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arxiv: 2507.12575 · v3 · submitted 2025-07-16 · ⚛️ physics.comp-ph · math.AP· math.PR

Shape optimization of metastable states

No\'e Blassel , Tony Leli\`evre , Gabriel Stoltz This is my paper

Pith reviewed 2026-05-19 03:53 UTC · model grok-4.3

classification ⚛️ physics.comp-ph math.APmath.PR
keywords metastable statesshape optimizationDirichlet eigenvaluesaccelerated molecular dynamicstimescale separationelliptic diffusionsbiomolecular simulationsspectral asymptotics
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The pith

Metastable states can be defined by optimizing their boundaries to maximize a local separation-of-timescales metric tied to accelerated molecular dynamics efficiency.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method for defining metastable states by optimizing the shapes of their boundaries rather than relying on energy minima. The goal is to maximize a metric of local timescale separation that directly controls the performance of certain accelerated sampling algorithms. Analytic expressions are derived for how Dirichlet eigenvalues vary under shape changes of the state boundaries, and these are used to build an ascent algorithm that handles multiple eigenvalues at once. To make the method workable in high dimensions, the authors introduce dynamical coarse-graining or low-temperature spectral asymptotics. Tests on a biomolecular benchmark system show that the resulting states improve sampling efficiency compared with conventional definitions.

Core claim

The central claim is that metastable states should be defined through shape optimization of a local separation of timescale metric that is directly linked to the efficiency of a family of accelerated molecular dynamics algorithms. Analytic expressions for shape-variations of Dirichlet eigenvalues are derived for a class of operators associated with reversible elliptic diffusions and are used to construct a local ascent algorithm that explicitly treats the case of multiple eigenvalues. Tractability for high-dimensional systems is obtained via dynamical coarse-graining or recently obtained low-temperature shape-sensitive spectral asymptotics, and the method is validated on a benchmark biomolec

What carries the argument

Shape-variations of Dirichlet eigenvalues for operators associated with reversible elliptic diffusions, used to drive a local ascent algorithm that optimizes state boundaries for timescale separation.

If this is right

  • Accelerated molecular dynamics algorithms become more efficient when supplied with these shape-optimized state definitions.
  • Entropic effects and rapid thermal barrier crossing are handled more reliably than with energy-minimization definitions.
  • The approach remains computationally feasible in high-dimensional systems through either dynamical coarse-graining or low-temperature asymptotics.
  • Long configurational trajectories are sampled more effectively once metastable states are defined this way.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same boundary-optimization logic could be applied to other reversible Markov processes outside molecular dynamics.
  • States identified this way may expose metastable regions whose boundaries are set more by entropy than by energy barriers.
  • The ascent algorithm might be combined with data-driven approximations to scale to systems with many thousands of degrees of freedom.

Load-bearing premise

The local separation-of-timescale metric continues to predict algorithmic efficiency even after the state boundaries are moved away from conventional energy-minima locations.

What would settle it

Running the accelerated molecular dynamics algorithms on the optimized states versus standard energy-minima states and checking whether the optimized boundaries produce measurably faster convergence or higher effective sampling rates.

Figures

Figures reproduced from arXiv: 2507.12575 by Gabriel Stoltz, No\'e Blassel, Tony Leli\`evre.

