skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Data-Driven Model Reduction for a Class of Semi-Explicit DAEs Using the Loewner Framewor
This paper introduces a modified version of the recent data-driven Loewner framework to compute reduced order models (ROMs) for a class of semi-explicit differential algebraic equation (DAE) systems, which include the semi-discretized linearized Navier-Stokes /Oseen equations. The modified version estimates the polynomial part of the original transfer function from data and incorporate this estimate into the Loewner ROM construction. Without this proposed modification the transfer function of the Loewner ROM is strictly proper, i.e., goes to zero as the magnitude of the frequency goes to infinity, and therefore may have a different behavior for large frequencies than the transfer function of the original system. The modification leads to a Loewner ROM with a transfer function that has a strictly proper and a polynomial part, just as the original model. This leads to better approximations for transfer functioncomponents in which the coefficients in the polynomial part are not too small. The construction of the improved Loewner ROM is described and the improvement is demonstrated on a large-scale system governed by the semi-discretized Oseen equations.  more » « less
Award ID(s):
1819144
PAR ID:
10292944
Author(s) / Creator(s):
; ;
Editor(s):
Reis, T.; Grundel, S.; Schöps, S.
Date Published:
Journal Name:
Progress in Differential-Algebraic Equations II
Page Range / eLocation ID:
185--210
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; (, Active Flow and Combustion Control 2021)
    King, R.; Peitsch, D. (Ed.)
    The Loewner framework is extended to compute reduced order models (ROMs) for systems governed by the incompressible Navier-Stokes (NS) equations. For quadratic ordinary differential equations (ODEs) it constructs a ROM directly from measurements of transfer function components derived from an expansion of the system’s input-to-output map. Given measurements, no explicit access to the system is required to construct the ROM. To extend the Loewner framework, the NS equations are transformed into ODEs by projecting onto the subspace defined by the incompressibility condition. This projection is used theoretically, but avoided computationally. This paper presents the overall approach. Currently, transfer function measurements are obtained via computational simulations; obtaining them from experiments is an open issue. Numerical results show the potential of the Loewner framework, but also reveal possible lack of stability of the ROM. A possible approach, which currently requires access to the NS system, to deal with these instabilities is outlined. 
    more » « less
  2. ; (, Realization and Model Reduction of Dynamical Systems - A Festschrift in Honor of the 70th Birthday of Thanos Antoulas)
    Beattie, C.A.; Benner, P.; Embree, M.; Gugercin, S.; Lefteriu, S. (Ed.)
    This paper introduces reduced order model (ROM) based Hessian approximations for use in inexact Newton methods for the solution of optimization problems implicitly constrained by a large-scale system, typically a discretization of a partial differential equation (PDE). The direct application of an inexact Newton method to this problem requires the solution of many PDEs per optimization iteration. To reduce the computational complexity, a ROM Hessian approximation is proposed. Since only the Hessian is approximated, but the original objective function and its gradient is used, the resulting inexact Newton method maintains the first-order global convergence property, under suitable assumptions. Thus even computationally inexpensive lower fidelity ROMs can be used, which is different from ROM approaches that replace the original optimization problem by a sequence of ROM optimization problem and typically need to accurately approximate function and gradient information of the original problem. In the proposed approach, the quality of the ROM Hessian approximation determines the rate of convergence, but not whether the method converges. The projection based ROM is constructed from state and adjoint snapshots, and is relatively inexpensive to compute. Numerical examples on semilinear parabolic optimal control problems demonstrate that the proposed approach can lead to substantial savings in terms of overall PDE solves required. 
    more » « less
  3. ; ; (, Financial Cryptography and Data Security)
    null (Ed.)
    We provide a modified version of the Ligero sublinear zero knowledge proof system for arithmetic circuits provided by Ames et. al. (CCS ‘17). Our modification "BooLigero" tailors Ligero for use in Boolean circuits to achieve a significant improvement in proof size. Although the original Ligero system could be used for Boolean circuits, Ligero generally requires allocating an entire field element to represent a single bit on a wire in a Boolean circuit. In contrast, our system performs operations over words of bits, allowing a proof size savings of between O(log(|F|)^1/4) and O(log(|F|)^1/2) compared to Ligero, where F is the field that leads to the optimal proof size in original Ligero. We achieve improvements in proof size of approximately 1.1-1.6x for SHA-2 and 1.7-2.8x for SHA-3. In addition to checking constraints of standard Boolean operations such as AND, XOR, and NOT over words, BooLigero also supports several other constraints such as multiplication in GF(2^w), bit masking, bit rearrangement within and across words, and bitwise outer product. Like Ligero, construction requires no trusted setup and no computational assumptions, which is ideal for blockchain applications. It is plausibly post-quantum secure in the standard model. Furthermore, it is public-coin, perfect honest-verifier zero knowledge, and can be made non-interactive in the random oracle model using the Fiat-Shamir transform. 
    more » « less
  4. ; (, Numerical Methods for Partial Differential Equations)
    Abstract In this article we study the stability of explicit finite difference discretization of advection–diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability of the system of ordinary differential equations that is obtained by discretizing the ADE in space and then extends to fully discretized methods in combination with explicit Runge–Kutta methods. In particular, we prove that all stable semi‐discretization of the ADE leads to a conditionally stable fully discretized method as long as the time‐integrator is at least first‐order accurate, whereas high‐order spatial discretization of the advection equation cannot yield a stable method if the temporal order is too low. In the second half of the article, the analysis and the stability results are extended to a partially dissipative wave system, which serves as a model for common practice in many fluid mechanics applications that incorporate a viscous stress in the momentum equation but no heat dissipation in the energy equation. Finally, the major theoretical predictions are verified by numerical examples. 
    more » « less
  5. ; ; (, Journal of Dynamics and Differential Equations)
    null (Ed.)
    This paper studies the local unstable manifold attached to an equilibrium solution of a system of delay differential equations (DDEs). Two main results are developed. The first is a general method for computing the formal Taylor series coefficients of a function parameterizing the unstable manifold. We derive linear systems of equations whose solutions are the Taylor coefficients, describe explicit formulas for assembling the linear equations for DDEs with polynomial nonlinearities. We also discuss a scheme for transforming non-polynomial DDEs into polynomial ones by appending auxiliary equations. The second main result is an a-posteriori theorem which—when combined with deliberate control of rounding errors— leads to mathematically rigorous computer assisted convergence results and error bounds for the truncated series. Our approach is based on the parameterization method for invariant manifolds and requires some mild non-resonance conditions between the unstable eigenvalues 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email