arXiv subjects

Richard Nickl

Publications and source records attributed to Richard Nickl.

At least 19 recordsLinked to original sources

arXiv2026

On statistical inference for non-linear dynamical systems evolving in their global attractor

We consider a two-dimensional periodic reaction-diffusion system under natural conditions on the reaction function and with initial condition $\theta$. We show that on the global attractor $\mathcal A$ of the resulting dynamical system $(u_\theta(t):t>0)$, a reverse Poincar\'e inequality holds true, and that as a consequence the map $\theta \mapsto u_\theta(t)$ satisfies a $L^2$-Lipschitz stability estimate on $\mathcal A$ for any $t>0$ fixed. We then show that statistical recovery of an initial condition $\theta$ in the attractor $\mathcal A$, as well as prediction of the states $u_\theta$, is possible from discrete measurements of the system at `fast' near parametric convergence rates.

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arXiv2026

On gradient stability in nonlinear PDE models and inference in interacting particle systems

We consider general parameter to solution maps $\theta \mapsto \mathcal G(\theta)$ of non-linear partial differential equations and describe an approach based on a Banach space version of the implicit function theorem to verify the gradient stability condition of Nickl&Wang (JEMS 2024) for the underlying non-linear inverse problem, providing also injectivity estimates and corresponding statistical identifiability results. We illustrate our methods in two examples involving a non-linear reaction diffusion system as well as a McKean--Vlasov interacting particle model, both with periodic boundary conditions. We apply our results to prove the polynomial time convergence of a Langevin-type algorithm sampling the posterior measure of the interaction potential arising from a discrete aggregate measurement of the interacting particle system.

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arXiv2025

Data assimilation with the 2D Navier-Stokes equations: Optimal Gaussian asymptotics for the posterior measure

A functional Bernstein - von Mises theorem is proved for posterior measures arising in a data assimilation problem with the two-dimensional Navier-Stokes equation where a Gaussian process prior is assigned to the initial condition of the system. The posterior measure, which provides the update in the space of all trajectories arising from a discrete sample of the (deterministic) dynamics, is shown to be approximated by a Gaussian random vector field arising from the solution to a linear parabolic PDE with Gaussian initial condition. The approximation holds in the strong sense of the supremum norm on the regression functions, showing that predicting future states of Navier-Stokes systems admits root(N)-consistent estimators even for commonly used nonparametric models. Consequences for coverage of credible bands and uncertainty quantification are discussed. A local asymptotic minimax theorem is derived that describes the lower bound for estimating the state of the nonlinear system, which is shown to be attained by the Bayesian data assimilation algorithm.

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arXiv2025

Inferring diffusivity from killed diffusion

We consider diffusion of independent molecules in an insulated Euclidean domain with unknown diffusivity parameter. At a random time and position, the molecules may bind and stop diffusing in dependence of a given `binding potential'. The binding process can be modeled by an additive random functional corresponding to the canonical construction of a `killed' diffusion Markov process. We study the problem of conducting inference on the infinite-dimensional diffusion parameter from a histogram plot of the `killing' positions of the process. We show first that these positions follow a Poisson point process whose intensity measure is determined by the solution of a certain Schr\"odinger equation. The inference problem can then be re-cast as a non-linear inverse problem for this PDE, which we show to be consistently solvable in a Bayesian way under natural conditions on the initial state of the diffusion, provided the binding potential is not too `aggressive'. In the course of our proofs we obtain novel posterior contraction rate results for high-dimensional Poisson count data that are of independent interest. A numerical illustration of the algorithm by standard MCMC methods is also provided.

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arXiv2024

On low frequency inference for diffusions without the hot spots conjecture

We remove the dependence on the `hot-spots' conjecture in two of the main theorems of the recent paper of Nickl (2024, Annals of Statistics). Specifically, we characterise the minimax convergence rates for estimation of the transition operator $P_{f}$ arising from the Neumann Laplacian with diffusion coefficient $f$ on arbitrary convex domains with smooth boundary, and further show that a general Lipschitz stability estimate holds for the inverse map $P_f\mapsto f$ from $H^2\to H^2$ to $L^1$.