Figure 1
Figure 1. Figure 1: The standard framework of the Hadamard shape derivative: a reference domain Ω is [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Directional shape perturbation of the triple Dirichlet eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Two-dimensional potentials (41), for decreasing values of the parameter [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Free energy profiles and effective diffusion coefficients for the CVs [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Domain-dependent eigenvalues (dotted lines) and their coarse-grained approximations [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Optimal domain for the effective dynamics, and optimal domain for the original gen [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Potential landscape and domains Ωα,β used in Figures 8a and 8b, as defined by (46), at the fixed value of the temperature parameter β = 10. The color coding is the same as that used in [PITH_FULL_IMAGE:figures/full_fig_p031_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Numerical validation of the low-temperature asymptotics of Theorems 3 and 2 from [9], [PITH_FULL_IMAGE:figures/full_fig_p032_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Asymptotic approach to the shape optimization problem for the potential (45) and the [PITH_FULL_IMAGE:figures/full_fig_p034_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Free energy landscape in the dihedral angles ( [PITH_FULL_IMAGE:figures/full_fig_p035_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Components of the effective diffusion tensor [PITH_FULL_IMAGE:figures/full_fig_p036_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: In Figure 12a, solid lines correspond to the boundaries of the optimized domains, [PITH_FULL_IMAGE:figures/full_fig_p037_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Effective separation of timescales throughout six runs of Algorithm 1, initialized with [PITH_FULL_IMAGE:figures/full_fig_p037_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Behavior of the four smallest Dirichlet eigenvalues throughout the six runs of Algo [PITH_FULL_IMAGE:figures/full_fig_p038_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Effect of the numerical degeneracy parameter [PITH_FULL_IMAGE:figures/full_fig_p038_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Convergence of the marginals of the Fleming–Viot process to the corresponding quasi [PITH_FULL_IMAGE:figures/full_fig_p041_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Empirical ξ-marginal for the stationary Fleming–Viot process for γ = 10 ps−1 . Top: free-energy basin. Bottom: numerically optimized domain. On both figures, sampled initial config￾urations for the Fleming–Viot process are overlaid on the stationary histogram, and distinguished by color according to the corresponding free-energy saddle point. To each of these metrics, we associate a corresponding measure … view at source ↗
Figure 18
Figure 18. Figure 18: Results of the Fleming–Viot simulations, showing that the optimized state consistently [PITH_FULL_IMAGE:figures/full_fig_p043_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Timescale separation ratios. Blue lines correspond to the free-energy basin, and red [PITH_FULL_IMAGE:figures/full_fig_p044_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: A trajectory sampled using Algorithm 3. Dotted lines correspond to step [PITH_FULL_IMAGE:figures/full_fig_p056_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Effect of the separation of timescales (7) and number [PITH_FULL_IMAGE:figures/full_fig_p058_21.png] view at source ↗
read the original abstract

The definition of metastable states is an ubiquitous task in the design and analysis of molecular simulations, and is a crucial input in a variety of acceleration methods for the sampling of long configurational trajectories. Although standard definitions based on local energy minimization procedures can sometimes be used, these definitions are typically suboptimal, or entirely inadequate when entropic effects are significant, or when the lowest energy barriers are quickly overcome by thermal fluctuations. In this work, we propose an approach to the definition of metastable states, based on the shape-optimization of a local separation of timescale metric directly linked to the efficiency of a family of accelerated molecular dynamics algorithms. To realize this approach, we derive analytic expressions for shape-variations of Dirichlet eigenvalues for a class of operators associated with reversible elliptic diffusions, and use them to construct a local ascent algorithm, explicitly treating the case of multiple eigenvalues. We propose two methods to make our method tractable in high-dimensional systems: one based on dynamical coarse-graining, the other on recently obtained low-temperature shape-sensitive spectral asymptotics. We validate our method on a benchmark biomolecular system, showcasing a significant improvement over conventional definitions of metastable states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes defining metastable states via shape optimization of a local separation-of-timescale metric based on Dirichlet eigenvalues of the reversible elliptic diffusion operator. Analytic shape-variation formulas are derived to enable a local ascent algorithm (explicitly handling multiple eigenvalues), with two tractability approximations (dynamical coarse-graining and low-temperature spectral asymptotics) for high-dimensional systems. The method is validated on a biomolecular benchmark, reporting significant improvement over conventional energy-minima definitions.

