Abstract Over the past decades, transformations between different classes of eigenvalue problems have played a central role in the development of numerical methods for eigenvalue computations. One of the most well-known and successful examples of this is the companion linearization for polynomial eigenvalue problems. In this paper, we construct a transformation that equivalently reframes a specific type of eigenvalue problem with eigenvector nonlinearities (NEPv) into an eigenvalue problem with eigenvalue nonlinearities (NEP). The NEPv class considered consists of nonlinearities expressed as sums of products of matrices and scalar functions, where the scalar functions depend nonlinearly on the eigenvector. Our transformation defines scalar eigenvalue nonlinearities through a polynomial system, resulting in NEP nonlinearities of algebraic type. We propose methods to solve the polynomial system, one of which involves a multiparameter eigenvalue problem (MEP). We adapt well-established NEP solvers to this setting, with the most effective strategy being a combination of deflation and a locally quadratically convergent iterative method. The efficiency and properties of the approach are illustrated by solving a problem related to a modification of a Gross-Pitaevskii equation (GPE). The simulations are reproducible and publicly available.

Abstract Sketching techniques have gained popularity in numerical linear algebra to accelerate the solution of least squares problems. The so-called $$\varepsilon $$ ε -subspace embedding property of a sketching matrix S has been largely used to characterize the problem residual norm, since the procedure is no longer optimal in terms of the (classical) Frobenius or Euclidean norm. By building on available results on the SVD of the sketched matrix SA derived by Gilbert, Park, and Wakin (Proc. of SPARS-2013), a novel decomposition of A , the $$S^\top \!S$$ S ⊤ S -SVD, is proposed, which holds with high probability, and in which the left singular vectors are orthonormal with respect to a (semi-)norm defined by the sketching matrix S . The new decomposition is less expensive to compute than the standard SVD, while preserving the singular values with probabilistic confidence. The $$S^\top \!S$$ S ⊤ S -SVD appears to be the right tool to analyze the quality of several sketching-based techniques in the literature, for which examples are reported. For instance, it is possible to simply bound the distance from (standard) orthogonality of sketching-based orthogonal matrices in state-of-the-art randomized algorithms for QR factorizations. As an application, the classical problem of the nearest orthogonal matrix is generalized to the new $$S^\top \!S$$ S ⊤ S -orthogonality, and the $$S^\top \!S$$ S ⊤ S -SVD is used to solve it. Probabilistic bounds on the quality of the solution are also derived.

In this paper we develop a Neumann-Neumann type domain decomposition method for elliptic problems on metric graphs. We describe the iteration in the continuous and discrete setting and rewrite the latter as a preconditioner for the Schur complement system. Then we formulate the discrete iteration as an abstract additive Schwarz iteration and prove that it converges to the finite element solution with a rate that is independent of the finite element mesh size. We also show that the condition number of the Schur complement is uniformly bounded with respect to the finite element mesh size. We provide an implementation and test it on various examples of interest and compare it to other preconditioners.