QMC: Transition Workshop - Probabilistic Integrators for Deterministic Differential Equations - Han Lie, May 8, 2018

Strong convergence of probabilistic solvers
for deterministic ordinary differential equations
(joint work with A. M. Stuart, T. J. Sullivan)
Han Cheng Lie
SAMSI QMC Transition Workshop
Research Triangle Park, NC
08.05.2018
Support by Excellence Initiative of German Research Foundation (DFG),
National Science Foundation grant DMS-1127914, SAMSI QMC

Introduction
Applied mathematics:
Computable approximations for continuous problems
Replacing continuous phenomena by ﬁnite sets of values

Introduction
The QMC Program: Placing points in space
Strategies: random (Monte Carlo), deterministic (quasi-Monte Carlo),
random-deterministic

Introduction
Classical numerical analysis for deterministic ODEs
Points belong to inﬁnite-dimensional space of trajectories
Quantity of interest: Dirac measure supported at true trajectory

Introduction
Classical numerical analysis for deterministic ODEs
Points belong to inﬁnite-dimensional space of trajectories
Quantity of interest: Dirac measure supported at true trajectory
Construct trajectories by interpolating between point evaluations
Main tasks: 1) Recursively construct sequences of point evalutions
Main tasks: 2) Quantify error in trajectory placement

Introduction
This project:
Use Monte Carlo to determine point evaluations
Approximate Dirac measure with nondegenerate probability measure
Establish contraction to Dirac measure at true trajectory
Further development of convergence analysis of Conrad et al.,
“Statistical analysis of differential equations: introducing probability
measures on numerical solutions”, Stat. Comput. (2017)

Solving ordinary differential equations
Initial value problem (IVP) for 0 ≤ t ≤ T with solution u = (u(t))0≤t≤T
d
dt u(t) = f(u(t)), u(0) = u0 ∈ Rd

d
dt u(t) = f(u(t)), u(0) = u0 ∈ Rd
Flow map semigroup (Fh
)h≥0, Fh
: Rd
→ Rd
satisﬁes
Fh
(u0) = u(h), F0
(u0) = u0
sequence of observations uk = (Fh
)k
(u0) = u(kh)

d
dt u(t) = f(u(t)), u(0) = u0 ∈ Rd
Flow map semigroup (Fh
)h≥0, Fh
: Rd
→ Rd
satisﬁes
Fh
(u0) = u(h), F0
(u0) = u0
sequence of observations uk = (Fh
)k
(u0) = u(kh)
Represent deterministic solver with time step h > 0 by Ψh
: Rd
→ Rd
[Ψh
]k
(u0) ≈ (Fh
)k
(u0) = uk , for u0 ∈ Rd
, k ∈ N0
Common task in classical numerical analysis: prove
supv∈Rd Ψh
(v) − Fh
(v) ≤ Chq+1
⇒ Asymptotically, worst-case error over trajectories decreases with h
at rate q

Motivation for probabilistic approach
Available information about solution values off time grid (kh)k comes
from on-grid values (Ψkh
(u0))k
Need for uncertainty quantiﬁcation (UQ) in off-grid values

(u0))k
In some cases, more solves on coarse grids yield more statistical
information than fewer solves on ﬁne grids
→ inverse problems, data assimilation, multilevel Monte Carlo

(u0))k
In some cases, more solves on coarse grids yield more statistical
information than fewer solves on ﬁne grids
→ inverse problems, data assimilation, multilevel Monte Carlo
In other cases, quantity of interest (QoI) is functional of solution of IVP
→ Can use estimates of off-mesh uncertainty due to Ψh
to estimate
uncertainty in QoI

Probabilistic solvers of Conrad et al. (2017)
Fix probability space (Ω, P), time step h > 0
Using deterministic solver Ψh
, generate (Uk )k with U0 = u0 by
Uk+1 := Ψh
(Uk ) + ξk (h), ξk (h) =
h
0
χk (s)ds

Uk+1 := Ψh
(Uk ) + ξk (h), ξk (h) =
h
0
χk (s)ds
♠ Stochastic process χk on [0, h] models off-grid behaviour
(Use Monte Carlo to determine point evaluations)

