Parameter estimation in hyperbolic linear SPDEs from multiple measurements

TL;DR

The paper tackles joint estimation of elastic and damping parameters in a general hyperbolic SPDE on a bounded domain from spatially local measurements. A novel augmented maximum likelihood estimator is constructed using localised kernels, and the asymptotic normality is established as the observation resolution shrinks, by leveraging a functional calculus framework built around generalized M,N- functions. The analysis reveals distinct convergence rates for elastic and damping parameters that depend on the orders of the differential operators and the damping strength, and shows asymptotic independence of the parameter blocks under the local-measurement regime. The work integrates operator-theoretic methods with stochastic analysis to derive the limiting Fisher information and CLT, providing a rigorous basis for high-resolution inference in higher-order SPDEs with both heat-like smoothing and wave-like oscillations.

Abstract

The coefficients of elastic and dissipative operators in a linear hyperbolic SPDE are jointly estimated using multiple spatially localised measurements. As the resolution level of the observations tends to zero, we establish the asymptotic normality of an augmented maximum likelihood estimator. The rate of convergence for the dissipative coefficients matches rates in related parabolic problems, whereas the rate for the elastic parameters also depends on the magnitude of the damping. The analysis of the observed Fisher information matrix relies upon the asymptotic behaviour of rescaled $M, N$-functions generalising the operator cosine and sine families appearing in the undamped wave equation. In contrast to the energetically stable undamped wave equation, the $M, N$-functions emerging within the covariance structure of the local measurements have additional smoothing properties similar to the heat kernel, and their asymptotic behaviour is analysed using functional calculus.

Figures (2)

• Figure 2.1: Realisation of the solution $u(t,x)$ to the clamped plate equation on $(0,1)$; (left) $\ddot u(t)=-0.3\Delta^2u(t)+0.3\Delta\dot u(t)+\dot W(t)$; (right) $\ddot u(t)=-0.3\Delta^2u(t)-0.3\dot u(t)+\dot W(t)$.
• Figure 3.1: $\log$-$\log$ plot of the RMSE for $\delta\rightarrow0$ and a maximal number of measurement locations in $d=1$ compared with the theoretical rate in black; (left) structurally damped \ref{['eq: plate struc']} with $\eta_1=0.3$, $\vartheta_1=0.3$; (right) weak damping \ref{['eq: plate weak']} with $\eta_1=-0.3,$$\vartheta_1=0.3$.

Theorems & Definitions (29)

• Example 2.1
• Remark 2.3: Accessibility of the measurements
• Theorem 3.2: Asymptotic behaviour of the joint estimator
• Remark 3.3: Parameter estimation under higher-order damping
• Example 3.4
• Lemma 4.1: Rescaling of operators
• proof : Proof of \ref{['lem: rescaling']}
• Lemma 4.2: Sufficient condition for positivity
• proof : Proof of \ref{['lem: positiveoperator']}
• Remark 4.3