The paper introduces the butterfly factorization as a data-sparse approximation for the matrices that satisfy a complementary low-rank property. The factorization can be constructed efficiently if either fast algorithms for applying the matrix and its adjoint are available or the entries of the matrix can be sampled individually. For an N × N matrix, the resulting factorization is a product of O(log N ) sparse matrices, each with O(N ) non-zero entries. Hence, it can be applied rapidly in O(N log N ) operations. Numerical results are provided to demonstrate the effectiveness of the butterfly factorization and its construction algorithms.

We develop an efficient algorithm, coordinate descent FCI (CDFCI), for the electronic structure ground state calculation in the configuration interaction framework. CDFCI solves an unconstrained non-convex optimization problem, which is a reformulation of the full configuration interaction eigenvalue problem, via an adaptive coordinate descent method with a deterministic compression strategy. CDFCI captures and updates appreciative determinants with different frequencies proportional to their importance. We show that CDFCI produces accurate variational energy for both static and dynamic correlation by benchmarking the binding curve of nitrogen dimer in the cc-pVDZ basis with 10 −3 mHa accuracy. We also demonstrate the efficiency and accuracy of CDFCI for strongly correlated chromium dimer in the Ahlrichs VDZ basis and produces state-of-theart variational energy. H = p,q t pqâ † pâ q + 1 2 p,r,q,s v prqsâ † pâ † râ qâs , (1)

This paper introduces the interpolative butterfly factorization for nearly optimal implementation of several transforms in harmonic analysis, when their explicit formulas satisfy certain analytic properties and the matrix representations of these transforms satisfy a complementary low-rank property. A preliminary interpolative butterfly factorization is constructed based on interpolative low-rank approximations of the complementary low-rank matrix. A novel sweeping matrix compression technique further compresses the preliminary interpolative butterfly factorization via a sequence of structure-preserving low-rank approximations. The sweeping procedure propagates the low-rank property among neighboring matrix factors to compress dense submatrices in the preliminary butterfly factorization to obtain an optimal one in the butterfly scheme. For an N × N matrix, it takes O(N log N ) operations and complexity to construct the factorization as a product of O(log N ) sparse matrices, each with O(N ) nonzero entries. Hence, it can be applied rapidly in O(N log N ) operations. Numerical results are provided to demonstrate the effectiveness of this algorithm.

This paper presents an efficient multiscale butterfly algorithm for computing Fourier integral operators (FIOs) of the form (Lf )(x) = R d a(x, ξ)e 2πıΦ(x,ξ) f (ξ)dξ, where Φ(x, ξ) is a phase function, a(x, ξ) is an amplitude function, and f (x) is a given input. The frequency domain is hierarchically decomposed into a union of Cartesian coronas. The integral kernel a(x, ξ)e 2πıΦ (x,ξ) in each corona satisfies a special low-rank property that enables the application of a butterfly algorithm on the Cartesian phase-space grid. This leads to an algorithm with quasi-linear operation complexity and linear memory complexity. Different from previous butterfly methods for the FIOs, this new approach is simple and reduces the computational cost by avoiding extra coordinate transformations. Numerical examples in two and three dimensions are provided to demonstrate the practical advantages of the new algorithm.

Leading eigenvalue problems for large scale matrices arise in many applications. Coordinatewise descent methods are considered in this work for such problems based on a reformulation of the leading eigenvalue problem as a non-convex optimization problem. The convergence of several coordinate-wise methods is analyzed and compared. Numerical examples of applications to quantum many-body problems demonstrate the efficiency and provide benchmarks of the proposed coordinate-wise descent methods.Keywords. Stochastic iterative method; coordinate-wise descent method; leading eigenvalue problem.A Proof of Theorem 2 32 B Proof of local convergence 35