Uk+1 := Ψh
(Uk ) + ξk (h), ξk (h) =
h
0
χk (s)ds
♠ Stochastic process χk on [0, h] models off-grid behaviour
(Use Monte Carlo to determine point evaluations)
Solution sequences: (u(kh))k (true solution)
([Ψh
]k
(u0))k (deterministic approximation)
(Uk )k (probabilistic approximation)

Two challenges, and their relevance
1) obtain probabilistic analogues of worst-case bounds over
trajectories, i.e. of
maxk uk − (Ψh
)k
(u0) ≤ Chq
♠ Sometimes, worst-case bound less useful than “most likely”-case
bound (e.g. weather prediction)

Two challenges, and their relevance
1) obtain probabilistic analogues of worst-case bounds over
trajectories, i.e. of
maxk uk − (Ψh
)k
(u0) ≤ Chq
♠ Sometimes, worst-case bound less useful than “most likely”-case
bound (e.g. weather prediction)
2) Describe conditions on (ξk )k so that P ◦ (Uk )−1
k provides good
approximation of Dirac measure at true solution
♠ Improving random approximations may be cheaper and more
effective than improving deterministic approximation

Strong convergence result of Conrad et. al.
Theorem
Assume: mean-zero, i.i.d. Gaussian noise (ξk (h))k
E[|ξk (h)|
2
] ≤ Cξ(h2
)p+1/2
, p ≥ 1
globally Lipschitz vector ﬁeld f, and Ψh
has local order q + 1, i.e.
supv∈Rn Ψh
(v) − Fh
(v) ≤ CΨhq+1
.
Then for some C > 0,
max
k
E |uk − Uk |
2
≤ Ch2 min{q,p}
.

Strong convergence result of Conrad et. al.
Theorem
Assume: mean-zero, i.i.d. Gaussian noise (ξk (h))k
E[|ξk (h)|
2
] ≤ Cξ(h2
)p+1/2
, p ≥ 1
globally Lipschitz vector ﬁeld f, and Ψh
has local order q + 1, i.e.
supv∈Rn Ψh
(v) − Fh
(v) ≤ CΨhq+1
.
max
k
E |uk − Uk |
2
≤ Ch2 min{q,p}
.
Compare with deterministic solver:
maxk uk − (Ψh
)k
(u0)
2
≤ Ch2q
♠ q = p ⇒ probabilistic solver has same convergence rate
⇒ max. amount of solution uncertainty consistent with order of Ψh

Recent results
Theorem (Stuart, Sullivan)
Assume: mean-zero, mutually independent noise (ξk (h))k such that
E[|ξk (h)|
2
] ≤ Cξ(h2
)p+1/2
, p ≥ 1,
globally Lipschitz ﬂow map Fh
, numerical solver Ψh
has local order
q + 1 and satisﬁes
X ∈ L2
P ⇒ Ψh
(X) ∈ L2
P.
E max
k
|uk − Uk |
2
≤ Ch2 min{q,p}
.

Recent results
E[|ξk (h)|
2
] ≤ Cξ(h2
)p+1/2
, p ≥ 1,
has local order
X ∈ L2
P ⇒ Ψh
(X) ∈ L2
P.
E max
k
|uk − Uk |
2
≤ Ch2 min{q,p}
.
♠ No Gaussianity assumption
♠ weaker Lipschitz condition (on ﬂow map instead of vector ﬁeld)

Recent results
E[|ξk (h)|
2
] ≤ Cξ(h2
)p+1/2
, p ≥ 1,
has local order
X ∈ L2
P ⇒ Ψh
(X) ∈ L2
P.
E max
k
|uk − Uk |
2
≤ Ch2 min{q,p}
.
♠ No Gaussianity assumption
♠ weaker Lipschitz condition (on ﬂow map instead of vector ﬁeld)
♠ stronger convergence, since supremum is inside expectation

Proof strategy: Use moment condition on Ψh
, zero mean,
independence to construct martingale
Apply martingale inequality, Lipschitz property of Fh
, uniform error
bound on Ψh
to bound
Ej := max
k≤j
uj − Uj
2
Take expectations, apply Gronwall’s lemma to sequence (E[Ej ])j

Proof strategy: Use moment condition on Ψh
, zero mean,
independence to construct martingale
Apply martingale inequality, Lipschitz property of Fh
, uniform error
bound on Ψh
to bound
Ej := max
k≤j
uj − Uj
2
Take expectations, apply Gronwall’s lemma to sequence (E[Ej ])j
Questions:
- Can we remove mean-zero and independence assumptions?
- Can we consider noise with fewer moments (heavier tails)?

