The course focuses on the design of numerical methods for molecular dynamics simulation, in both its deterministic and stochastic forms. The primary source of material is my book with Charlie Matthews Molecular Dynamics, supplemented by the articles mentioned below. Copies of the book will be available on-site. The course is divided into four lectures. The lectures will be given using data projector. Slides will be posted here as soon as available. It is recommended to obtain the free software MD.M (works with MATLAB, Octave) and to perform some of the demo simulations on small systems, especially if this is your first exposure to molecular dynamics.
Lecture 1. Molecular dynamics models goals and purposes, numerical methods by splitting, analysis using Baker-Campbell-Hausdorff expansion, and examples.
Materials for Lecture 1. Chapters 1-3 of Molecular Dynamics.
Slides for Lecture 1 [uploaded 3.4.2017].
Lecture 2. Ensembles for molecular simulation (microcanonical, canonical), stochastic differential equations (Brownian/Langevin dynamics), numerical methods for Langevin dynamics (especially splitting methods), error analysis, examples in biomolecular dynamics.
Materials for Lecture 2. Chapters 6-7 of Molecular Dynamics, ref. .
Slides for Lecture 2 [uploaded 3.4.2017].
Lecture 3. The timestep problem in MD, holonomic constraints, SHAKE and RATTLE discretization, stochastic schemes, geodesic integration, isokinetic MD, examples.
Materials for Lecture 3. Chapters 4, 6 of Molecular Dynamics, ref. [2,3].
Slides for Lecture 3 [uploaded 5.4.2017].
Lecture 4. Extended systems, thermostats, Nose dynamics, pairwise adaptive schemes and shear flows, an application in Bayesian inference.
Materials for Lecture 4. Chapter 8 of Molecular Dynamics, ref. [4,5].
PhD 1988, MS 1986, University of Illinois
BS 1983, Purdue University
|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|
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 2014 I was heavily involved with the EPSRC (Science and Innovation) Centre for Numerical Algorithms and Intelligent Software (NAIS) which links Edinburgh, Heriot-Watt and Strathclyde Universities. I was also 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.
In recent years I have been working mor eand more in the areas of data science. I was involved with setting up the Alan Turing Institute and I am currently a Faculty Fellow of the ATI. This means that I spend part of my time at the new Alan Turing Institute headquarters in the British Library. The directions in data science that I am exploring include the use of stochastic differential equations to explore Bayesian parameterization of complicated distributions for data analysis. This work has potential high impact in neural network training and applications in "big data". Much of my work on molecular simulation can be applied in data science through this approach.
I currently hold a new EPSRC grant in Data-Driven Coarse-Graining using Space-Time Diffusion Maps. This is an exciting project that bridges the data science and molecular modelling worlds. I also have an ERC-funded collaboration in Biological Modelling with V. Danos in Informatics.
Associations, Positions, Roles
|2016--||Research Director, School of Mathematics|
|2016--||Co-Director, Maxwell Institute for Mathematical Sciences|
|2016--||Faculty Fellow, Alan Turing Institute|
|2015-2016||Science Committee, Alan Turing Institute|
|2014-15||IMA Leslie Fox Prize Adjudication Committee|
|2014--||Fellowships subpanel, Royal Society of Edinburgh|
|2014--||Fellow of the Royal Society of Edinburgh (FRSE)|
|2012--||Fellow of the Institute of Mathematics and its Applications (FIMA)|
|2012--||2014 Co-Director, Maxwell Institute for Mathematical Sciences|
|2012||Senior Research Fellow, Dutch Science Foundation|
|2011||JT Oden Fellowship, University of Texas|
|2009--||Member, Steering Committee of the Centre for Numerical Algorithms (NAIS)|
|2009--||11 Director, NAIS|
|2009||SIAM Dahlquist Prize Selection Committee|
|2008-11||Deputy Director, Maxwell Institute for Mathematical Sciences|
|2007-2015||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|
|2014--||Proceedings of the Royal Society A|
|2013--||European Journal of Applied Mathematics|
