stochastic methods for sampling driven molecular systems

A. Jones and B. Leimkuhler, Journal of Chemical Physics, 135: Preprint(2011) . JOURNAL

Dimensional reductions for the computation of time-dependent quantum expectations

B. Leimkuhler and G. Mazzi, SIAM Journal on Scientific Computing, 33: 2024-2038, 2011. JOURNAL

Stochastic-dynamical thermostats for constraints and stiff restraints

J. Bajars, J. Frank and B. Leimkuhler, The European Physical Journal, 200: 131-152, 2011. JOURNAL PDF

Simplified modelling of a thermal bath, with application to a fluid vortex system

S. Dubinkina, J. Frank, and B. Leimkuhler, SIAM Multiscale Modelling and Simulation, 8: 1882-1901, 2010. JOURNAL PDF

Generalized Bulgac-Kusnezov methods for sampling of the Gibbs-Boltzmann measure

B. Leimkuhler, Physical Review E, 81: Preprint(2010) . JOURNAL PDF

A gentle stochastic thermostat for molecular dynamics

B. Leimkuhler, E. Noorizadeh and F. Theil, Journal of Statistical Physics, 135: 261-277, 2009. JOURNAL PDF

A Metropolis-adjusted Nose-Hoover thermostat

B. Leimkuhler and S. Reich, ESAIM:Mathematical modelling and numerical analysis, 43: 743-755, 2009. JOURNAL PDF

A temperature control technique for nonequilibrium molecular simulation

B. Leimkuhler, F. Legoll and E. Noorizadeh, Journal of Chemical Physics, 128: Preprint(2008) . JOURNAL PDF

Molecular dynamics and the accuracy of numerically computed averages

S. Bond and B. Leimkuhler, Acta Numerica, 16: 1-65, 2007. JOURNAL PDF

Stabilized integration of Hamiltonian systems with hard-sphere inequality constraints

S. Bond and B. Leimkuhler, SIAM Journal on Scientific Computing, 30: 134-147, 2007. JOURNAL PDF

Molecular simulation in the canonical ensemble and beyond

Z. Jia and B. Leimkuhler, ESAIM:Mathematical modelling and numerical analysis, 41: 333-350, 2007. JOURNAL

Rapid thermal equilibration of coarse-grained molecular dynamics

S. Gill, Z. Jia, B. Leimkuhler and A. Cocks, Physical Review B, 73: Preprint(2006) . JOURNAL PDF

Geometric integrators for multiple timescale simulation

Z. Jia and B. Leimkuhler, Journal of Physics A, 39: 5379-5403, 2006. JOURNAL

Approach to thermal equilibrium in biomolecular simulation

E. Barth, B. Leimkuhler, and C. Sweet, New Algorithms for Macromolecular Simulation (Springer Lecture Notes in Computational Science and Engineering), 49: 125-140, 2006. JOURNAL PDF

A projective thermostatting technique

Z. Jia and B. Leimkuhler, SIAM Multiscale Modelling and Simulation, 4: 563-583, 2005. JOURNAL PDF

A Hamiltonian formulation for recursive multiple thermostats in a common timescale

B. Leimkuhler and C. Sweet, SIAM Journal on Applied Dynamical Systems, 4: 187-216, 2005. JOURNAL PDF

An efficient geometric integrator for thermostatted anti-/ferromagnetic Models

T. Arponen and B. Leimkuhler, BIT Numerical Mathematics, 44: 403-424, 2004. JOURNAL PDF

The canonical ensemble via symplectic integration using Nose and Nose-Poincare chains

B. Leimkuhler and C. Sweet, Journal of Chemical Physics, 121: 108-117, 2004. JOURNAL PDF

A parallel multiple time-scale reversible integrator for dynamics simulation

Z. Jia and B. Leimkuhler, Future Generation Computer Systems, 19: 415-424, 2003. JOURNAL PDF

Generating generalized distributions from dynamical simulation

E. Barth, B. Laird and B. Leimkuhler, Journal of Chemical Physics, 118: 5759-5769, 2003. JOURNAL PDF

Generalized dynamical thermostatting technique

B. Laird and B. Leimkuhler, Physical Review E 68, 016704 (2003), 68: Preprint(2003) . JOURNAL PDF

On the approximation of Feynman-Kac path integrals

S. Bond, B. Laird, and B. Leimkuhler, Journal of Computational Physics, 185: 472-483, 2003. JOURNAL PDF

An efficient multiple time-scale reversible integrator for the gravitational N-body problem

B. Leimkuhler, Applied Numerical Mathematics, 43: 175-190, 2002. JOURNAL PDF

A separated form of Nosé dynamics for constant temperature and pressure simulation