Recent results
Uniform moment bound assumption: For some p, Cξ ≥ 1, R ∈ N,
E |ξk (h)|
r
≤ Cξhp+1/2 r
, 1 ≤ r ≤ R

Recent results
E |ξk (h)|
r
≤ Cξhp+1/2 r
, 1 ≤ r ≤ R
♠ No mean-zero or i.i.d. assumptions; fewer moments if R = 1

Recent results
E |ξk (h)|
r
≤ Cξhp+1/2 r
, 1 ≤ r ≤ R
Theorem (L.)
Assume that the uniform moment bound holds, that Ψh
has order
q + 1, and Fh
is globally Lipschitz. Then for some C > 0,
E max
k
|uk − Uk |
R
≤ C(hR
)min{q,p−1/2}

Recent results
E |ξk (h)|
r
≤ Cξhp+1/2 r
, 1 ≤ r ≤ R
Theorem (L.)
has order
q + 1, and Fh
E max
k
|uk − Uk |
R
≤ C(hR
)min{q,p−1/2}
♠ No assumption that X ∈ L2
P ⇒ Ψh
(X) ∈ L2
P

Recent results
E |ξk (h)|
r
≤ Cξhp+1/2 r
, 1 ≤ r ≤ R
Theorem (L.)
has order
q + 1, and Fh
E max
k
|uk − Uk |
R
≤ C(hR
)min{q,p−1/2}
♠ No assumption that X ∈ L2
P ⇒ Ψh
(X) ∈ L2
P
♣ p = q + 1
2 max. uncertainty consistent with order of Ψh
(tradeoff: zero mean + independence vs. faster moment decay)

Signiﬁcance for uncertainty quantiﬁcation
♠ No independence assumption allows for state-dependent noise ξk
Uk+1 := Ψh
(Uk ) + ξk (h, (Uj )j≤k )

Uk+1 := Ψh
(Uk ) + ξk (h, (Uj )j≤k )
Bayesian approach: choose where to place next evaluation points,
based on which evaluation points have already been chosen

Uk+1 := Ψh
(Uk ) + ξk (h, (Uj )j≤k )
♠ Better use of regularity of noise than result of Conrad et al.
(convergence in LR
P topology, for ξk (h) ∈ LR
P )

Uk+1 := Ψh
(Uk ) + ξk (h, (Uj )j≤k )
♠ Better use of regularity of noise than result of Conrad et al.
(convergence in LR
P topology, for ξk (h) ∈ LR
P )
Consequence: if p ≥ q + 1
2 , R ≥ 3, then
P max
k
|uk − Uk | ≥ h ≤ C(hR
)q
< C(h2
)q
Nondegenerate prob. measure P ◦ (Uk )−1
k improves approximation
of Dirac measure at true solution sequence (uk )k more rapidly

Comparison with classical numerical analysis
Assume: uniform moment bound condition, for some R ∈ N.

Decay rates of error incurred by one step:
(deterministic) sup
v∈Rn
Ψh
(v) − Fh
(v)
R
hR(q+1)
(probabilistic) E[|ξk (h)|
R
] (hR
)p+1/2
Recall: p = q + 1

(deterministic) sup
v∈Rn
Ψh
(v) − Fh
(v)
R
hR(q+1)
R
] (hR
)p+1/2
Recall: p = q + 1
♠ Relative to deterministic solvers, probabilistic solver with uniform
moment bound has ‘same performance’