|2012--||Journal of Computational Dynamics (AIMS)|
|2008--||IMA Journal on Numerical Analysis|
|2009-13||SIAM Journal on Numerical Analysis|
|2001-7||SIAM Journal on Scientific Computing|
Grants and Projects
|2016--||EPSRC International Centre for Mathematical Sciences (co-I)|
|2016 --||EPSRC Data-Driven Coarse-Graining using Space-Time Diffusion Maps (PI)|
|2014||Adaptive Collective Variables: Automatic Identification and Application of Multiresolution Modelling (co-I)|
|2014||Adaptive QM/MM simulations (co-I)|
|2014||Highly efficient time-domain quantum chemistry algorithms (co-I)|
|2013 -- 2016||NSF-EPSRC SI2-CHE:ExTASY Extensible Tools for Advanced Sampling and analYsis (Edinburgh PI)|
|2013 -- 2018||ERC Advanced Grant in Ruled-Based Modelling for Biology (Co-I)|
|2009 -- 2014||EPSRC (S&I)/SFC Numerical Algorithms and Intelligent Software for the Evolving HPC Platform (co-I)|
|2008 -- 2010||EPSRC Network (Bath, Bristol, Edinburgh, Warwick) “Mathematical Challenges of Molecular Dynamics”|
|2004 -- 2007||Australian RC Geometric Integration (co-I)|
|2004||US NIH Algorithms for Macromolecular Modelling (co-I)|
|2004 -- 2005||EPSRC Algorithms for Macromolecular Modelling (PI)|
|2004 -- 2007||EPSRC Developing an Efficient Method for Locating Periodic Orbits (co-I)|
|2003 -- 2005||SRIF Advanced Computing Facility (HPC) (co-I)|
|2002 -- 2004||Australian RC Geometric Numerical Integration (co-I)|
|2001 -- 2004||EPSRC A Mixed Atomistic and Continuum Model for Crossing Multiple Length and Time Scales (co-I)|
|2001 -- 2004||EPSRC Geometric Integrators for Switched and Multiple Time-scale Dynamics (PI)|
|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
Principal Organiser, Multiscale Molecular Modelling, Edinburgh, 2010
Principal Organiser, State-of-the-art Algorithms for Molecular Dynamics, Edinburgh 2012
Co-organiser, Complex Molecular Systems, Lorentz Center, Leiden, 2012
Organiser, Multiscale Computational Methods in Materials Modelling, Edinburgh 2014
Co-organiser, three Alan Turing Institute ``scoping workshops'', 2015-16
Co-organiser, Stochastic numerical algorithms, multiscale modeling and high-dimensional data analytics, ICERM/Brown University, 2016
Co-organiser, Trends and advances in Monte Carlo sampling algorithms, SAMSI/Duke University, 2017
Molecular Dynamics (Springer, 2015) one of the first mathematical books on the subject.
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.
Please see the separate "books" page for more details on these.
Most of my articles are uploaded to the arXiv. For complete, up-to-date lists of publications, please refer to my papers page or see my page in Google Scholar.
B. Leimkuhler and C. Matthews
Springer, Berlin, 2015.
Link to Publisher's Website: Springer-Verlag
|This book describes the mathematical underpinnings of algorithms used for molecular dynamics simulation, including both deterministic and stochastic numerical methods. Molecular dynamics is one of the most versatile and powerful methods of modern computational science and engineering and is used widely in chemistry, physics, materials science and biology. Understanding the foundations of numerical methods means knowing how to select the best one for a given problem (from the wide range of techniques on offer) and how to create new, efficient methods to address particular challenges as they arise in complex applications. Aimed at a broad audience, this book presents the basic theory of Hamiltonian mechanics and stochastic differential equations, as well as topics including symplectic numerical methods, the handling of constraints and rigid bodies, the efficient treatment of Langevin dynamics, thermostats to control the molecular ensemble, multiple time-stepping, and the dissipative particle dynamics method.|
1. Introduction; 2. Numerical integrators; 3. Analyzing geometric integrators; 4. The stability threshold; 5. Phase space distributions and microcanonical averages; 6. The canonical distribution and stochastic differential equations; 7. Numerical methods for stochastic differential equations; 8. Extended variable methods.
Simulating Hamiltonian Dynamics
B. Leimkuhler and S. Reich
Cambridge University Press, 2005.
