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When working with PDEs the reconstruction of a previous state often provesdifficult. Good prior knowledge and fast computational methods are crucial tobuild a working reconstruction. We want to identify the heat sources on a threedimensional domain from later measurements under the assumption of small,distinct sources, such as hot chippings from a milling tool. This leads us tothe need for a Prior reflecting this a priori information. Sparsity-inducinghyperpriors have proven useful for similar problems with sparse signal or imagereconstruction. We combine the method of using a hierarchical Bayesian modelwith gamma hyperpriors to promote sparsity with low-rank computations for PDEsystems in tensor train format.

A low-rank tensor method to reconstruct sparse initial states for PDEs with Isogeometric Analysis