(deterministic) sup
v∈Rn
Ψh
(v) − Fh
(v)
R
hR(q+1)
R
] (hR
)p+1/2
Recall: p = q + 1
♠ Relative to deterministic solvers, probabilistic solver with uniform
moment bound has ‘same performance’
Cost: constants depend on additive noise model (ξk )k
Gain: solver for UQ more ﬂexible than worst-case analysis

Related results:
Corollary (Exponential integrability): If uniform moment bound
condition holds for R = +∞, then
E exp ρ max
k
|uk − Uk | < ∞, ∀ρ ∈ R.
♠ Similar techniques yield same result for maxk |Uk |

Related results:
E exp ρ max
k
|uk − Uk | < ∞, ∀ρ ∈ R.
Corollary (Convergence in continuous time) If for some p, R, Cξ ≥ 1
E sup
0≤t≤h
|ξk (t)|
R
≤ Cξhp+1/2
R
,
ﬂow map Fh
is globally Lipschitz, Ψh
has order q + 1, then
E sup
0≤t≤T
|u(t) − U(t)|
R
≤ C hR
min{q,p−1/2}
.

Related results:
E exp ρ max
k
|uk − Uk | < ∞, ∀ρ ∈ R.
Corollary (Convergence in continuous time) If for some p, R, Cξ ≥ 1
E sup
0≤t≤h
|ξk (t)|
R
≤ Cξhp+1/2
R
,
flow map Fh
is globally Lipschitz, Ψh
has order q + 1, then
E sup
0≤t≤T
|u(t) − U(t)|
R
≤ C hR
min{q,p−1/2}
.
Analogous results for locally Lipschitz vector fields and flow maps,
implicit Euler Ψh

Summary
QMC Program: placing points in space
This project: Use Monte Carlo to place points in ∞-dim. space
Motivation: uncertainty quantiﬁcation in off-grid values

Summary
We provide theoretical support for probabilistic integrators, by
Showing that they can have same performance as deterministic
solvers (modulo constants)

Summary
Leveraging more information in the noise get better
approximations, but faster

Summary
Leveraging more information in the noise get better
approximations, but faster
Removing mean-zero and independence assumptions many
more possibilities for placement of evaluation points

Acknowledgements
Ilse Ipsen for supporting this programme
SAMSI staff for efﬁcient administration: Thomas Gehrmann, Rebecca
Gunn, Sue McDonald, Karem Jackson, and others
Chris Oates, Tim Sullivan, and members of WG Probabilistic
Numerics

Some literature on probabilistic methods for evolution equations
[Caveat: Not a complete list]
Abdulle and Garegnani, “Random time step probabilistic methods for uncertainty quantification in chaotic and
geometric numerical integration”, arXiv:1801.01340
Chkrebtii, Campbell, Calderhead, and Girolami, “Bayesian solution uncertainty quantification for differential
equations”, Bayesian Analysis (2016)
Cockayne, Oates, Sullivan, and Girolami, “Probabilistic numerical methods for PDE-constrained Bayesian inverse
problems”, 36th Int. Workshop on Bayesian Inference and Maximum Entropy Methods, AIP Conf. Proc. (2016)
Conrad, Girolami, Särkka, Stuart, and Zygalakis, “Statistical analysis of differential equations: introducing probability
measures on numerical solutions”, Stat. Comput. (2017)
Kersting and Hennig, “Active uncertainty calibration in Bayesian ODE solvers”, Proc. Conf. Uncertainty in A.I. (2016)
Lie, Stuart and Sullivan, “Strong convergence rates of probabilistic solvers for ordinary differential equations”,
arXiv:1703.03680
Lie, Sullivan and Teckentrup, “Random forward models and log-likelihoods in Bayesian inverse problems”,
arXiv:1712.05717
Magnani, Kersting, Schober, and Hennig, “Bayesian filtering for ODEs with bounded derivatives”, arXiv:1709.08471
Schober, Särkka, and Hennig, “A probabilistic model for the numerical solution of initial value problems”, Stat.
Comput. (2017)
Schober, Duvenaud, and Hennig, “Probabilistic ODE solvers with Runge-Kutta means”, NIPS (2014)
Teymur, Zygalakis, and Calderhead, “Probabilistic Linear Multistep Methods”, NIPS (2016)

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