Significance. If the optimized states demonstrably improve the practical efficiency of the target accelerated MD family (beyond metric improvement alone), the approach would offer a principled alternative to energy-based state definitions when entropic barriers or rapid crossings dominate. The derivations of shape gradients and the handling of multiplicity are potentially reusable for other spectral optimization problems in diffusion processes.

major comments (2)
  1. [Validation/benchmark section] Validation/benchmark section: the reported 'significant improvement' is quantified solely via the separation-of-timescale metric on the biomolecular example; no direct measurements (e.g., decorrelation times, transition rates, or effective sampling speedup) are shown for the downstream accelerated MD integrators after boundary variation. This leaves open whether the local metric remains a faithful proxy once states deviate from energy minima.
  2. [Derivation of shape variations] Derivation of shape variations (around the Dirichlet eigenvalue formulas): while the local ascent is constructed from the derived gradients, the manuscript does not provide an a-priori error bound or sensitivity analysis showing that the metric improvement persists under the dynamical coarse-graining approximation when entropic effects are strong.
minor comments (2)
  1. [Introduction] Notation for the family of accelerated algorithms and the precise definition of the 'local separation of timescale metric' should be introduced earlier and used consistently to avoid ambiguity when linking to algorithmic efficiency.
  2. [Figures] Figure captions for the benchmark results should explicitly state which quantity (metric value, eigenvalue gap, or actual sampling statistic) is plotted and include error bars or variability across runs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and the positive assessment of the significance of the derivations and the handling of eigenvalue multiplicity. Below we respond point-by-point to the two major comments. We agree that both points identify areas where the manuscript can be strengthened and will incorporate revisions accordingly.

read point-by-point responses

  1. Referee: [Validation/benchmark section] Validation/benchmark section: the reported 'significant improvement' is quantified solely via the separation-of-timescale metric on the biomolecular example; no direct measurements (e.g., decorrelation times, transition rates, or effective sampling speedup) are shown for the downstream accelerated MD integrators after boundary variation. This leaves open whether the local metric remains a faithful proxy once states deviate from energy minima.

    Authors: We agree that direct measurements of decorrelation times or sampling speedup in the downstream accelerated MD integrators would constitute stronger practical validation. The separation-of-timescale metric is derived precisely because it is the quantity that controls the efficiency of the target family of accelerated MD methods; the manuscript therefore treats improvement of this metric as the primary figure of merit for the state definitions. Nevertheless, to address the referee’s concern we will add an explicit discussion in the revised manuscript that recalls the theoretical link between the metric and the expected acceleration, and we will acknowledge that full end-to-end integrator benchmarks remain an important direction for subsequent work. revision: partial

  2. Referee: [Derivation of shape variations] Derivation of shape variations (around the Dirichlet eigenvalue formulas): while the local ascent is constructed from the derived gradients, the manuscript does not provide an a-priori error bound or sensitivity analysis showing that the metric improvement persists under the dynamical coarse-graining approximation when entropic effects are strong.

    Authors: The dynamical coarse-graining approximation is presented as one of two tractability routes, and its use on the biomolecular benchmark (where entropic barriers are relevant) already provides empirical support that the optimized states improve the metric. We acknowledge, however, that a formal a-priori error bound or dedicated sensitivity study for strong entropic regimes is absent. In the revision we will insert a new subsection that states the assumptions underlying the coarse-graining step, derives a first-order sensitivity estimate with respect to the coarse-graining parameter, and reports a numerical sensitivity check on the benchmark system to quantify how the metric improvement behaves when entropic effects are pronounced. revision: yes

Circularity Check

0 steps flagged

Derivation chain remains self-contained with independent mathematical derivations and benchmark validation

full rationale

The paper derives new analytic expressions for shape variations of Dirichlet eigenvalues of reversible elliptic diffusion operators and constructs a local ascent algorithm from them to optimize a separation-of-timescale metric. These steps are presented as original mathematical contributions, supplemented by two tractability methods (dynamical coarse-graining and low-temperature spectral asymptotics) and validated on an external biomolecular benchmark that demonstrates improvement over conventional energy-minima definitions. No quoted reduction shows any prediction or central claim equaling its inputs by construction, no fitted parameter is renamed as a prediction, and no load-bearing self-citation chain is required for the core result; the benchmark provides an independent external check against actual sampling performance.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on standard properties of reversible elliptic diffusions and their Dirichlet spectra; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract description.

axioms (1)
  • standard math Reversible elliptic diffusions admit well-defined Dirichlet eigenvalues whose shape derivatives can be expressed analytically.
    Invoked to justify the local ascent algorithm (abstract, paragraph 3).

