Some advances on inverse particle transport problems with applications to homeland security Gregory Thoreson 1 , Jean C. Ragusa 1 , Wolfgang Bangerth 2 Texas A&M University, 1 Dept. of Nuclear Engineering, 2 Dept. of Mathematics College Station, TX 77843-3133, USA E-mails: gthoreson@tamu.edu ; ragusa@ne.tamu.edu, bangerth@math.tamu.edu INTRODUCTION Generally speaking, inverse problems consist of the recovery of coefficients in a domain from data in the domain (invasive inverse problems) and/or data on its boundaries (noninvasive inverse problems). In particle inverse problems, photons (including X-rays), neutrons or both are used. The most commonly found inverse problems include computed tomography/radiography with applications to medical fields, oceanography, homogenization, aerospace engineering, object detection (including special nuclear material, chemical or biological agents) etc [1,2,3]. For instance, tomographic imaging is a non-contact, non-destructive investigation method that provides cross-sectional images of objects from transmission data measured by illuminating the objects from one or more different directions and locations. A set of mathematical techniques reconstruct (i.e., infer) the composition of the interior of the objects from their cross- sectional images. The common feature of these so-called inverse problems is that the unknowns are the object material properties and the givens are the data collected when illuminating the object with a known source. Data can be collected inside the object (invasive method) or at its boundaries (noninvasive method). For the purpose of homeland security applications, such as cargo and container screening, we will mostly consider noninvasive inverse problems where measurements are performed at the boundaries of the object under scrutiny. The uniqueness of solutions for such problems is discussed in [5,6]; noninvasive problems have in general no unique solution except in the case of 1-group transport with isotropic of mildly anisotropic scattering (see [6]). An unconstrained optimization approach combined with duality principles was previously devised to solve the noninvasive inverse problem [7]. In this paper, we show equivalence of solutions of the previous unconstrained optimization framework with a more conventional constrained optimization approach. We also provide 1-group 2-D sensitivity studies for a sarin vile hidden in a high scatterer object. THEORY The noninvasive reconstruction problem is a nonlinear optimization problem. It consists of finding the distribution of the optical properties of the object under investigation so that the neutron fluxes recovered at the boundary Σ X ∂ of the domain X best match the measured fluxes. We briefly recall earlier results based on the unconstrained problem, describe the constrained problem, and show the equivalence between these two frameworks. Unconstrained minimization problem Assuming that some initial guess of the material property is given, one computes the functional F measuring the discrepancy between the predicted particle fluxes ( ) Ψ Σ at the boundary, obtained by solving the transport equation, and the measured fluxes * Ψ . The functional F to be minimized reads as follows: ( ) 2 * 0 0 1 , () () 2 X n F dE dS d ∞ ∂ Ω⋅ > ⎡ ⎤ ΣΨΣ = Ω ⋅ Ψ Σ −Ψ ⎣ ⎦ ∫ ∫ ∫ G G Ω n (1). Tikhonov regularization terms could be added to F to ensure existence of a solution; we omit them here solely for clarity in demonstrating the equivalence between the two frameworks. Furthermore, for conciseness, we define , b f g dx ± Γ± = ∫ fg where b dx dSdEd = Ω ⋅ Ω n and [ ] ( ) 0, 2 X π ± Γ± = ∂ × ∞× , hence we have: ( ) * 1 , () () , () 2 F * + ΣΨΣ = Ψ Σ −Ψ Ψ Σ −Ψ (2). Once the functional has been defined and an initial guess for the material properties has been chosen, a direct problem { } on , on inc B S D Ψ = Ψ=Ψ Γ− (3) is solved for flux ( ) Ψ Σ due to an extraneous source S, if any, and an incident flux (the illuminating source). inc Ψ B H P = ⋅∇+Σ− − Ω is the transport operator, with H the scattering operator and P the production operator. The phase-space is [ ] ( ) 0, 4 D X π = × ∞× . If the value of the misfit function for this ( ) Ψ Σ is zero, then the material properties are the correct converged values. Otherwise, we need to update the material properties using an iterative method. This is done by evaluating the gradient of this functional with respect to the material properties, i.e., F Σ ∇ , which indicates how material properties influence the particle fluxes. This gradient is: * ( ), () F Σ Σ + ∇ =∂ΨΣ Ψ Σ −Ψ (4). Using duality principles [4,3], Eq. 4 is replaced by: ( ) † ( ),( ) () F B Σ Σ ∇ =Ψ Σ ∂ ΨΣ (5)