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arXiv2024

Bernstein-von Mises theorems for time evolution equations

We consider a class of infinite-dimensional dynamical systems driven by non-linear parabolic partial differential equations with initial condition $\theta$ modelled by a Gaussian process `prior' probability measure. Given discrete samples of the state of the system evolving in space-time, one obtains updated `posterior' measures on a function space containing all possible trajectories. We give a general set of conditions under which these non-Gaussian posterior distributions are approximated, in Wasserstein distance for the supremum-norm metric, by the law of a Gaussian random function. We demonstrate the applicability of our results to periodic non-linear reaction diffusion equations \begin{align*} \frac{\partial}{\partial t} u - \Delta u &= f(u) \\ u(0) &= \theta \end{align*} where $f$ is any smooth and compactly supported reaction function. In this case the limiting Gaussian measure can be characterised as the solution of a time-dependent Schr\"odinger equation with `rough' Gaussian initial conditions whose covariance operator we describe.

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arXiv2024

Bayesian Nonparametric Inference in McKean-Vlasov models

We consider nonparametric statistical inference on a periodic interaction potential $W$ from noisy discrete space-time measurements of solutions $\rho=\rho_W$ of the nonlinear McKean-Vlasov equation, describing the probability density of the mean field limit of an interacting particle system. We show how Gaussian process priors assigned to $W$ give rise to posterior mean estimators that exhibit fast convergence rates for the implied estimated densities $\bar \rho$ towards $\rho_W$. We further show that if the initial condition $\phi$ is not too smooth and satisfies a standard deconvolvability condition, then one can consistently infer Sobolev-regular potentials $W$ at convergence rates $N^{-\theta}$ for appropriate $\theta>0$, where $N$ is the number of measurements. The exponent $\theta$ can be taken to approach $1/2$ as the regularity of $W$ increases corresponding to `near-parametric' models.

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arXiv2024

Consistent inference for diffusions from low frequency measurements

Let $(X_t)$ be a reflected diffusion process in a bounded convex domain in $\mathbb R^d$, solving the stochastic differential equation $$dX_t = \nabla f(X_t) dt + \sqrt{2f (X_t)} dW_t, ~t \ge 0,$$ with $W_t$ a $d$-dimensional Brownian motion. The data $X_0, X_D, \dots, X_{ND}$ consist of discrete measurements and the time interval $D$ between consecutive observations is fixed so that one cannot `zoom' into the observed path of the process. The goal is to infer the diffusivity $f$ and the associated transition operator $P_{t,f}$. We prove injectivity theorems and stability inequalities for the maps $f \mapsto P_{t,f} \mapsto P_{D,f}, t<D$. Using these estimates we establish the statistical consistency of a class of Bayesian algorithms based on Gaussian process priors for the infinite-dimensional parameter $f$, and show optimality of some of the convergence rates obtained. We discuss an underlying relationship between the degree of ill-posedness of this inverse problem and the `hot spots' conjecture from spectral geometry.

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arXiv2023

On posterior consistency of data assimilation with Gaussian process priors: the 2D Navier-Stokes equations

We consider a non-linear Bayesian data assimilation model for the periodic two-dimensional Navier-Stokes equations with initial condition modelled by a Gaussian process prior. We show that if the system is updated with sufficiently many discrete noisy measurements of the velocity field, then the posterior distribution eventually concentrates near the ground truth solution of the time evolution equation, and in particular that the initial condition is recovered consistently by the posterior mean vector field. We further show that the convergence rate can in general not be faster than inverse logarithmic in sample size, but describe specific conditions on the initial conditions when faster rates are possible. In the proofs we provide an explicit quantitative estimate for backward uniqueness of solutions of the two-dimensional Navier-Stokes equations.

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arXiv2023

On log-concave approximations of high-dimensional posterior measures and stability properties in non-linear inverse problems

The problem of efficiently generating random samples from high-dimensional and non-log-concave posterior measures arising from nonlinear regression problems is considered. Extending investigations from arXiv:2009.05298, local and global stability properties of the model are identified under which such posterior distributions can be approximated in Wasserstein distance by suitable log-concave measures. This allows the use of fast gradient based sampling algorithms, for which convergence guarantees are established that scale polynomially in all relevant quantities (assuming `warm' initialisation). The scope of the general theory is illustrated in a non-linear inverse problem from integral geometry for which new stability results are derived.