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Works this paper leans on

76 extracted references · 76 canonical work pages

  1. [1]

    Allaire and M

    G. Allaire and M. Schoenauer , Conception optimale de structures , vol. 58 of SMAI Math´ ematiques et Applications, Springer, 2007

    work page 2007

  2. [2]

    Allen and D

    M. Allen and D. Tildesley , Computer Simulation of Liquids , Oxford University Press, 2017

    work page 2017

  3. [3]

    Aristoff, Generalizing Parallel Replica dynamics: Trajectory fragments, asynchronous computing, and PDMPs , SIAM/ASA Journal on Uncertainty Quantification, 7 (2019), pp

    D. Aristoff, Generalizing Parallel Replica dynamics: Trajectory fragments, asynchronous computing, and PDMPs , SIAM/ASA Journal on Uncertainty Quantification, 7 (2019), pp. 685–719

    work page 2019

  4. [4]

    Aristoff, M

    D. Aristoff, M. Johnson, and D. Perez , Arbitrarily accurate, nonparametric coarse graining with Markov renewal processes and the Mori–Zwanzig formulation , AIP Advances, 13 (2023), p. 095131

    work page 2023

  5. [5]

    Ashbaugh and R

    M. Ashbaugh and R. Benguria , A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions , Annals of Mathematics, 135 (1992), pp. 601–628

    work page 1992

  6. [6]

    Binder, T

    A. Binder, T. Leli`evre, and G. Simpson, A generalized parallel replica dynamics, Journal of Computational Physics, 284 (2015), pp. 595–616

    work page 2015

  7. [7]

    Blassel, Paper code repository: https: // github

    N. Blassel, Paper code repository: https: // github. com/ noeblassel/ sosad, 2025

    work page 2025

  8. [8]

    org/ records/ 15727493, 2025

    , Paper data repository: https: // zenodo. org/ records/ 15727493, 2025

    work page 2025

  9. [9]

    Blassel, T

    N. Blassel, T. Leli `evre, and G. Stoltz , Quantitative low-temperature asymptotics for reversible diffusions in temperature-dependent domains , arXiv preprint:2501.16082, (2024)

    work page arXiv 2024

  10. [10]

    Bolhuis, C

    P. Bolhuis, C. Dellago, and D. Chandler , Reaction coordinates of biomolecular iso- merization, Proceedings of the National Academy of Sciences, 97 (2000), pp. 5877–5882

    work page 2000

  11. [11]

    Bovier, V

    A. Bovier, V. Gayrard, and M. Klein , Metastability in reversible diffusion processes. II: Precise asymptotics for small eigenvalues , Journal of the European Mathematical Society, 7 (2005), pp. 69–99

    work page 2005

  12. [12]

    Chen, Collective variable-based enhanced sampling and machine learning , The European Physical Journal B, 94 (2021), pp

    M. Chen, Collective variable-based enhanced sampling and machine learning , The European Physical Journal B, 94 (2021), pp. 1–17

    work page 2021

  13. [13]

    Chenais, On the existence of a solution in a domain identification problem , Journal of Mathematical Analysis and Applications, 52 (1975), pp

    D. Chenais, On the existence of a solution in a domain identification problem , Journal of Mathematical Analysis and Applications, 52 (1975), pp. 189–219

    work page 1975

  14. [14]

    Comer, J

    J. Comer, J. Gumbart, J. H ´enin, T. Leli `evre, A. Pohorille, and C. Chipot , The adaptive biasing force method: Everything you always wanted to know but were afraid to ask , The Journal of Physical Chemistry B, 119 (2015), pp. 1129–1151

    work page 2015

  15. [15]

    T. Cui, X. Tong, and O. Zahm, Optimal Riemannian metric for Poincar´ e inequalities and how to ideally precondition Langevin dynamics , arXiv preprint arXiv:2404.02554, (2024)

    work page arXiv 2024

  16. [16]

