In this talk we shall review the historical developments of direct sampling methods for solving general nonlinear inverse problems of partial dierential equations. Nonlinear inverse problems are mostly ill-posed and much harder to solve than direct problems. Popular least-squares minimizations with Tikhonov regularizations are generally nonlinear, non-convex, non-smooth, and computationally very expensive in applications.

Direct sampling methods (DSMs) were initially proposed for inverse acoustic scattering problems, using the far-eld data [8] and the near-eld data [6]. The DSM was derived and justied in [7] using a dierent approach from [8], especially the smallness” assumption was removed that is crucial to the derivation of the method in [8]. The DSM was further developed for inverse Maxwell scattering problems [4], and then for non-wave type inverse problems, e.g., for EIT in [3], for DOT in [2] and for moving inclusions in [1]. The DSMs are computationally very cheap, focusing on signicant applications with very limited data available and existing methods being too expensive or not applicable.

General motivations, principles and justications of DSMs are presented in this talk, and choices of the key ingredients of the DSMs are discussed. Numerical experiments are also demonstrated for various inverse problems. The main results of the talk are selected from the joint works with Yat Tin Chow (UCR), Kazufumi Ito (NCSU), Bangti Jin (UCL) and Jingzhi Li (SUSTC). The research projects have been substantially supported by NSFC/Hong Kong RGC Joint Scheme 2016/17 (Project N CUHK437/16) and Hong Kong RGC General Research Fund (Projects 14304517).

Laser‐based diagnostics, including line‐of‐sight attenuation (LOSA) spectroscopy, multiangle light scattering (MALS) and laser‐induced incandescence (LII), are mainstay tools for measuring soot in combustion‐related applications.

They have also been applied to characterize aerosols of metal and metalloid nanoparticles to develop gas phase synthesis routes, and, increasingly, to characterize fundamental thermodynamic properties of materials at extreme temperatures. In all these applications, it is crucial to quantify the uncertainty in the derived parameters, particularly since the inference procedures for these diagnostics are often mathematically ill‐posed.

This talk reviews how Bayesian data analysis can be used to quantify uncertainty caused by measurement noise and model error, how data from multiple measurement modalities and prior information can be combined to mitigate illposedness, and how Bayesian model selection can be used to identify the most probable measurement model.

The Laplace transformation is commonly used for modeling dynamical systems. It allows transforming an ordinary differential equation (ODE) into a non differential one. If time is the independent variable, and if both the equation and its initial conditions are linear, with time invariant coefficients (LTI), the Laplace transform of the solution is proportional either to the transform of the source term, that is the variation of its intensity with time (case of a forced response) ot to the non-zero initial condition(s) (relaxation case). If one gets back to the time domain, the forced solution is a convolution product between the original (inverse transform) of the transfer function, that is the impulse response, and the intensity of the source. This property can be extended to space discretized problems that are governed by a system of ODEs’ with initial conditions (electrical circuits in transient regime for example).

For partial differential equations (PDE) systems under LTI assumptions, such as conduction heat transfer, some analytical solutions, in the Laplace domain, can be derived for configurations where boundary conditions apply over simple geometrical surfaces (slabs, cylinders, spheres, ..), see the Thermal Quadrupole method for example [1]. They are now complemented by numerical inversion algorithms of the corresponding transforms at discrete times to recover the transient detailed temperature/flux fields.

The Laplace transformation can also be applied to physical systems where heat transfer is governed by a PDE system, but with non simple geometries: it is the case for a heterogeneous physical system (including solids and flowing fluid, even with linearized radiation in a cavity configuration) under LTI assumptions (time constant, but not necessary uniform, velocity field), in any 3D geometry. This also requires each single transient source (the input) to be separable, which means it can be written, in the transient heat equation and in its boundary conditions as a product of a time part, its intensity, by a space part, its geometrical support [2]. The temperature response at any point (output) is still a convolution product in time between its cause (source or input), and the impulse response.
In practice, the impulse response has to be found through solving an inverse problem, here a deconvolution. One can either use a numerical temperature solution of a Finite Element simulation code for a given source (model reduction), or the experimental noisy temperature signal delivered by a local sensor for a measured source in a calibration experiment (model identification, which requires some kind of regularization in the inversion). Examples of experimentally identified impulse responses, for characterizing heat exchangers [3], are given in this presentation: impedances, between a temperature and a thermal power, or transmittances, linking temperatures at two different points.

References:
1. Maillet, S. André, J.-C. Batsale, A. Degiovanni, C. Moyne, Thermal Quadrupoles – Solving the Heat Equation through Integral Transforms, Wiley, Chichester, 2000.
2. W. Al Hadad, D. Maillet, Y. Jannot, Modeling unsteady diffusive and advective heat transfer for linear dynamical systems: A transfer function approach, Int. J. Heat Mass Transfer (2017) 115, 304313.
3. W. Al Hadad, D. Maillet, Y. Jannot, Experimental transfer functions identification: Thermal impedance and transmittance in a channel heated by an upstream unsteady volumetric heat source, Int. J. Heat Mass Transfer (2018) 116, 931-339.

An inverse technique is applied to retrieve heat diffusivity/conductivity using an original technique based on active thermography. The heat load is produced by a laser flash, while the spatial and temporal variation of temperature is recorded by an IR camera. The used models of the direct solver range from simple analytical heat conduction solutions, to numerical 3D CFD solvers. The former uses the concept of semi-infinite domain with laser heating modeled by Dirac’s delta, leading to a simple, closed-form based on appropriate Green’s function. The latter works in realistic geometry of the body accounting for the finite time and space dimension of the laser impulse. In this case, the geometry may be captured using laser scanning. The numerical direct solver used in the inverse problem loop requires prohibitive computation times. This question is handled resorting to Reduced Order Method based on the Proper Orthogonal Decomposition coupled with the Radial Basis Functions.

An important question in IR temperature measurement is the knowledge of surface emissivity, property that is difficult and expensive to measure. Two ways of circumventing this problem are proposed: one is the usage of an auxiliary variable defined as a ratio of temperature excesses over the initial conditions, taken at the same point but two different times. Inverse procedure (Levenberg Marquard) compares then not the simulated and measured temperatures, but these auxiliary variables. As a side effect of this approach, the dependence of the solution from the difficult to assess, the absorbed amount of laser ray is obtained. Strictly, this is true only in the analytical models, while for the numerical solver it works under the assumption, that the distribution of energy across the laser ray, is uniform. Another, the more general approach is to use the time instant, at which the measured by the IR camera attains maximum, instead of working with temperatures.

The paper will also show the application of the Bayesian technique with Monte Carlo Markov Chain to generate the subsequent elements of the direct solutions. Also, in this case, the acceleration of the process can be achieved by resorting to the Reduced Order Model.

The paper will discuss also the extension of the method to anisotropic media, treatment of internal channels and some types of layered media. Comparison with results obtained by commercial equipment implementing Parker’s flash method shows good accuracy of the proposed approach.