WIAS Preprint No. 441, (1998)

Determination of point wave sources by pointwise observations: stability and reconstruction



Authors

  • Bruckner, Gottfried
  • Yamamoto, Masahiro

2010 Mathematics Subject Classification

  • 35L05 35R30 65R30

Keywords

  • Point source reconstruction, wave equation, uniqueness, stability, regularization

Abstract

We consider a wave equation with point source terms: $$ leftaligned fracpartial^2 upartial t^2(x,t) & = fracpartial^2 upartial x^2 (x,t) + lambda(t)sum^N_k=1alpha_kdelta(x-x_k), qquad 0 0$. par We prove uniqueness and stabilty in determining point sources in terms of the norm in $H^1(0,T)$ of observations. The uniqueness result requires that $eta$ is an irrational number and $T ge 1$, and our stability result further needs a-priori (but reasonable) informations of unknown $ x_1, ..., x_N $. Moreover, we establish two schemes for reconstructing $ x_1, ...., x_N $ which are stable against errors in $L^2(0,T)$.

Appeared in

  • Inverse Problems, 16(2000), pp. 723-748

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