    Dapogny, P

    C. Dapogny, P. Frey, F. Omn `es, and Y. Privat , Geometrical shape optimization in fluid mechanics using FreeFem++, Structural and Multidisciplinary Optimization, 58 (2018), pp. 2761–2788. 59

    work page 2018

  17. [17]

    De Gournay , Velocity extension for the level-set method and multiple eigenvalues in shape optimization, SIAM Journal on Control and Optimization, 45 (2006), pp

    F. De Gournay , Velocity extension for the level-set method and multiple eigenvalues in shape optimization, SIAM Journal on Control and Optimization, 45 (2006), pp. 343–367

    work page 2006

  18. [18]

    Del Moral and L

    P. Del Moral and L. Miclo , A Moran particle system approximation of Feynman–Kac formulae, Stochastic processes and their applications, 86 (2000), pp. 193–216

    work page 2000

  19. [19]

    Di Ges`u, T

    G. Di Ges`u, T. Leli`evre, D. Le Peutrec, and B. Nectoux , Sharp asymptotics of the first exit point density , Annals of PDE, 5 (2019), pp. 1–174

    work page 2019

  20. [20]

    , The exit from a metastable state: Concentration of the exit point distribution on the low energy saddle points, part 1 , Journal de Math´ ematiques Pures et Appliqu´ ees, 138 (2020), pp. 242–306

    work page 2020

  21. [21]

    Evans, Partial Differential Equations, vol

    L. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, Amer- ican Mathematical Society, 2022

    work page 2022

  22. [22]

    Ferguson, Machine learning and data science in soft materials engineering , Journal of Physics: Condensed Matter, 30 (2018), p

    A. Ferguson, Machine learning and data science in soft materials engineering , Journal of Physics: Condensed Matter, 30 (2018), p. 043002

    work page 2018

  23. [23]

    Fiorin, M

    G. Fiorin, M. Klein, and J. H´enin, Using collective variables to drive molecular dynamics simulations, Molecular Physics, 111 (2013), pp. 3345–3362

    work page 2013

  24. [24]

    Gkeka, G

    P. Gkeka, G. Stoltz, A. Fariman, Z. Belkacemi, M. Ceriotti, J. Chodera, A. Din- ner, A. Ferguson, J.-B. Maillet, H. Minoux, C. Peter, F. Pietrucci, A. Silveira, A. Tkatchenko, Z. Trstanova, R. Wiewiora, and T. Leli `evre, Machine learning force fields and coarse-grained variables in molecular dynamics: application to materials and biological systems, Jour...

    work page 2020

  25. [25]

    Glielmo, B

    A. Glielmo, B. Husic, A. Rodriguez, C. Clementi, F. No ´e, and A. Laio , Un- supervised learning methods for molecular simulation data , Chemical Reviews, 121 (2021), pp. 9722–9758

    work page 2021

  26. [26]

    Gonz ´alez, Measurement of areas on a sphere using Fibonacci and latitude–longitude lattices, Mathematical Geosciences, 42 (2010), pp

    A. Gonz ´alez, Measurement of areas on a sphere using Fibonacci and latitude–longitude lattices, Mathematical Geosciences, 42 (2010), pp. 49–64

    work page 2010

  27. [27]

    Grisvard, Elliptic problems in nonsmooth domains , vol

    P. Grisvard, Elliptic problems in nonsmooth domains , vol. 69 of Classics in Applied Math- ematics, SIAM, 2011

    work page 2011

  28. [28]

    Hadamard, M´ emoire sur le probl` eme d’analyse relatif ` a l’´ equilibre des plaques ´ elastiques encastr´ ees, vol

    J. Hadamard, M´ emoire sur le probl` eme d’analyse relatif ` a l’´ equilibre des plaques ´ elastiques encastr´ ees, vol. 33 of M´ emoires pr´ esent´ es par divers savants ` a l’Acad´ emie des sciences de l’Institut de France., Imprimerie nationale, 1908

    work page 1908

  29. [29]

