An official website of the United States government
Hypergraphs and tensors extend classic graph and matrix theories to account for multiway relationships, which are ubiquitous in engineering, biological, and social systems. While the Kronecker product is a potent tool for analyzing the coupling of systems in a graph or matrix context, its utility in studying multiway interactions, such as those represented by tensors and hypergraphs, remains elusive. In this article, we present a comprehensive exploration of algebraic, structural, and spectral properties of the tensor Kronecker product. We express Tucker and tensor train decompositions and various tensor eigenvalues in terms of the tensor Kronecker product. Additionally, we utilize the tensor Kronecker product to form Kronecker hypergraphs, which are tensor-based hypergraph products, and investigate the structure and stability of polynomial dynamics on Kronecker hypergraphs. Finally, we provide numerical examples to demonstrate the utility of the tensor Kronecker product in computing Z-eigenvalues, performing various tensor decompositions, and determining the stability of polynomial systems.
more »
« less
Scalable Computation of $\mathcal{H}_{\infty}$ Energy Functions for Polynomial Drift Nonlinear Systems
This paper presents a scalable tensor-based approach to computing controllability and observability-type energy functions for nonlinear dynamical systems with polynomial drift and linear input and output maps. Using Kronecker product polynomial expansions, we convert the Hamilton- Jacobi-Bellman partial differential equations for the energy functions into a series of algebraic equations for the coefficients of the energy functions. We derive the specific tensor structure that arises from the Kronecker product representation and analyze the computational complexity to efficiently solve these equations. The convergence and scalability of the proposed energy function computation approach is demonstrated on a nonlinear reaction-diffusion model with cubic drift nonlinearity, for which we compute degree 3 energy function approximations in n = 1023 dimensions and degree 4 energy function approximations in n = 127 dimensions.
more »
« less
- Award ID(s):
- 2130727
- PAR ID:
- 10552207
- Publisher / Repository:
- IEEE
- Date Published:
- ISBN:
- 979-8-3503-8265-5
- Page Range / eLocation ID:
- 2521 to 2526
- Format(s):
- Medium: X
- Location:
- Toronto, ON, Canada
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Abstract We propose and analyse a mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Our approach is based on the introduction of a pseudostress tensor related to the velocity gradient and pressure, leading to a mixed formulation where the pseudostress tensor and the velocity are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators and a regularization technique. We then present well posedness and error analysis for semidiscrete continuous-in-time and fully discrete finite element approximations on simplicial grids with spatial discretization based on the Raviart–Thomas spaces of degree $$k$$ for the pseudostress tensor and discontinuous piecewise polynomial elements of degree $$k$$ for the velocity and backward Euler time discretization. We provide several numerical results to confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method for a range of model parameters.more » « less
-
This paper presents a novel method for generating a single polynomial approximation that produces correctly rounded results for all inputs of an elementary function for multiple representations. The generated polynomial approximation has the nice property that the first few lower degree terms produce correctly rounded results for specific representations of smaller bitwidths, which we call progressive performance. To generate such progressive polynomial approximations, we approximate the correctly rounded result and formulate the computation of correctly rounded polynomial approximations as a linear program similar to our prior work on the RLIBM project. To enable the use of resulting polynomial approximations in mainstream libraries, we want to avoid piecewise polynomials with large lookup tables. We observe that the problem of computing polynomial approximations for elementary functions is a linear programming problem in low dimensions, i.e., with a small number of unknowns. We design a fast randomized algorithm for computing polynomial approximations with progressive performance. Our method produces correct and fast polynomials that require a small amount of storage. A few polynomial approximations from our prototype have already been incorporated into LLVM’s math library.more » « less
-
We consider the problem of matrix approximation and denoising induced by the Kronecker product decomposition. Specifically, we propose to approximate a given matrix by the sum of a few Kronecker products of matrices, which we refer to as the Kronecker product approximation (KoPA). Because the Kronecker product is an extensions of the outer product from vectors to matrices, KoPA extends the low rank matrix approximation, and includes it as a special case. Comparing with the latter, KoPA also offers a greater flexibility, since it allows the user to choose the configuration, which are the dimensions of the two smaller matrices forming the Kronecker product. On the other hand, the configuration to be used is usually unknown, and needs to be determined from the data in order to achieve the optimal balance between accuracy and parsimony. We propose to use extended information criteria to select the configuration. Under the paradigm of high dimensional analysis, we show that the proposed procedure is able to select the true configuration with probability tending to one, under suitable conditions on the signal-to-noise ratio. We demonstrate the superiority of KoPA over the low rank approximations through numerical studies, and several benchmark image examples.more » « less
-
The stabilizer rank of a quantum state ψ is the minimal r such that | ψ ⟩ = ∑ j = 1 r c j | φ j ⟩ for c j ∈ C and stabilizer states φ j . The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the n -th tensor power of single-qubit magic states.We prove a lower bound of Ω ( n ) on the stabilizer rank of such states, improving a previous lower bound of Ω ( n ) of Bravyi, Smith and Smolin \cite{BSS16}. Further, we prove that for a sufficiently small constant δ , the stabilizer rank of any state which is δ -close to those states is Ω ( n / log n ) . This is the first non-trivial lower bound for approximate stabilizer rank.Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of F 2 n , and we use tools from analysis of boolean functions and complexity theory. The proof of the first result involves a careful analysis of directional derivatives of quadratic polynomials, whereas the proof of the second result uses Razborov-Smolensky low degree polynomial approximations and correlation bounds against the majority function.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.23919/ACC60939.2024.10644363
