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boundary observation operator

A boundary observation operator is a mathematical mapping that extracts measurable physical quantities along the boundary of a domain from the full internal solution or state of a system governed by partial differential equations. In mathematical physics, system control, and inverse problems, this operator models the practical measurement process where sensors collect data, such as field values, surface potentials, or normal flux derivatives, strictly at the perimeter rather than inside the medium. Formally realized through trace or restriction operations, the boundary observation operator relates the continuous interior behavior of a physical system to observable boundary datasets, serving as a fundamental component in reconstructing unknown interior parameters, material properties, or source terms from external measurements.

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