    Haug and B

    E. Haug and B. Rousselet , Design sensitivity analysis in structural mechanics. I. Static response variations, Journal of Structural Mechanics, 8 (1980), pp. 17–41

    work page 1980

  30. [30]

    , Design sensitivity analysis in structural mechanics. II. Eigenvalue variations , Journal of Structural Mechanics, 8 (1980), pp. 161–186

    work page 1980

  31. [31]

    , Design sensitivity analysis in structural mechanics. III. Effects of shape variation , Journal of Structural mechanics, 10 (1982), pp. 273–310

    work page 1982

  32. [32]

    Hecht, O

    F. Hecht, O. Pironneau, A. Le Hyaric, and K. Ohtsuka , FreeFem++ Manual, Lab- oratoire Jacques Louis Lions, (2005), pp. 1–188

    work page 2005

  33. [33]

    H´edin and T

    F. H´edin and T. Leli`evre, Gen. parRep: A first implementation of the Generalized Parallel Replica dynamics for the long time simulation of metastable biochemical systems , Computer Physics Communications, 239 (2019), pp. 311–324. 60

    work page 2019

  34. [34]

    Helffer and F

    B. Helffer and F. Nier , Quantitative analysis of metastability in reversible diffusion pro- cesses via a Witten complex approach: The case with boundary , Matem´ atica contemporˆ anea, 26 (2004), pp. 41–85

    work page 2004

  35. [35]

    Helffer and J

    B. Helffer and J. Sj¨ostrand, Puits multiples en m´ ecanique semi-classique. IV:´Etude du complexe de Witten, Communications in partial differential equations, 10 (1985), pp. 245–340

    work page 1985

  36. [36]

    Henrot, Extremum Problems for Eigenvalues of Elliptic Operators , Springer Science & Business Media, 2006

    A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators , Springer Science & Business Media, 2006

    work page 2006

  37. [37]

    Henrot and M

    A. Henrot and M. Pierre, Variation et optimisation de formes : une analyse g´ eom´ etrique, vol. 48 of Math´ ematiques et applications, Springer, 2005

    work page 2005

  38. [38]

    Hottovy, A

    S. Hottovy, A. McDaniel, G. Volpe, and J. Wehr , The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction , Communications in Mathematical Physics, 336 (2015), pp. 1259–1283

    work page 2015

  39. [39]

    Journel and P

    L. Journel and P. Monmarch ´e, Uniform convergence of the Fleming–Viot process in a hard killing metastable case , arXiv preprint arXiv:2207.02030, (2022)

    work page arXiv 2022

  40. [40]

    Kato, Perturbation Theory for Linear Operators , vol

    T. Kato, Perturbation Theory for Linear Operators , vol. 132, Springer Science & Business Media, 2013

    work page 2013

  41. [41]

    Kirsch, H

    W. Kirsch, H. Cycon, R. Froese, and B. Simon , Schr¨ odinger Operators, Springer, Berlin, 1987

    work page 1987

  42. [42]

    Krantz and H

    S. Krantz and H. Parks , Geometric Integration Theory, Cornerstones, Springer Science & Business Media, 2008

    work page 2008

  43. [43]

    Krumscheid, M

    S. Krumscheid, M. Pradas, G. Pavliotis, and S. Kalliadasis , Data-driven coarse graining in action: Modeling and prediction of complex systems, Physical Review E, 92 (2015), p. 042139

    work page 2015

  44. [44]

    Lagard`ere, L

    L. Lagard`ere, L. Jolly, F. Lipparini, F. Aviat, B. Stamm, Z. Jing, M. Harger, H. Torabifard, G. Cisneros, and M. Schnieders , Tinker-HP: A massively parallel molecular dynamics package for multiscale simulations of large complex systems with advanced point dipole polarizable force fields, Chemical Science, 9 (2018), pp. 956–972

    work page 2018

  45. [45]

