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Hölder-Logarithmic Stability and Convergence Rates for an Inverse Random Source Problem

Philipp Mickan, Thorsten Hohage

Abstract

In this paper, we investigate an inverse random source problem concerned with recovering the strength of a random, uncorrelated acoustic source from correlation measurements of emitted time-harmonic acoustic waves. Such problems arise in applications including aeroacoustics and seismic imaging. Unlike their deterministic counterparts, inverse random source problems are known to be uniquely solvable in the absence of noise. Nevertheless, due to their inherent ill-posedness, regularization is required to stably reconstruct the source strength. We derive conditional Hölder-logarithmic stability estimates under Sobolev smoothness assumptions by employing complex geometrical optics solutions. Moreover, by establishing a variational source condition, we obtain Hölder-logarithmic convergence rates for spectral regularization methods. At fixed frequency, the exponents in the logarithmic stability and convergence estimates grow unboundedly as the Sobolev regularity of the source increases. Finally, we present numerical experiments supporting our theoretical findings.

Hölder-Logarithmic Stability and Convergence Rates for an Inverse Random Source Problem

Abstract

In this paper, we investigate an inverse random source problem concerned with recovering the strength of a random, uncorrelated acoustic source from correlation measurements of emitted time-harmonic acoustic waves. Such problems arise in applications including aeroacoustics and seismic imaging. Unlike their deterministic counterparts, inverse random source problems are known to be uniquely solvable in the absence of noise. Nevertheless, due to their inherent ill-posedness, regularization is required to stably reconstruct the source strength. We derive conditional Hölder-logarithmic stability estimates under Sobolev smoothness assumptions by employing complex geometrical optics solutions. Moreover, by establishing a variational source condition, we obtain Hölder-logarithmic convergence rates for spectral regularization methods. At fixed frequency, the exponents in the logarithmic stability and convergence estimates grow unboundedly as the Sobolev regularity of the source increases. Finally, we present numerical experiments supporting our theoretical findings.
Paper Structure (8 sections, 7 theorems, 75 equations, 5 figures, 1 table)

This paper contains 8 sections, 7 theorems, 75 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Variational source conditions
  4. Proof of \ref{['prop:VSC']}
  5. Extension to convected Helmholtz equation
  6. Numerical tests of convergence rates
  7. Wave number dependence for flat sources
  8. Conclusion

Key Result

Theorem 2.3

Suppose ass:SourceAssumptionsass:Measurement are satisfied. Let $m\geq 0$ and $s,C_{\mathrm{s}}>0$ with $m<s$. Moreover, take $Y=\mathop{\mathrm{HS}}\nolimits(L^2(\mathop{\mathrm{\mathbb{M}}}\nolimits))$ and $X=H^m(D)$. Then there exists a constant $C>0$ depending on $m,s,C_{\mathrm{s}}, \kappa$ and holds true. More precisely, we have a Hölder-logarithmic conditional stability estimate with index

Figures (5)

  • Figure 1: Each row shows on the left the exact solution $q^\dagger$ given by multivariate splines of degree $1$ (top) and $3$ (bottom) approximating a half-sphere and hollow dot. In the middle and right are reconstructions using $\Vert \cdot\Vert_{L^2}$ as regularization penalty for the respective exact solution, fixed wave length $\kappa=6$, and different sample sizes $N$.
  • Figure 2: Displayed on the left the exact source strength with randomly choose linear spline coefficients $c^{\lambda}_{k,l,m}\sim |X^{\lambda}_{k,l,m}|$ and on the right reconstructions for fixed wave number $\kappa=12$, distance $R=4$ and regularization penalty $\Vert \cdot\Vert_{L^2}$ , and different sample numbers.
  • Figure 3: Each of the four panels shows the reconstruction error on a logarithmic scale over the data noise level $\delta$ on a double logarithmic scale for one kind of exact solution. For each line synthetic data were generated for 18 values of the sample size $N$ between 550 and 92500. The measurement radius $R$, the wave number $\kappa$, and the regularization norm are indicated by line type, marker type, and color, respectively.
  • Figure 4: The reconstruction of a flat ground truth given by a unfinished circle with a dot very close at the bottom and bubbles of decreasing size on the left for different wave number $\kappa$
  • Figure 5: Rates of the relative error in the $L^2$ norm for the surface source for different wave numbers.

Theorems & Definitions (13)

  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5: Hölder-logarithmic rates
  • Proposition 3.1
  • Lemma 4.1
  • Proof 1
  • Lemma 4.2
  • Proof 2
  • Remark 4.3
  • Proof 3: Proof of \ref{['prop:VSC']}
  • ...and 3 more