We present progress in fast, high-resolution imaging, material classification, and fault detection using hyperspectral X-ray measurements. Classical X-ray CT approaches rely on data from many projection angles, resulting in long acquisition and reconstruction times. Additionally, conventional CT cannot distinguish between materials with similar densities. However, in additive manufacturing, the majority of materials used are known a priori. This knowledge allows to vastly reduce the data collected and increase the accuracy of fault detection. In this context, we propose an imaging method for non-destructive testing of materials based on the combination of spectral X-ray CT and discrete tomography. We explore the use of spectral X-ray attenuation models and measurements to recover the characteristic functions of materials in heterogeneous media with piece-wise uniform composition. We show by means of numerical simulation that using spectral measurements from a small number of angles, our approach can alleviate the typical deterioration of spatial resolution and the appearance of streaking artifacts.

We consider an Uncertainty Quantification (UQ) problem for the low-frequency, time-harmonic Maxwell equations with conductivity that is modelled by a fixed layer and a lognormal random field layer. We formulate and prove the well-posedness of the stochastic and the parametric problem; the latter obtained using a Karhunen-Loève expansion for the random field with covariance function belonging to the anisotropic Whittle-Matérn class. For the approximation of the infinitedimensional integrals in the forward UQ problem, we employ the Sparse Quadrature (SQ) method and we prove dimension-independent convergence rates for this model. These rates depend on the sparsity of the parametric representation for the random field and can exceed the convergence rate of the Monte-Carlo method, thus enabling a computationally tractable calculation for Quantities of Interest. To further reduce the computational cost involved in large-scale models, such as those occurring in the Controlled-Source Electromagnetic Method, this work proposes a combined SQ and model reduction approach using the Reduced Basis (RB) and Empirical Interpolation (EIM) methods. We develop goal-oriented, primal-dual based, a posteriori error estimators that enable an adaptive, greedy construction of the reduced problem using training sets that are selected from a sparse grid algorithm. The performance of the SQ algorithm is tested numerically and shown to agree with the estimates. We also give numerical evidence for the combined SQ-EIM-RB method that suggests a similar convergence rate. Finally, we report numerical results that exhibit the behaviour of quantities in the algorithm.

We consider a sketched implementation of the finite element method for elliptic partial differential equations on high-dimensional models. Motivated by applications in real-time simulation and prediction we propose an algorithm that involves projecting the finite element solution onto a lowdimensional subspace and sketching the reduced equations using randomised sampling. We show that a sampling distribution based on the leverage scores of a tall matrix associated with the discrete Laplacian operator, can achieve nearly optimal performance and a significant speedup. We derive an expression of the complexity of the algorithm in terms of the number of samples that are necessary to meet an error tolerance specification with high probability, and an upper bound for the distance between the sketched and the high-dimensional solutions. Our analysis shows that the projection not only reduces the dimension of the problem but also regularises the reduced system against sketching error. Our numerical simulations suggest speed improvements of two orders of magnitude in exchange for a small loss in the accuracy of the prediction.1991 Mathematics Subject Classification. 65F05, 65M60, 68W20.

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