B. Leimkuhler, Computer Physics Communications, 148: 206-213, 2002. JOURNAL PDF

A test set for molecular dynamics

E. Barth, B. Leimkuhler, and S. Reich, Lecture Notes in Computational Science and Engineering, 24: 73-103, 2002. JOURNAL PDF

A reversible averaging integrator for multiple timescale dynamics

B. Leimkuhler and S. Reich, Journal of Computational Physics, 171: 95-114, 2001. JOURNAL PDF

Explicit, time-reversible and variable stepsize integration

T. Holder, B. Leimkuhler, and S. Reich, Applied Numerical Mathematics, 39: 367-377, 2001. JOURNAL PDF

Scaling invariance and adaptivity

C. Budd, B. Leimkuhler and M. Piggott, Applied Numerical Mathematics, 39: 261-288, 2001. JOURNAL PDF

Molecular dynamics algorithms for mixed hard-core/continuous potentials

Y. Houndonougbo, B. Laird, and B. Leimkuhler, Journal of Molecular Physics, 98: 309-316, 2000. JOURNAL PDF

A time-reversible, regularized, switching integrator for the N-body problem

A. Kvaerno and B. Leimkuhler, SIAM Journal on Scientific Computing, 22: 1016-1035, 2000. JOURNAL PDF

Geometric integrators based on scaling and switching

B. Leimkuhler, Proceedings of the Equadiff Conference (Berlin, 1999), World Scientific, 988-993, 2000. PDF

A Semi-explicit, variable-stepsize integrator for constrained dynamics

E. Barth, B. Leimkuhler, and S. Reich, SIAM Journal on Scientific Computing, 21: 1027-1044, 1999. JOURNAL PDF

Reversible adaptive regularization: perturbed Kepler motion and classical atomic trajectories

B. Leimkuhler, Philosophical Transactions of the Royal Society of London A, 357: 1101-1134, 1999. JOURNAL PDF

Reversible adaptive regularization methods for atomic N-body problems in applied fields

B. Leimkuhler, Applied Numerical Mathematics, 29: 31-43, 1999. JOURNAL PDF

Asymptotic error analysis of the Adaptive Verlet method

S. Cirilli, E. Hairer and B. Leimkuhler, BIT Numerical Mathematics, 39: 25-33, 1999. JOURNAL PDF

The Nose-Poincare method for constant temperature molecular dynamics

S. Bond, B. Leimkuhler and B. Laird, Journal of Computational Physics, 151: 114-134, 1999. JOURNAL PDF

Timestep acceleration of waveform relaxation

B. Leimkuhler, SIAM Journal on Numerical Analysis, 35: 31-50, 1998. JOURNAL PDF

The Adaptive Verlet method

W. Huang and B. Leimkuhler, SIAM Journal on Scientific Computing, 18: 239-256, 1997. JOURNAL PDF

Geometric integrators for classical spin systems

J. Frank, W. Huang and B. Leimkuhler, Journal of Computational Physics, 133: 160-172, 1997. JOURNAL PDF

Symplectic splitting methods for rigid body molecular dynamics

A. Dullweber, B. Leimkuhler and R. McLachlan, Journal of Chemical Physics, 107: 5840-5851, 1997. JOURNAL PDF

A symplectic integrator for Riemannian manifolds

B. Leimkuhler and G. Patrick, Journal of Nonlinear Science, 6: 367-384, 1996. JOURNAL PDF

Orthosymplectic integration of linear Hamiltonian systems

B. Leimkuhler and E. Van Vleck, Numerische Mathematik, 77: 269-282, 1996. JOURNAL PDF

Symplectic integration of constrained Hamiltonian systems

B. Leimkuhler and S. Reich, Mathematics of Computation, 63: 589-605, 1994. JOURNAL PDF

Algorithms for constrained molecular dynamics

E. Barth, K. Kuczera, B. Leimkuhler and R. Skeel, Journal of Computational Chemistry, 16: 1192-1209, 1995. JOURNAL PDF

Symplectic numerical integrators in constrained Hamiltonian systems

B. Leimkuhler and R. Skeel, Journal of Computational Physics, 112: 117-125, 1994. JOURNAL PDF

Estimating waveform relaxation convergence

B. Leimkuhler, SIAM Journal on Scientific Computing, 14: 872-889, 1993. JOURNAL PDF

Numerical solution of differential-algebraic equations for constrained mechanical motion

C. Fuehrer and B. Leimkuhler, Numerische Mathematik, 59: 55-69, 1991. JOURNAL PDF

Automatic integration of Euler-Lagrange equations with constraints

C.W. Gear, B. Leimkuhler, G.K. Gupta, Journal of Computational and Applied Mathematics, 12-13: 77-90, 1985. JOURNAL PDF