Parameter estimation for semilinear SPDEs from local measurements

Altmeyer, Randolf; Cialenco, Igor; Pasemann, Gregor

doi:10.48550/arxiv.2004.14728

Cited by 6 publications

(15 citation statements)

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“…Note that the existence of continuous trajectories is a crucial requirement for dealing with fully discrete observations. Even in the linear setting f ≡ 0, function valued solutions to equation (2) only exist in dimension one. In order to deal with larger dimensions, it would be necessary to consider a more regular noise process than a cylindrical Brownian motion.…”

Section: Basic Assumptions and Hölder Regularity Of The Solution Processmentioning

confidence: 99%

“…This theory hast been generalized by Pasemann and Stannat [41] as well as Pasemann et al [40] to more general equations. Diffusivity estimation based on the local measurements approach due to Altmeyer and Reiß [3] was studied in a semilinear framework, see Altmeyer et al [1,2]. Here, the observations are given by X t , K h , t ∈ [0, T ], for a kernel function K h that localizes in space as h → 0.…”

Section: Introductionmentioning

confidence: 99%

“…Similar to [2,41], our techniques can principally be applied to more general types of semilinear SPDEs, e.g., Burgers' equation where the nonlinearity is given by F (u) = −u ∂ ∂x u. To facilitate the analysis, we would have to consider a more regular noise process than a cylindrical Brownian motion.…”

Section: Introductionmentioning

confidence: 99%

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Nonparametric calibration for stochastic reaction-diffusion equations based on discrete observations

Hildebrandt¹,

Trabs²

2021

Preprint

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Nonparametric estimation for semilinear SPDEs, namely stochastic reaction-diffusion equations in one space dimension, is studied. We consider observations of the solution field on a discrete grid in time and space with infill asymptotics in both coordinates. Firstly, based on a precise analysis of the Hölder regularity of the solution process and its nonlinear component, we show that the asymptotic properties of diffusivity and volatility estimators derived from realized quadratic variations in the linear setup generalize to the semilinear SPDE. In particular, we obtain a rate-optimal joint estimator of the two parameters. Secondly, we derive a nonparametric estimator for the reaction function specifying the underlying equation. The estimate is chosen from a finite-dimensional function space based on a simple least squares criterion. Oracle inequalities with respect to both the empirical and usual L 2 -risk provide conditions for the estimator to achieve the usual nonparametric rate of convergence. Adaptivity is provided via model selection.

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Section: Basic Assumptions and Hölder Regularity Of The Solution Processmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonparametric calibration for stochastic reaction-diffusion equations based on discrete observations

Hildebrandt¹,

Trabs²

2021

Preprint

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“…We present several types of semilinear SPDEs whose nonlinearity F satisfies (5.5), which in particular guarantees that all results from this section hold for the solutions to these classes of equations. For technical details see [PS20,ACP20]. 1) (fractional) Heat equation: In the case F = 0, (5.1) becomes linear, sometimes called fractional head equation, and (5.5) is trivially satisfied for any η > 0.…”

Section: Semilinear Spdes On a Bounded Domainmentioning

confidence: 99%

Statistical analysis of discretely sampled semilinear SPDEs: a power variation approach

Cialenco¹,

Kim²,

Pasemann³

2021

Preprint

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Motivated by problems from statistical analysis for discretely sampled SPDEs, first we derive central limit theorems for higher order finite differences applied to stochastic process with arbitrary finitely regular paths. These results are proved by using the notion of ∆-power variations, introduced herein, along with the Hölder-Zygmund norms. Consequently, we prove a new central limit theorem for ∆-power variations of the iterated integrals of a fractional Brownian motion (fBm). These abstract results, besides being of independent interest, in the second part of the paper are applied to estimation of the drift and volatility coefficients of semilinear stochastic partial differential equations in dimension one, driven by an additive Gaussian noise white in time and possibly colored in space. In particular, we solve the earlier conjecture from [CKL20] about existence of a nontrivial bias in the estimators derived by naive approximations of derivatives by finite differences. We give an explicit formula for the bias and derive the convergence rates of the corresponding estimators. Theoretical results are illustrated by numerical examples.

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“…Owing to the physical restriction of being able to measure only local averages, [3] have introduced local measurements and constructed estimators in a linear SPDE for the diffusion term which are provably rate-optimal. Even more, the proposed estimators apply in a nonparametric setting of spatially varying diffusion and are robust to misspecification of the noise or when subject to certain nonlinearities [2].…”

Section: Introductionmentioning

confidence: 99%

Parameter Estimation in an SPDE Model for Cell Repolarisation

Altmeyer¹,

Bretschneider²,

Janák³

et al. 2020

Preprint

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As a concrete setting where stochastic partial differential equations (SPDEs) are able to model real phenomena, we propose a stochastic Meinhardt model for cell repolarisation and study how parameter estimation techniques developed for simple linear SPDE models apply in this situation. We establish the existence of mild SPDE solutions and we investigate the impact of the driving noise process on pattern formation in the solution. We then pursue estimation of the diffusion term and show asymptotic normality for our estimator as the space resolution becomes finer. The finite sample performance is investigated for synthetic and real data resembling experimental findings for cell orientation.

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Parameter estimation for semilinear SPDEs from local measurements

Cited by 6 publications

References 0 publications

Nonparametric calibration for stochastic reaction-diffusion equations based on discrete observations

Nonparametric calibration for stochastic reaction-diffusion equations based on discrete observations

Statistical analysis of discretely sampled semilinear SPDEs: a power variation approach

Parameter Estimation in an SPDE Model for Cell Repolarisation

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