MSLMS

Numerical Methods & Algorithms

Advanced numerical methods and algorithms can help applied scientists speed up simulations, improve their reliability and use most efficiently available High Performance computing resources

Stochastic and deterministic enhanced sampling algorithms

One of the computational grand challenges is the development of methodologies for efficient and accurate sampling of high-dimensional spaces. If met, significant progress will be made in understanding many important phenomena in life and physical sciences.

Development of novel efficient Hamiltonian Monte Carlo (HMC) and Modified Hamiltonian Monte Carlo (MHMC) methods, which combine stochastic Markov Chain Monte Carlo (MCMC) with deterministic Hamiltonian Dynamics, is one of our long-standing research interests. Our first methods [1, 2] of these two classes, patented in the US and GB, have clearly demonstrated their superiority in sampling performance over traditional sampling techniques, such as conventional MCMC, molecular dynamics (MD), Hybrid/Hamiltonian Monte Carlo. Today we work on adapting our methods to various simulation scales and statistical ensembles to bridge the time, space and precision scale gaps between simulations and the phenomena of interest. The resulting methods - canonical, constant-pressure, grand-canonical, meso- and multiple-time-stepping variants of HMC and MHMC [2 - 4], as well as the Mix & Match HMC [5], developed specifically for Bayesian inference aim at a wide range of applications.

In addition, we re-design original Monte Carlo algorithms to address particular challenges in applications of interest, e.g. polymerization reactions with delays, quantum measurements, population dynamics, metabolic modelling [6-9].   

We also continue to improve performance of the developed samplers by introducing novel adaptive schemes, numerical integrators and parallel algorithms.

Numerical Integration

It often happens that differential equations, widely used in applied mathematics to describe various phenomena, do not have solutions expressible in closed form, and require numerical methods for finding approximate solutions.

Construction of robust and computationally efficient problem-specific integration schemes is, therefore, an important step in facilitating progress in fields as diverse as Classical and Quantum Mechanics, Materials Science, Biology or Engineering. 

Common threads of topics we address are geometric integration and splitting algorithms.

Our objectives are to further theoretical knowledge on splitting integrators and other geometric methods, to construct new schemes and apply them to a number of fields. With that in mind, we develop new adaptive multi-stage splitting methods for molecular dynamics and (modified) Hamiltonian/Hybrid Monte Carlo sampling in molecular simulations [10-11] and computational statistics [12], and also devise splitting schemes for solving Population Balance Equations (PBE) [13].

Combined with novel sampling techniques and computational models being introduced by the group, the integrators will make possible accurate modelling of very large complex systems of varied nature [14-16]. Currently, many such systems are beyond the capabilities of the existing modelling approaches.

Machine-learning and data-driven approaches

We incorporate in our modelling methodologies various machine-learning-driven algorithms, to enable large-scale high-resolution simulation in energy and health applications. Bayesian and frequentist machine learning methods assist in performing global optimizations of molecular structures, and help train atomistic force fields in simulation of energy materials. Machine learning algorithms also find their use in tuning performance and accuracy of the sampling techniques, applied for predictive modelling of microbiome and human metabolism, as well as in the analysis of clinical and RNA-Seq data for cancer research [17].  

We have also explored the development of data-mining methodologies for the unsupervised analysis of atomistic trajectories. In particular, our recent density-based CrySF method [18] enables the automated characterization of structure and diffusion from the simulation of ordered and disordered solids. The method and its implementation are both robust and versatile, requiring few tuning parameters, and is applicable across a wide range of materials.

Metaheuristic schemes for structural optimization

Atomistic simulations have substantially advanced our understanding of structure–performance relationships in materials science. Yet even small variations in the initial atomic distribution can lead to pronounced differences in the predicted properties. This challenge becomes particularly severe in systems with a combinatorially large configuration space, where exhaustive sampling of all possible states is practically unattainable. The resulting combinatorial explosion (often termed the “exponential wall”) restricts the effectiveness of conventional methods, especially for systems featuring high chemical or structural complexity.

To overcome these limitations, we introduce a novel algorithm designed for efficient exploration of configuration space, leveraging customizable classical energy models in combination with metaheuristic optimization strategies. Implemented within the Search for Optimal Structure (SOS) code, our algorithm balances speed and accuracy while remaining lightweight and straightforward to integrate into existing workflows. It is fully compatible with any metaheuristic solver and accommodates a broad spectrum of energy models, ranging from simple electrostatic interactions to more sophisticated pairwise potentials.

