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We derive a CUR approximate matrix factorization based on the Discrete Empirical Interpolation Method (DEIM). For a given matrix A, such a factorization provides a low rank approximate decomposition of the form A ≈ CUR, where C and R are subsets of the columns and rows of A, and U is constructed to make CUR a good approximation. Given a low-rank singular value decomposition A ≈ VSW T , the DEIM procedure uses V and W to select the columns and rows of A that form C and R. Through an error analysis applicable to a general class of CUR factorizations, we show that the accuracy tracks the optimal approximation error within a factor that depends on the conditioning of submatrices of V and W. For very large problems, V and W can be approximated well using an incremental QR algorithm that makes only one pass through A. Numerical examples illustrate the favorable performance of the DEIM-CUR method compared to CUR approximations based on leverage scores. IntroductionThis work presents a new CUR matrix factorization based upon the Discrete Empirical Interpolation Method (DEIM). A CUR factorization is a low rank approximation of a matrix A ∈ R m×n of the form A ≈ CUR, where C = A(:, q) ∈ R m×k is a subset of the columns of A and R = A(p, :) ∈ R k×n is a subset of the rows of A.(We generally assume m ≥ n throughout.) The k × k matrix U is constructed to assure that CUR is a good approximation to A. Assuming the best rank-k singular value decomposition (SVD) A ≈ VSW T is available, the algorithm uses the DEIM index selection procedure, q = DEIM(V) and p = DEIM(W), to determine C and R. The resulting approximate factorization is nearly as accurate as the best rank-k SVD, withwhere σ k+1 is the first neglected singular value of A, η p ≡ V(p, : ) −1 , and η q ≡ W(q, : ) −1 .Here and throughout, · denotes the vector 2-norm and the matrix norm it induces, and · F is the Frobenius norm. We use MATLAB notation to index vectors and matrices, so that, e.g., A(p, :) denotes the k rows of A *

Abstract. We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as λ → ∞, dim(σ(H λ )) · log λ converges to an explicit constant (≈ 0.88137). We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schrödinger dynamics generated by the Fibonacci Hamiltonian.

We consider discrete Schrödinger operators with pattern Sturmian potentials. This class of potentials strictly contains the class of Sturmian potentials, for which the spectral properties of the associated Schrödinger operators are well understood. In particular, it is known that for every Sturmian potential, the associated Schrödinger operator has zero-measure spectrum and purely singular continuous spectral measures. We conjecture that the same statements hold in the more general class of pattern Sturmian potentials. We prove partial results in support of this conjecture. In particular, we confirm the conjecture for all pattern Sturmian potentials that belong to the family of Toeplitz sequences. Date