A Time-Dependent Direct Sampling Method for Recovering Moving Potentials in a Heat Equation

Abstract

We are concerned with a numerical reconstruction of the moving potential/absorption coefficient in a heat conduction process when only a single set of boundary measurements of the thermal reflection is available. We propose an efficient direct sampling method (DSM) to locate moving extended objects, represented by time-dependent potentials in a heat equation, and track the trajectories of the moving objects. This appears to be the first DSM for recovering and tracking moving inhomogeneous inclusions in a time-dependent PDE system. Our new method is essentially different from the existing DSMs for solving various stationary or time-harmonic inverse problems but still preserves several important features: it is robust against the noise in the data, easy to implement, and inexpensive computationally. Mathematical justifications are provided to verify the validity of this new method, and insightful mathematical analysis is performed to understand the behavior of the key probing functions proposed. Numerical experiments are presented to demonstrate the effectiveness and efficiency of the method. The DSM provides a new promising numerical strategy for the ill-posed inverse problem of recovering time-dependent moving inhomogeneous media.

Direct Sampling for Recovering Sound Soft Scatterers from Point Source Measurements

“…Once we have shown this, we can employ similar analysis of the direct sampling methods studied in [2]. Recall, that the single layer potential 12)-( 13) with f = S ϕ as defined in (7). Note that by the mapping properties of the single layer potential (see Chapter 6 of [19]), we have that the range of S is a subset of H 1/2 (Γ).…”

“…by the above definition of the operators in ( 6) and (7). Note that the Range(S) ⊂ H 1/2 (∂D) and S : H −1/2 (∂D) −→ H 1/2 (Γ) by the mapping properties in [19].…”

“…Therefore, these methods can be advantageous to use in applications such as non-destructive testing and medical imaging. Similar methods were also studied in diffuse optical tomography [5], electrical impedance tomography [6] and thermodynamics [7]. These methods were studied in detail for far-field data, but little was done for the case of near-field data.…”

Least-Squares Method for Inverse Medium Problems

“…For this purpose, we shall propose to adopt the direct sampling-type method (DSM) that we have been developing in recent years (cf. [17,18,26,27]). Using the index functions provided by DSM, we shall determine a computational domain that is often much smaller than the original physical domain, then the restricted index functions on the computational domain serve as the initial guesses of the unknown coefficients.…”

Monostatic sampling methods in limited-aperture configuration

“…Furthermore, it does not require any additional operations, such as singular value decomposition, and it is highly tolerant of the noise of measured data [8,9,10]. Consequently, the DSM has been applied to various imaging modalities, such as impedance tomography [11], diffusive optical tomography [12], radar imaging [13], and recovering moving potentials in heat equations [14]. In monostatic configuration, an intuitive indicator function of the DSM was proposed in [13] without a theoretical explanation.…”