The recently introduced polynomial time integration framework proposes a novel way to construct time integrators for solving systems of first-order ordinary differential equation by using interpolating polynomials in the complex time plane. In this work we continue to develop the framework by introducing several additional types of polynomials and proposing a general class of construction strategies for polynomial block methods with imaginary nodes. The new construction strategies do not involve algebraic order conditions and are instead motivated by geometric arguments similar to those used for constructing traditional spatial finite differences. Moreover, the newly proposed methods address several shortcomings of previously introduced polynomial block methods including the ability to solve dispersive equations and the lack of efficient serial methods when parallelism cannot be used. To validate our new methods, we conduct two numerical experiments that compare the performance of polynomial block methods against backward difference methods and implicit Runge-Kutta schemes.
We propose an approach to constructing iterative methods for finding polynomial roots simultaneously. One feature of this approach is using the fundamental theorem of symmetric polynomials. Within this framework, we reconstruct many of the existing root finding methods. The new results presented in this paper are some modifications of the Durand-Kerner method.
In this paper we extend the polynomial time integration framework to include exponential integration for both partitioned and unpartitioned initial value problems. We then demonstrate the utility of the exponential polynomial framework by constructing a new class of parallel exponential polynomial block methods (EPBMs) based on the Legendre points. These new integrators can be constructed at arbitrary orders of accuracy and have improved stability compared to existing exponential linear multistep methods. Moreover, if the ODE right-hand side evaluations can be parallelized efficiently, then hi…
Low-rank tensors are an established framework for high-dimensional least-squares problems. We propose to extend this framework by including the concept of block-sparsity. In the context of polynomial regression each sparsity pattern corresponds to some subspace of homogeneous multivariate polynomials. This allows us to adapt the ansatz space to align better with known sample complexity results. The resulting method is tested in numerical experiments and demonstrates improved computational resource utilization and sample efficiency.
This paper describes and develops a fast and accurate path following algorithm that computes the field of values boundary curve for every conceivable complex or real square matrix $A$. It relies on a matrix flow decomposition method that finds a proper block-diagonal flow representation for the associated hermitean matrix flow ${\cal F}_A(t) = \cos(t) H + \sin(t) K$. Here ${\cal F}_A(t)$ is a 1-parameter-varying linear combination of the real and skew part matrices $H = (A+A^*)/2$ and $K = (A-A^*)/(2i)$ of $A$. For decomposing flows ${\cal F}_A(t)$, the algorithm decomposes a given dense gener…
