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Convexification of Restricted Dirichlet-to-Neumann Map
Numerical Analysis · Submitted 6 Aug 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We develop, in a unified way, a general and a radically new numerical concept for CIPs with restricted DN data for a broad class of PDEs of the second order, such as, e.g. elliptic, parabolic and hyperbolic ones.
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By our definition, "restricted Dirichlet-to-Neumann map" (DN) means that the Dirichlet and Neumann boundary data for a Coefficient Inverse Problem (CIP) are generated by a point source running along an interval of a straight line. On the other hand, the conventional DN data can be generated, at least sometimes, by a point source running along a hypersurface. CIPs with the restricted DN data are non-overdetermined in the $n-$D case with $n \geq 2$. We develop, in a unified way, a general and a radically new numerical concept for CIPs with restricted DN data for a broad class of PDEs of the second order, such as, e.g. elliptic, parabolic and hyperbolic ones. Namely, using Carleman Weight Functions, we construct globally convergent numerical methods. H\"{o}lder stability and uniqueness are also proved. The price we pay for these features is a well acceptable one in the Numerical Analysis: we truncate a certain Fourier-like series with respect to some functions depending only on the position of that point source. At least three applications are: imaging of land mines, crosswell imaging and electrical impedance tomography.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | By our definition, "restricted Dirichlet-to-Neumann map" (DN) means that the Dirichlet and Neumann boundary data for a Coefficient Inverse Problem (CIP) are generated by a point source running along an interval of a straight line. | background 0.93 | 0.33 | 0.03 | asserted 1.00 |
| s1 | On the other hand, the conventional DN data can be generated, at least sometimes, by a point source running along a hypersurface. | background 1.00 | 0.13 | 0.02 | hedged possibility 0.99 |
| s2 | CIPs with the restricted DN data are non-overdetermined in the $n-$D case with $n \geq 2$. | result 0.56 | 0.50 | 0.08 | asserted 1.00 |
| s3 | We develop, in a unified way, a general and a radically new numerical concept for CIPs with restricted DN data for a broad class of PDEs of the second order, such as, e.g. elliptic, parabolic and hyperbolic ones. | contribution 1.00 | 0.95 | 0.34 | asserted 1.00 |
| s4 | Namely, using Carleman Weight Functions, we construct globally convergent numerical methods. | method 0.69 | 0.94 | 0.36 | asserted 1.00 |
| s5 | H\"{o}lder stability and uniqueness are also proved. | result 0.88 | 0.90 | 0.12 | asserted 1.00 |
| s6 | The price we pay for these features is a well acceptable one in the Numerical Analysis: we truncate a certain Fourier-like series with respect to some functions depending only on the position of that point source. | limitation 0.80 | 0.85 | 0.06 | asserted 0.99 |
| s7 | At least three applications are: imaging of land mines, crosswell imaging and electrical impedance tomography. | contribution 0.67 | 0.53 | 0.06 | asserted 0.79 |
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Original abstract
By our definition, "restricted Dirichlet-to-Neumann map" (DN) means that the Dirichlet and Neumann boundary data for a Coefficient Inverse Problem (CIP) are generated by a point source running along an interval of a straight line. On the other hand, the conventional DN data can be generated, at least sometimes, by a point source running along a hypersurface. CIPs with the restricted DN data are non-overdetermined in the $n-$D case with $n \geq 2$. We develop, in a unified way, a general and a radically new numerical concept for CIPs with restricted DN data for a broad class of PDEs of the second order, such as, e.g. elliptic, parabolic and hyperbolic ones. Namely, using Carleman Weight Functions, we construct globally convergent numerical methods. H\"{o}lder stability and uniqueness are also proved. The price we pay for these features is a well acceptable one in the Numerical Analysis: we truncate a certain Fourier-like series with respect to some functions depending only on the position of that point source. At least three applications are: imaging of land mines, crosswell imaging and electrical impedance tomography.
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The question “How do we know both functions $g_0$ and $g_1$ in ([ref]), ([ref] ) if only the function $u$ is usually actually measured in experiments?" is addressed in [ref] via the so-called “data propagation procedure", see a detailed description in [ref] .
Imaging of buried objects from multi-frequency experimental data using a globally convergent inversion method · Helmholtz equation · 0.86
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