Link to Publisher's Website: Cambridge University Press
Geometric integrators are time-stepping methods, designed such that they exactly satisfy conservation laws, symmetries or symplectic properties of a system of differential equations. In this book the authors outline the principles of geometric integration and demonstrate how they can be applied to provide efficient numerical methods for simulating conservative models. Beginning from basic principles and continuing with discussions regarding the advantageous properties of such schemes, the book introduces methods for the N-body problem, systems with holonomic constraints, and rigid bodies. More advanced topics treated include high-order and variable stepsize methods, schemes for treating problems involving multiple time-scales, and applications to molecular dynamics and partial differential equations. The emphasis is on providing a unified theoretical framework as well as a practical guide for users. The inclusion of examples, background material and exercises enhance the usefulness of the book for self-instruction or as a text for a graduate course on the subject.
• Thorough treatment of a relatively new subject, covers theory, applications and also gives practical advice on implementing the techniques • Emphasis on 'efficient' numerical methods • Large number of examples and exercises
1. Introduction; 2. Numerical methods; 3. Hamiltonian mechanics; 4. Geometric integrators; 5. The modified equations; 6. Higher order methods; 7. Contained mechanical systems; 8. Rigid Body dynamics; 9. Adaptive geometric integrators; 10. Highly oscillatory problems; 11. Molecular dynamics; 12. Hamiltonian PDEs.
|Stochastic dynamics on large networks: prediction and inference||MPI Dresden||15.10.2018|
|Large scale activated event simulations||Erwin Schroedinger Institute||1.10.2018|
|Royal Society Theo Murphy Conference on Multiresolution Simulations of Intracellular Processes||Chicheley Hall||24.9.2018|
|Applied Math Summer School||Peking University||23.7.2018|
|Data driven modelling of complex systems||ATI-London||8.5.2018|
|Symposium on Data-driven Methods in Molecular Simulations of Soft-Matter Systems (SYMS)||EPS-DPG Joint Conference Berlin||11.3.2018|
|Methods for particle systems with multiple scales||WIAS/Berlin||29.5.2017|
|Stochastic dynamics out of equilibrium||CRM/Marseille||3.4.2017|
|Numerical aspects of nonequilibrium dynamics||IHP/Paris||25.4.2017|
|New trends in Mathematical Physics at the interface of Analysis and Probability||University College London||15.2.2017|
|GLE2017 (Generalized Langevin Equation)||Kings College, London||12.1.2017|
|Collective variables in classical mechanics||IPAM/UCLA||24.10.2016|
|Multiscale Simulation Methods in Soft Matter Systems II||CECAM/TU Darmstadt||4.10.2016|
|Stochastic numerical algorithms, multiscale modeling and high-dimensional data analytics||ICERM/Brown University||18.7.2016|
|Big data for the physical sciences||Turing Institute/London||13.1.2016|
|Data Intensive and Extreme Scale Numerical Simulation||Turing Institute/London||5.1.2016|
|Stochastic Dynamical Systems in Biology||Newton Institute/Cambridge||4.1.2016|
|Challenges in Stat Mech||Imperial College||7.12.2015|
|Predictive Multiscale Materials||Cambridge (Turing Gateway)||1.12.2015|
|Probabilistic Numerics||Turing Institute/London||19.11.2015|
|Evolution Equations||Maxwell Institute/Edinburgh||16.9.2015|
|International Conference on Industrial and Applied Mathematics (ICIAM)||Beijing||10.8.2015|
|Mathematics in Data Science||ICERM/Brown University||28.7.2015|
|Free Energy Calculations||Banff/Oaxaca||19.7.2015|
|Mathematical Methods in Quantum Molecular Dynamics||Oberwolfach||31.5.2015|
|Nosé Dynamics 30 Years||Tokyo||10.11.2014|
|Advances in Enhanced Sampling Methods||Telluride, Colorado||1.7.2014|
|Searching for Reaction Coordinates and Order Parameters||Telluride, Colorado||7.7.2014|
|Multiscale computational methods in materials modelling||Edinburgh||18.6.2014|
|Computational methods for statistical mechanics||ICMS/Edinburgh||2.6.2014|
|Symposium on Statistical Mechanics: Computational coarse-graining of many-body systems||Warwick||9.12.2013|
|Complex Molecular Systems||Lorentz Centre/Leiden||13.8.2013|