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Nonparametric velocity estimation in stochastic convection-diffusion equations from multiple local measurements

Claudia Strauch, Anton Tiepner

Abstract

We investigate pointwise estimation of the function-valued velocity field of a second-order linear SPDE. Based on multiple spatially localised measurements, we construct a weighted augmented MLE and study its convergence properties as the spatial resolution of the observations tends to zero and the number of measurements increases. By imposing Hölder smoothness conditions, we recover the pointwise convergence rate known to be minimax-optimal in the linear regression framework. The optimality of the rate in the current setting is verified by adapting the lower bound ansatz based on the RKHS of local measurements to the nonparametric situation.

Nonparametric velocity estimation in stochastic convection-diffusion equations from multiple local measurements

Abstract

We investigate pointwise estimation of the function-valued velocity field of a second-order linear SPDE. Based on multiple spatially localised measurements, we construct a weighted augmented MLE and study its convergence properties as the spatial resolution of the observations tends to zero and the number of measurements increases. By imposing Hölder smoothness conditions, we recover the pointwise convergence rate known to be minimax-optimal in the linear regression framework. The optimality of the rate in the current setting is verified by adapting the lower bound ansatz based on the RKHS of local measurements to the nonparametric situation.
Paper Structure (20 sections, 25 theorems, 180 equations, 1 figure)

This paper contains 20 sections, 25 theorems, 180 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Pointwise estimation approach
  4. The weighted augmented MLE
  5. Convergence in probability: Upper bound results
  6. Lower bounds
  7. Technical supplement: Auxiliary results and proofs
  8. The rescaled semigroup
  9. Properties of multiple local measurements
  10. Proof of the upper bound
  11. The Fisher information and the martingale part
  12. The remainder term
  13. Proof of the upper bound statement
  14. Proof of the lower bound
  15. Remaining proofs
  16. ...and 5 more sections

Key Result

Theorem 3.1

Under Assumption ass: total, the weighted augmented MLE satisfies In particular, this bound is independent of the spatial location $x\in\mathcal{J}$ in the sense that, for any $\varepsilon>0$, there exist some $M>0$, $\delta'>0$ such that, for any $x\in\mathcal{J}$ and for any $\delta\leq\delta'$, we have

Figures (1)

  • Figure 3.1: (top-left) typical realisation of the solution $X(t,x)$ in $d=1$ with domain $\Lambda=(0,1)$; (top-right) trajectory of $\widehat{\vartheta}_\delta(x)$ compared to $\vartheta(x)=-0.3+1.5x^2$ in the interval $[0.2,0.8]\subset\Lambda$ with weights $w_k(x)$ based on the Epanechnikov kernel; (bottom) $\log$-$\log$ plot of the root mean squared error for estimating $\vartheta$ at $x=0.5$ with $\delta\rightarrow0$, $h\asymp \delta^{2/5}$ (left); $\delta$ fix, $h\rightarrow0$ (right).

Theorems & Definitions (46)

  • Remark 2.1: higher order approximations
  • Theorem 3.1
  • Remark 3.2: convergence rate
  • Corollary 3.3
  • Corollary 3.4
  • Remark 3.5: discussion of Corollary \ref{['cor: intrisk']}
  • Lemma 3.6
  • Example 3.7
  • Theorem 4.1
  • Theorem 4.2
  • ...and 36 more