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arXiv2022

On free energy barriers in Gaussian priors and failure of cold start MCMC for high-dimensional unimodal distributions

We exhibit examples of high-dimensional unimodal posterior distributions arising in non-linear regression models with Gaussian process priors for which MCMC methods can take an exponential run-time to enter the regions where the bulk of the posterior measure concentrates. Our results apply to worst-case initialised (`cold start') algorithms that are local in the sense that their step-sizes cannot be too large on average. The counter-examples hold for general MCMC schemes based on gradient or random walk steps, and the theory is illustrated for Metropolis-Hastings adjusted methods such as pCN and MALA.

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arXiv2022

On polynomial-time computation of high-dimensional posterior measures by Langevin-type algorithms

The problem of generating random samples of high-dimensional posterior distributions is considered. The main results consist of non-asymptotic computational guarantees for Langevin-type MCMC algorithms which scale polynomially in key quantities such as the dimension of the model, the desired precision level, and the number of available statistical measurements. As a direct consequence, it is shown that posterior mean vectors as well as optimisation based maximum a posteriori (MAP) estimates are computable in polynomial time, with high probability under the distribution of the data. These results are complemented by statistical guarantees for recovery of the ground truth parameter generating the data. Our results are derived in a general high-dimensional non-linear regression setting (with Gaussian process priors) where posterior measures are not necessarily log-concave, employing a set of local `geometric' assumptions on the parameter space, and assuming that a good initialiser of the algorithm is available. The theory is applied to a representative non-linear example from PDEs involving a steady-state Schrödinger equation.

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arXiv2021

On some information-theoretic aspects of non-linear statistical inverse problems

Results by van der Vaart (1991) from semi-parametric statistics about the existence of a non-zero Fisher information are reviewed in an infinite-dimensional non-linear Gaussian regression setting. Information-theoretically optimal inference on aspects of the unknown parameter is possible if and only if the adjoint of the linearisation of the regression map satisfies a certain range condition. It is shown that this range condition may fail in a commonly studied elliptic inverse problem with a divergence form equation, and that a large class of smooth linear functionals of the conductivity parameter cannot be estimated efficiently in this case. In particular, Gaussian `Bernstein von Mises'-type approximations for Bayesian posterior distributions do not hold in this setting.

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arXiv2021

Statistical guarantees for Bayesian uncertainty quantification in non-linear inverse problems with Gaussian process priors

Bayesian inference and uncertainty quantification in a general class of non-linear inverse regression models is considered. Analytic conditions on the regression model $\{\mathscr G(θ): θ\in Θ\}$ and on Gaussian process priors for $θ$ are provided such that semi-parametrically efficient inference is possible for a large class of linear functionals of $θ$. A general semi-parametric Bernstein-von Mises theorem is proved that shows that the (non-Gaussian) posterior distributions are approximated by certain Gaussian measures centred at the posterior mean. As a consequence posterior-based credible sets are valid and optimal from a frequentist point of view. The theory is illustrated with two applications with PDEs that arise in non-linear tomography problems: an elliptic inverse problem for a Schrödinger equation, and inversion of non-Abelian X-ray transforms. New analytical techniques are deployed to show that the relevant Fisher information operators are invertible between suitable function spaces

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arXiv2020

Consistent Inversion of Noisy Non-Abelian X-Ray Transforms

For $M$ a simple surface, the non-linear statistical inverse problem of recovering a matrix field $Φ: M \to \mathfrak{so}(n)$ from discrete, noisy measurements of the $SO(n)$-valued scattering data $C_Φ$ of a solution of a matrix ODE is considered ($n\geq 2$). Injectivity of the map $Φ\mapsto C_Φ$ was established by [Paternain, Salo, Uhlmann; Geom.Funct.Anal. 2012]. A statistical algorithm for the solution of this inverse problem based on Gaussian process priors is proposed, and it is shown how it can be implemented by infinite-dimensional MCMC methods. It is further shown that as the number $N$ of measurements of point-evaluations of $C_Φ$ increases, the statistical error in the recovery of $Φ$ converges to zero in $L^2(M)$-distance at a rate that is algebraic in $1/N$, and approaches $1/\sqrt N$ for smooth matrix fields $Φ$. The proof relies, among other things, on a new stability estimate for the inverse map $C_Φ\to Φ$. Key applications of our results are discussed in the case $n=3$ to polarimetric neutron tomography, see [Desai et al., Nature Sc.Rep. 2018] and [Hilger et al., Nature Comm. 2018]

