We consider dynamical low-rank approximations to parabolic problems on higher-order tensor manifolds in Hilbert spaces. In addition to existence of solutions and their stability with respect to perturbations to the problem data, we show convergence of spatial discretizations. Our framework accommodates various standard low-rank tensor formats for multivariate functions, including tensor train and hierarchical tensors.
In this paper, we present modifications of the iterative hard thresholding (IHT) method for recovery of jointly row-sparse and low-rank matrices. In particular a Riemannian version of IHT is considered which significantly reduces computational cost of the gradient projection in the case of rank-one measurement operators, which have concrete applications in blind deconvolution. Experimental results are reported that show nearoptimal recovery for Gaussian and rank-one measurements, and that adaptive stepsizes give crucial improvement. A Riemannian proximal gradient method is derived for the special case of unknown sparsity.
The existence of weak solutions of dynamical low-rank evolution for parabolic partial differential equations in two spatial dimensions is shown, covering also non-diagonal diffusion in the elliptic part. The proof is based on a variational time-stepping scheme on the low-rank manifold. Moreover, this scheme is shown to be closely related to practical methods for computing such low-rank evolutions.
A Jordan net (resp. web) is an embedding of a unital Jordan algebra of dimension 3 (resp. 4) into the space $${\mathbb{S}}^n$$ S n of symmetric $$n\times n$$ n × n matrices. We study the geometries of Jordan nets and webs: we classify the congruence orbits of Jordan nets (resp. webs) in $${\mathbb{S}}^n$$ S n for $$n\le 7$$ n ≤ 7 (resp. $$n\le 5$$ n ≤ 5 ), we find degenerations between these orbits and list obstructions to the existence of such degenerations. For Jordan nets in $$\mathbb{S}^n$$ S n for $$n\le 5$$ n ≤ 5 , these obstructions show that our list of degenerations is complete . For $$n=6$$ n = 6 , the existence of one degeneration is still undetermined. To explore further, we used an algorithm that indicates numerically whether a degeneration between two orbits exists. We verified this algorithm using all known degenerations and obstructions and then used it to compute the degenerations between Jordan nets in $$\mathbb {S}^7$$ S 7 and Jordan webs in $$\mathbb {S}^n$$ S n for $$n=4,5$$ n = 4 , 5 .
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