 

Computational Toolkits

Accurate prediction of complex physical and social processes is often impossible without full utilisation of the capabilities of modern HPC

The methodologies derived in MSLMS are implemented in the in-house or well-established open-source software packages. MSLMS in-house computer codes include MultiHMC-GROMACS [4, 10, 11] (Figure 5) - enhanced sampling molecular dynamics and Hybrid Monte Carlo methods in GROMACS package; CrySF [18] (Figure 6 ) - postprocessing analysis of molecular dynamics trajectories; SOS (Figure 7) - efficient structural optimization of solid-state systems with a large number of possible configurations; HaiCS [5, 12] (Figure 8) - statistical sampling and parameter estimation through Bayesian inference using Hamiltonian Monte Carlo methods; CRadPol [19] - simulation of controlled radical polymerization using experimental observations; Ddpm [20] - simulation of dynamic development of particles morphology; complex angular momenta (CAM) analysis suite (PADE II [21], ICS_Regge [22], DCS_Regge [23]); Bayes-cancer [17] - analysis of integrated cell-patient data.

Together with our collaborators from Lawrence Berkeley National Laboratory (USA) we contributed to the development of BayFlux [9] - quantitative metabolic modelling and Biodesign (iGR1773) [25] - biosystems design for production of biofuels.

 

Modelling and Simulation

Modelling and simulation play a crucial role in theorizing and understanding complex systems. Our group employs the most efficient simulation techniques (in-house or external) to perform multiscale simulation of physical systems and Bayesian inference (Figure 9).

The computational models, proposed by the MSLMS are implemented in the MSLMS software packages and include the PBE model for study of formation and evolution of multi-phase polymer particles morphology [20] (Figure 10); the effective medium model for estimation of conductivities of cubic/tetragonal phase mixtures of solid electrolytes [26]; the CAM (Complex Angular Momenta Analysis) models for elastic, inelastic and reactive integral and differential cross sections in elementary chemical reactions [21-24] (Figure 11).

 

References

[1] Akhmatskaya E., Reich S. GSHMC: An Efficient Method for Molecular Simulation. Journal of Computational Physics 227 (2008) 4934 – 4954.

[2] Akhmatskaya E., Reich S. New Hybrid Monte Carlo Methods for Efficient Sampling: from Physics to Biology and Statistics. Progress in Nuclear Science and Technology 2 (2011) 447-462.

[3] Escribano B., Akhmatskaya E., Reich S., Azpiroz J.M. Multiple-time-stepping Generalized Hybrid Monte Carlo Methods. Journal of Computational Physics 280 1 (2015) 1 – 20.

[4] Fernández-Pendás M., Escribano B., Radivojević T., Akhmatskaya E. Constant Pressure Hybrid Monte Carlo Simulations in GROMACS. Journal of Molecular Modeling 20 (2014) 2487.

[5] Radivojevic T., Akhmatskaya E. Modified Hamiltonian Monte Carlo for Bayesian Inference. Statistics and Computing 30 (2020) 377-404.

[6] Sokolovski D., Rusconi S., Brouard S, Akhmatskaya E. Re-examination of Continuous Fuzzy Measurement on Two-level Systems. Phys. Rev. A 95 (2017) 042111.

[7] Sokolovski D., Rusconi S., Akhmatskaya E., Asua J.M. Non-Markovian Effects in the Growth of a Polymer Chain. Proceedings of the Royal Society A 471 (2015) 20140899.

[8] Rusconi S., Akhmatskaya E., Sokolovski D., Ballard N., de la Cal J.C. Relative Frequencies of Constrained Events in Stochastic Processes: an Analytical Approach. Phys. Rev. E 92 (4) (2015) 043306.

[9] Backman T.W.H., Schenk C., Radivojevic T., Ando D., Singh J., Czajka J.J., Costello Z., Keasling J.D., Tang Y., Akhmatskaya E., Garcia Martin H. BayFlux: A Bayesian Method to Quantify Metabolic Fluxes and their Uncertainty at the Genome Scale. PLOS Computational Biology 19(11) (2023) e1011111.

[10] Fernández-Pendás M., Akhmatskaya E., Sanz-Serna J.M. Adaptive Multi-stage Integrators for Optimal Energy Conservation in Molecular Simulations. Journal of Computational Physics 327 (2016) 434-449.

[11] Akhmatskaya E., Fernández-Pendás M., Radivojevic T., Sanz-Serna J.M. Adaptive Splitting Integrators for Enhancing Sampling Efficiency of Modified Hamiltonian Monte Carlo Methods in Molecular Simulation. Langmuir 33 (43) (2017) 11530-11542.

