Convergence of position based dynamics
for first-order particle systems with volume exclusion
Steffen Plunder
Kyoto University
ICIAM 2023 Tokyo
Minisymposium on Evolution Equations
for Interacting Species: Applications and Analysis
Particle systems with volume exclusion
Feasible states: "swiss cheese set"
With penalty terms:
Going non-smooth:
set of orthogonally outward pointing vectors
Hard spheres
(second order dynamics)
Overdamped hard spheres
(first-order dynamics)
Comparing position based dynamics (PBD) and a penalty method
Models:
PBD is fast,
stable
and easy to implement! 🥳
... there must be a catch?! 🤨
Motivation: Agent-based modelling in mathematical biology
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Typical requirements:
Numerical implications:
large number of cells
computationally taxing
volume exclusion
stiffness issues
very uncertain parameters
simulation runtime
impacts
quality of parameter fitting
History of PBD
(2006)
Müller, Heidelberger, Hennix, Ratcliff
Position Based Dynamics
(2013)
Macklin, Müller:
Position Based Fluids
(2015)
Wang et al:
Chebyshev Semi-Iterative Approach
for Accelerating Projective and Position-based Dynamics
(2016)
Macklin, Müller:
XPBD
(2019)
Macklin et al:
Small Steps in Physics Simulation
PBD is used in movies, animation, games... But ignored by math community.
Publications on PBD per domain
Outline
Mathematical foundation:
Differential inclusions on uniformly prox-regular sets
Position based dynamics
: A rigorous method or just heuristic show-off?
Convergence of PBD for first-order systems
Application to
particle systems with volume exclusion
Preliminaries from variational analysis
Example
Well-
posedness
has a
unique
, absolutely continuous solution.
(2010)
Frédéric Bernicot, Juliette Venel:
Differential inclusions with proximal normal cones in Banach spaces
, J. of Convex Analysis
(2006)
Jean Fenel Edmond, Lionel Thibault:
BV solutions of nonconvex sweeping process differential inclusion with perturbation
, J of Diff. Eqs.
(2020)
Bernard Brogliato, Aneel Tanwani:
Dynamical Systems Coupled with Monotone Set-Valued Operators:
Formalisms, Applications, Well-Posedness, and Stability
, SIAM Review
Classical numerical methods for non-smooth dynamical systems
Moreau's catch-up scheme:
Typical approach for time-stepping schemes:
Use Lagranian multipliers
Taylor (+ index reduction)
Solve for multipliers with
iterative methods.
Projected Nonlinear Gauss-Seidel (PNGS):
First-order position-based dynamics (PBD)
Classical convergence proofs do not apply.
Seemingly just a "termination of PNGS without error bounds".
However, it is extremely easy to implement!
Large, accurate time-steps (PNGS)
vs
small, inaccurate time-steps (PBD)
Convergence result
Thm. [SP, Sara Merino (2023); "soon on arxiv"]
Sketch of the proof (1/2)
Need notion of convergence for
Answer:
Scalary upper semicontinuity!
Sketch of the proof (1/2): Consistency
(2006)
Jean Fenel Edmond, Lionel Thibault:
BV solutions of nonconvex sweeping
process differential inclusion with perturbation
, J of Diff. Eqs.
Consistency in a scalarly upper semicontinuous sense
Def. (SP, Sara Merino, 2023)
Thm. (SP, Sara Merino):
PBD is consistent in a susc sense!
Sketch of the proof (2/2)
How can we avoid that projection errors accumulate?
1.
Metric calmness of intersection
(subregularity):
2.
Hypomonotonicity
of normal cones for unif. prox-reg. sets::
"direction of projection cannot diverge too much"
Main technical lemma. (SP, Sara Merino, 2023)
cp. (2013)
Robert Hesse, Russell D. Luke:
Nonconvex Notions of Regularity and Convergence
of Fundamental Algorithms for Feasibility Problems
SIAM Journal on Optimization
(2006)
Jean Fenel Edmond, Lionel Thibault:
BV solutions of nonconvex sweeping
process differential inclusion with perturbation
, J of Diff. Eqs.
By adapting the existence proof from
Consistency (in a susc sense) + stability = convergence.
we get:
Cor. (SP, Sara Merino, 2023).
Hence, PBD convergeces for first-order dynamics!
Wrapping things up...
Application to particle systems
Thm. [SP, Sara Merino (2023)]
cp. (2011)
Bertrand Maury, Juliette Venel:
A discrete contact model for crowd motion,
ESAIM
Analysis of intersection of sets and
sum of gradients
kind of dual
to each other:
Hint: RHS is nicer (equal for volume exclusion)!
computational cost
Suitable for complex models
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New Julia package:
github.com/SteffenPL/
DifferentialInclusions.jl
Work in progress!
Supports:
- Sparse constraints
- Integration into SciML
Julia's ecosystem
Alternative projects:
https://github.com/siconos/siconos
Next steps?
Order of convergence
; stochastic differential inclusions; sweeping processes...
For particles/macroscopic models:
PBD particle method for PDE models with volume exclusion?
How to fix PBD failure of convergence for second-order systems?
PBD might be the only explict scheme for DAEs.
Numerical analysis
Thanks for your attention!
Sara Merino-Aceituno
University Vienna
Seirin-Lee, BiMed-Math Lab
Kyoto University,
Institute for Advanced Study of Human Biology