MSLMS

Created in September 2010 and led by Ikerbasque Research Professor E. Akhmatskaya, the MSLMS group has been extended in 2021 with the ESM - BCAM Severo Ochoa Strategic Lab on Modelling with PDEs in Mathematical Biology headed by Prof. J.A. Carrillo (University of Oxford, UK). The research of the group and Lab is complementary (Figure 4).

Figure 4. MSLMS-BCAM-So Lab synergy

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.

Enhanced sampling stochastic and deterministic algorithms

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] 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 (Figure 5). Today we work on adapting our methods to various simulation scales and statistical ensembles, in order 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 [2] of HMC and MHMC as well as the Mix & Match HMC [3], 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 [4-5].    

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

Figure 5. In-house MHMC (GSHMC) method vs. traditional sampling techniques for molecular simulation. 

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 [6-7].

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 [8] (Figure 6). Currently, many such systems are beyond the capabilities of the existing modelling approaches.
   

Figure 6. Ga³⁺ vs. Al³⁺ doping in LLZO: Using our enhanced MHMC sampler (GSHMC) combined with the adaptive modified splitting approach MAIA developed in MSLMS allowed us to explore, for the first time, the low-temperature region in atomistic simulations of metal-substituted garnet structured solid electrolytes. The study provides new insights into ion transport and conductivity in the promising solid electrolytes for next-generation solid-state Li batteries [8].

Machine-learning-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 to 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.  

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 [6-7] (enhanced sampling Hybrid Monte Carlo methods in GROMACS package), LICA (postprocessing analysis of molecular dynamics trajectories), HaiCS [3] (statistical sampling and parameter estimation through Bayesian inference using Hamiltonian Monte Carlo methods), CRadPol (simulation of Controlled Radical Polymerization using experimental observations), Ddpm (simulation of dynamic development of particles morphology), ICS_Regge [9], DCS_Regge [10] (numerical Regge poles analysis of resonance structures in elastic, inelastic and reactive state-to-state integral and differential cross sections), PADE II [11] (type‐II Padé reconstruction of a scattering matrix element).

The computational models, proposed by the MSLMS, e.g., the PBE model for study of formation and evolution of multi-phase polymer particles morphology [12] (Figure 7); the effective medium model for estimation of conductivities of cubic/tetragonal phase mixtures of solid electrolytes [13]; the CAM (Complex Angular Momenta Analysis) models for elastic, inelastic and reactive integral and differential cross sections in elementary chemical reactions [9-11] (Figure 8), and the PDE magnetotelluric model for Bayesian inversion are implemented in the MSLMS software packages along with well-established models. 

Figure 7. Our novel Population Balance Model for Latex Particles Morphology Formation of reduced complexity (r-LPMF) maintains the same level of accuracy for the identified range of parameters  as a full LPMF model and requires up to 2 orders of magnitude less computational effort [12].

Figure 8. In-house package DCS_Regge allowed us to explain the origin of the forward peak, observed in the F+H₂-> HF+H reaction [10].

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) in order to perform multiscale simulation of physical systems and Bayesian inference.

Figure 8. Modelling and simulation in MSLMS. 

References

[1] 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.

[2] Bonilla M.R., García Daza F., Fernández-Pendás M., Carrasco J., Akhmatskaya E. Multiscale Modelling and Simulation of Advanced Battery Materials. Progress in Industrial Mathematics: Success Stories, M. Cruz, C. Pares, P. Quintela (Eds), Springer, 69 – 113 (2021), ISBN 978-3-030-61843-8

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

[4] Sokolovski D., Rusconi S., Brouard S, Akhmatskaya E. Reexamination of continuous fuzzy measurement on two-level systems. Phys. Rev. A 95 (2017) 042111.

[5] 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.

[6] 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.

[7] 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.

[8] 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.

[9] 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.

[10] 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.

[11] 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.

[12] 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).

[13] 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.