scholarly journals Inexact methods for the low rank solution to large scale Lyapunov equations

2020 ◽  
Vol 60 (4) ◽  
pp. 1221-1259 ◽  
Author(s):  
Patrick Kürschner ◽  
Melina A. Freitag
2021 ◽  
Vol 8 (3) ◽  
pp. 526-536
Author(s):  
L. Sadek ◽  
◽  
H. Talibi Alaoui ◽  

In this paper, we present a new approach for solving large-scale differential Lyapunov equations. The proposed approach is based on projection of the initial problem onto an extended block Krylov subspace by using extended nonsymmetric block Lanczos algorithm then, we get a low-dimensional differential Lyapunov matrix equation. The latter differential matrix equation is solved by the Backward Differentiation Formula method (BDF) or Rosenbrock method (ROS), the obtained solution allows to build a low-rank approximate solution of the original problem. Moreover, we also give some theoretical results. The numerical results demonstrate the performance of our approach.


2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Mohammad-Sahadet Hossain ◽  
M. Monir Uddin

We have presented the efficient techniques for the solutions of large-scale sparse projected periodic discrete-time Lyapunov equations in lifted form. These types of problems arise in model reduction and state feedback problems of periodic descriptor systems. Two most popular techniques to solve such Lyapunov equations iteratively are the low-rank alternating direction implicit (LR-ADI) method and the low-rank Smith method. The main contribution of this paper is to update the LR-ADI method by exploiting the ideas of the adaptive shift parameters computation and the efficient handling of complex shift parameters. These approaches efficiently reduce the computational cost with respect to time and memory. We also apply these iterative Lyapunov solvers in balanced truncation model reduction of periodic discrete-time descriptor systems. We illustrate numerical results to show the performance and accuracy of the proposed methods.


2020 ◽  
Vol 2 (1) ◽  
pp. 1-13
Author(s):  
Linus Balicki

The low-rank alternating direction implicit (LR-ADI) iteration is an effective method for solving large-scale Lyapunov equations. In the software library pyMOR, solutions to Lyapunov equations play an important role when reducing a model using the balanced truncation method. In this article we introduce the LR-ADI iteration as well as pyMOR, while focusing on its features which are relevant for integrating the iteration into the library. We compare the run time of the iteration's pure pyMOR implementation with those achieved by external libraries available within the pyMOR framework.


SIAM Review ◽  
2004 ◽  
Vol 46 (4) ◽  
pp. 693-713 ◽  
Author(s):  
Jing-Rebecca Li ◽  
Jacob White
Keyword(s):  
Low Rank ◽  

2021 ◽  
Vol 47 (2) ◽  
pp. 1-34
Author(s):  
Umberto Villa ◽  
Noemi Petra ◽  
Omar Ghattas

We present an extensible software framework, hIPPYlib, for solution of large-scale deterministic and Bayesian inverse problems governed by partial differential equations (PDEs) with (possibly) infinite-dimensional parameter fields (which are high-dimensional after discretization). hIPPYlib overcomes the prohibitively expensive nature of Bayesian inversion for this class of problems by implementing state-of-the-art scalable algorithms for PDE-based inverse problems that exploit the structure of the underlying operators, notably the Hessian of the log-posterior. The key property of the algorithms implemented in hIPPYlib is that the solution of the inverse problem is computed at a cost, measured in linearized forward PDE solves, that is independent of the parameter dimension. The mean of the posterior is approximated by the MAP point, which is found by minimizing the negative log-posterior with an inexact matrix-free Newton-CG method. The posterior covariance is approximated by the inverse of the Hessian of the negative log posterior evaluated at the MAP point. The construction of the posterior covariance is made tractable by invoking a low-rank approximation of the Hessian of the log-likelihood. Scalable tools for sample generation are also discussed. hIPPYlib makes all of these advanced algorithms easily accessible to domain scientists and provides an environment that expedites the development of new algorithms.


2018 ◽  
Vol 75 (10) ◽  
pp. 3313-3330 ◽  
Author(s):  
Hauke Schulz ◽  
Bjorn Stevens

Measurements from the Barbados Cloud Observatory are analyzed to identify the processes influencing the distribution of moist static energy and the large-scale organization of tropical convection. Five years of water vapor and cloud profiles from a Raman lidar and cloud radar are composed to construct the structure of the observed atmosphere in moisture space. The large-scale structure of the atmosphere is similar to that now familiar from idealized studies of convective self-aggregation, with shallow clouds prevailing over a moist marine layer in regions of low-rank humidity, and deep convection in a nearly saturated atmosphere in regions of high-rank humidity. With supplementary reanalysis datasets the overall circulation pattern is reconstructed in moisture space, and shows evidence of a substantial lower-tropospheric component to the circulation. This shallow component of the circulation helps support the differentiation between the moist and dry columns, similar to what is found in simulations of convective self-aggregation. Radiative calculations show that clear-sky radiative differences can explain a substantial part of this circulation, with further contributions expected from cloud radiative effects. The shallow component appears to be important for maintaining the low gross moist stability of the convecting column. A positive feedback between a shallow circulation driven by differential radiative cooling and the low-level moisture gradients that help support it is hypothesized to play an important role in conditioning the atmosphere for deep convection. The analysis suggests that the radiatively driven shallow circulations identified by modeling studies as contributing to the self-aggregation of convection in radiative–convective equilibrium similarly play a role in shaping the intertropical convergence zone and, hence, the large-scale structure of the tropical atmosphere.


Sign in / Sign up

Export Citation Format

Share Document