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arXiv2020

On statistical Calderón problems

For $D$ a bounded domain in $\mathbb R^d, d \ge 2,$ with smooth boundary $\partial D$, the non-linear inverse problem of recovering the unknown conductivity $γ$ determining solutions $u=u_{γ, f}$ of the partial differential equation \begin{equation*} \begin{split} \nabla \cdot(γ\nabla u)&=0 \quad \text{ in }D, \\ u&=f \quad \text { on } \partial D, \end{split} \end{equation*} from noisy observations $Y$ of the Dirichlet-to-Neumann map \[f \mapsto Λ_γ(f) = {γ\frac{\partial u_{γ,f}}{\partial ν}}\Big|_{\partial D},\] with $\partial/\partial ν$ denoting the outward normal derivative, is considered. The data $Y$ consists of $Λ_γ$ corrupted by additive Gaussian noise at noise level $\varepsilon>0$, and a statistical algorithm $\hat γ(Y)$ is constructed which is shown to recover $γ$ in supremum-norm loss at a statistical convergence rate of the order $\log(1/\varepsilon)^{-δ}$ as $\varepsilon \to 0$. It is further shown that this convergence rate is optimal, up to the precise value of the exponent $δ>0$, in an information theoretic sense. The estimator $\hat γ(Y)$ has a Bayesian interpretation in terms of the posterior mean of a suitable Gaussian process prior and can be computed by MCMC methods.

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arXiv2020

Consistency of Bayesian inference with Gaussian process priors in an elliptic inverse problem

For $\mathcal{O}$ a bounded domain in $\mathbb{R}^d$ and a given smooth function $g:\mathcal{O}\to\mathbb{R}$, we consider the statistical nonlinear inverse problem of recovering the conductivity $f>0$ in the divergence form equation $$ \nabla\cdot(f\nabla u)=g\ \textrm{on}\ \mathcal{O}, \quad u=0\ \textrm{on}\ \partial\mathcal{O}, $$ from $N$ discrete noisy point evaluations of the solution $u=u_f$ on $\mathcal O$. We study the statistical performance of Bayesian nonparametric procedures based on a flexible class of Gaussian (or hierarchical Gaussian) process priors, whose implementation is feasible by MCMC methods. We show that, as the number $N$ of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate $N^{-λ}, λ>0,$ for the reconstruction error of the associated posterior means, in $L^2(\mathcal{O})$-distance.

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arXiv2019

Convergence rates for Penalised Least Squares Estimators in PDE-constrained regression problems

We consider PDE constrained nonparametric regression problems in which the parameter $f$ is the unknown coefficient function of a second order elliptic partial differential operator $L_f$, and the unique solution $u_f$ of the boundary value problem \[L_fu=g_1\text{ on } \mathcal O, \quad u=g_2 \text{ on }\partial \mathcal O,\] is observed corrupted by additive Gaussian white noise. Here $\mathcal O$ is a bounded domain in $\mathbb R^d$ with smooth boundary $\partial \mathcal O$, and $g_1, g_2$ are given functions defined on $\mathcal O, \partial \mathcal O$, respectively. Concrete examples include $L_fu=Δu-2fu$ (Schrödinger equation with attenuation potential $f$) and $L_fu=\text{div} (f\nabla u)$ (divergence form equation with conductivity $f$). In both cases, the parameter space \[\mathcal F=\{f\in H^α(\mathcal O)| f > 0\}, ~α>0, \] where $H^α(\mathcal O)$ is the usual order $α$ Sobolev space, induces a set of non-linearly constrained regression functions $\{u_f: f \in \mathcal F\}$. We study Tikhonov-type penalised least squares estimators $\hat f$ for $f$. The penalty functionals are of squared Sobolev-norm type and thus $\hat f$ can also be interpreted as a Bayesian `MAP'-estimator corresponding to some Gaussian process prior. We derive rates of convergence of $\hat f$ and of $u_{\hat f}$, to $f, u_f$, respectively. We prove that the rates obtained are minimax-optimal in prediction loss. Our bounds are derived from a general convergence rate result for non-linear inverse problems whose forward map satisfies a modulus of continuity condition, a result of independent interest that is applicable also to linear inverse problems, illustrated in an example with the Radon transform.

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