    Le Bris, T

    C. Le Bris, T. Leli`evre, M. Luskin, and D. Perez, A mathematical formalization of the parallel replica dynamics, Monte–Carlo Methods and Applications, 18 (2012), pp. 119–146

    work page 2012

  46. [46]

    Le Peutrec and B

    D. Le Peutrec and B. Nectoux, Small eigenvalues of the Witten Laplacian with Dirichlet boundary conditions: the case with critical points on the boundary, Analysis & PDE, 14 (2021), pp. 2595–2651

    work page 2021

  47. [47]

    Legoll and T

    F. Legoll and T. Leli`evre, Effective dynamics using conditional expectations, Nonlinear- ity, 23 (2010), pp. 21–31

    work page 2010

  48. [48]

    Lehoucq, D

    R. Lehoucq, D. Sorensen, and C. Yang , ARPACK users’ guide: Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods , Software, Environments, and Tools, SIAM, 1998

    work page 1998

  49. [49]

    Leimkuhler and C

    B. Leimkuhler and C. Matthews , Molecular Dynamics, vol. 39 of Interdisciplinary Ap- plied Mathematics, Springer Cham, 2016

    work page 2016

  50. [50]

    Leli`evre and F

    T. Leli`evre and F. Nier , Low temperature asymptotics for quasistationary distributions in a bounded domain , Analysis & PDE, 8 (2015), pp. 561–628

    work page 2015

  51. [51]

    Optimizing the diffusion coefficient of overdamped

    T. Leli`evre, G. Pavliotis, G. Robin, R. Santet, and G. Stoltz, Optimizing the diffu- sion coefficient of overdamped Langevin dynamics , arXiv preprint arXiv:2404.12087, (2024). 61

    work page arXiv 2024

  52. [52]

    Leli`evre, M

    T. Leli`evre, M. Rachid, and G. Stoltz, A spectral approach to the narrow escape problem in the disk , arXiv preprint, (2024)

    work page 2024

  53. [53]

    Leli`evre, M

    T. Leli`evre, M. Rousset, and G. Stoltz , Free Energy Computations - A Mathematical Perspective, Imperial College Press, 2010

    work page 2010

  54. [54]

    Leli`evre, R

    T. Leli`evre, R. Santet, and G. Stoltz , Improving sampling by modifying the effective diffusion, arXiv preprint arXiv:2410.00525, (2024)

    work page arXiv 2024

  55. [55]

    Leli`evre and G

    T. Leli`evre and G. Stoltz, Partial differential equations and stochastic methods in molec- ular dynamics, Acta Numerica, 25 (2016), pp. 681–880

    work page 2016

  56. [56]

    Nateghi and F

    V. Nateghi and F. N ¨uske, Kinetically consistent coarse graining using kernel-based ex- tended dynamic mode decomposition, arXiv preprint:2409.16396, (2024)

    work page arXiv 2024

  57. [57]

    Nocedal and S

    J. Nocedal and S. Wright , Numerical Optimization, Springer, 1999

    work page 1999

  58. [58]

    N¨uske, P

    F. N¨uske, P. Koltai, L. Boninsegna, and C. Clementi , Spectral properties of effective dynamics from conditional expectations, Entropy, 23 (2021), p. 134

    work page 2021

  59. [59]

    Oudet, Numerical minimization of eigenmodes of a membrane with respect to the domain, ESAIM: Control, Optimisation and Calculus of Variations, 10 (2004), pp

    E. Oudet, Numerical minimization of eigenmodes of a membrane with respect to the domain, ESAIM: Control, Optimisation and Calculus of Variations, 10 (2004), pp. 315–330

    work page 2004

  60. [60]

    Payne, G

    L. Payne, G. P ´olya, and H. Weinberger , On the ratio of consecutive eigenvalues , J. Math. Phys, 35 (1956), pp. 289–298

    work page 1956

  61. [61]

    Perez, E

    D. Perez, E. Cubuk, A. Waterland, E. Kaxiras, and A. Voter, Long-time dynamics through parallel trajectory splicing, Journal of Chemical Theory and Computation, 12 (2016), pp. 18–28

    work page 2016

  62. [62]