[12] Nagar L., Fernández-Pendás M., Sanz-Serna J.M., Akhmatskaya E.  Adaptive Multi-stage Integration Schemes for Hamiltonian Monte Carlo. Journal of Computational Physics 502 (2024) 112800.

[13] Rusconi S., Dutykh D., Zarnescu A., Sokolovski D., Akhmatskaya E. An Optimal Scaling to Computationally Tractable Dimensionless Models: Study of Latex Particles Morphology Formation. Computer Physics Communications 247 (2020) 106944.

[14] García Daza F.A., Bonilla M.R., Llordés A., Carrasco J., Akhmatskaya E. Atomistic Insight into Ion Transport and Conductivity in Ga/Al-Substituted Li₇La₃Zr₂O₁₂ Solid Electrolytes. ACS Appl. Mater. Interfaces 11(1) (2019) 753-765.

[15] Bonilla M.R., García-Daza F., Cortés H.A., Carrasco J., Akhmatskaya E. On the Interfacial Lithium dynamics in Li₇La₃Zr₂O₁₂: Poly(ethylene oxide) (LiTFSI) Composite Polymer-ceramic Solid Electrolytes under Strong Polymer Phase Confinement. J. Colloid Interface Sci. 623 (2022) 870-882.

[16] Cortés H.A., Bonilla M.R., Marinero E., Carrasco J., Akhmatskaya E. Anion Trapping and Ionic Conductivity Enhancement in PEO-based Composite Polymer-Li7La3Zr2O12 Electrolytes: the Role of the Garnet Li Molar Content. Macromolecules 56 (2023) 4256-4266.

[17] Parga-Pazos M., Cusimano N., Rábano M., Akhmatskaya E., dM. Vivanco M. A Novel Mathematical Approach for the Analysis of Integrated Cell-patient Data Uncovers a 6-gene Signature Linked to Endocrine Therapy Resistance. Laboratory Investigation 104 (2024) 100286.

[18] Cortés H.A., Bonilla M.R., Akhmatskaya E. Unsupervised Density-based Method for Analysing Ion Mobility in Crystalline Solid-state Electrolytes. npj Comput Mater 11 (2025) 368.

[19] Ballard N., Rusconi S., Akhmatskaya E., Sokolovski D., de la Cal J., Asua J.M. The Impact of Competitive Processes on Controlled Radical Polymerization. Macromolecules 47 (19) (2014) 6580–6590.

[20] Rusconi S., Schenk C., Zarnescu A., Akhmatskaya E. Reducing Model Complexity by Means of the Optimal Scaling: Population Balance Model for Latex Particles Morphology Formation. Applied Mathematics and Computation (2022) 443 (2023) 127756.

[21] Sokolovski D., Akhmatskaya E., Sen S. Extracting Resonance Poles from Numerical Scattering Data: Type-II Padé Reconstruction. Computer Physics Communications 182 (2) (2011) 448-466.

[22] Akhmatskaya E., Sokolovski D., Echeverría-Arrondo C. Numerical Regge Pole Analysis of Resonance Structures in Elastic, Inelastic and Reactive State-to-state Integral Cross Sections. Computer Physics Communications 185 (7) (2014) 2127–2137.

[23] Akhmatskaya E., Sokolovski D. Numerical Regge Pole Analysis of Resonance Structures in State-to-state Reactive Differential Cross Sections. Computer Physics Communications 277 (2022) 108370.

[24] Sokolovski D., De Fazio D., Akhmatskaya E. A Transition State Resonance Radically Reshapes Angular Distributions of the F + H₂ -> F H (vf = 3) + H Reaction in the 62.09-101.67 meV Energy Range. ACS Physical Chemistry Au 5(2) (2025) 219-226.

[25] Roell G., Schenk C., Anthony W., Carr R., Ponukumati A., Kim J., Akhmatskaya E., Foston M., Dantas G., Moon T.S., Tang Y., Garcia Martin H. A High-Quality Genome-Scale Model for Rhodococcus opacus Metabolism. ACS Synthetic Biology 12 (6) (2023) 1632-1644.

[26] Bonilla M.R., García Daza F., Carrasco J., Akhmatskaya E. Exploring Li-ion Conductivity in Cubic,Tetragonal and Mixed-phase Al-substituted LLZO Using Atomistic Simulations and Effective Medium Theory. Acta Materialia 175 (2019) 426-435.