    Perez, B

    D. Perez, B. Uberuaga, and A. Voter , The parallel replica dynamics method–Coming of age, Computational Materials Science, 100 (2015), pp. 90–103

    work page 2015

  63. [63]

    Ramil, T

    M. Ramil, T. Leli `evre, and J. Reygner , Mathematical foundations for the Parallel Replica algorithm applied to the underdamped Langevin dynamics , MRS Communications, 12 (2022), pp. 454–459

    work page 2022

  64. [64]

    Rousselet, Shape design sensitivity of a membrane , Journal of Optimization Theory and Applications, 40 (1983), pp

    B. Rousselet, Shape design sensitivity of a membrane , Journal of Optimization Theory and Applications, 40 (1983), pp. 595–623

    work page 1983

  65. [65]

    Rousset , On the control of an interacting particle estimation of Schr¨ odinger ground states, SIAM journal on mathematical analysis, 38 (2006), pp

    M. Rousset , On the control of an interacting particle estimation of Schr¨ odinger ground states, SIAM journal on mathematical analysis, 38 (2006), pp. 824–844

    work page 2006

  66. [66]

    Ryckaert, G

    J. Ryckaert, G. Ciccotti, and H. Berendsen , Numerical integration of the Cartesian equations of motion of a system with constraints: molecular dynamics of n-alkanes , Journal of Computational Physics, 23 (1977), pp. 327–341

    work page 1977

  67. [67]

    Simon, Semiclassical analysis of low lying eigenvalues

    B. Simon, Semiclassical analysis of low lying eigenvalues. I. Non-degenerate minima: Asymp- totic expansions, Annales de l’IHP: Physique Th´ eorique, 38 (1983), pp. 295–308

    work page 1983

  68. [68]

    Simpson and M

    G. Simpson and M. Luskin, Numerical analysis of parallel replica dynamics, ESAIM: Math- ematical Modelling and Numerical Analysis, 47 (2013), pp. 1287–1314

    work page 2013

  69. [69]

    Sørensen and A

    M. Sørensen and A. Voter, Temperature-accelerated dynamics for simulation of infrequent events, The Journal of Chemical Physics, 112 (2000), pp. 9599–9606

    work page 2000

  70. [70]

    Tao, Topics in Random Matrix Theory , vol

    T. Tao, Topics in Random Matrix Theory , vol. 132 of Graduate Studies in Mathematics, American Mathematical Soc., 2012

    work page 2012

  71. [71]

    Teschl, Mathematical Methods in Quantum Mechanics, vol

    G. Teschl, Mathematical Methods in Quantum Mechanics, vol. 157, American Mathematical Society, 2014. 62

    work page 2014

  72. [72]

    Uberuaga, S

    B. Uberuaga, S. Stuart, and A. Voter , Parallel replica dynamics for driven systems: Derivation and application to strained nanotubes, Physical Review B—Condensed Matter and Materials Physics, 75 (2007), p. 014301

    work page 2007

  73. [73]

    Villemonais , General approximation method for the distribution of Markov processes conditioned not to be killed , ESAIM: Probability and Statistics, 18 (2014), pp

    D. Villemonais , General approximation method for the distribution of Markov processes conditioned not to be killed , ESAIM: Probability and Statistics, 18 (2014), pp. 441–467

    work page 2014

  74. [74]

    Voter, Hyperdynamics: Accelerated molecular dynamics of infrequent events , Physical Review Letters, 78 (1997), p

    A. Voter, Hyperdynamics: Accelerated molecular dynamics of infrequent events , Physical Review Letters, 78 (1997), p. 3908

    work page 1997

  75. [75]

    9599–9606

    , Parallel replica method for dynamics of infrequent events, Physical Review B, 57 (1998), pp. 9599–9606

    work page 1998

  76. [76]

    Zhang, C

    W. Zhang, C. Hartmann, and C. Sch ¨utte, Effective dynamics along given reaction coordinates, and reaction rate theory, Faraday Discussions, 195 (2016), pp. 365–394. 63

    work page 2016