Projection-based Model Order Reduction Methods Research Articles

Reduction and Observer Design for a Grey-Box Model in Continuous Pharmaceutical Manufacturing

In this contribution, a novel Reduced Order Model (ROM) formulation of the grey-box model proposed in Elkhashap et al. (2020a) for the pharmaceutical continuous vibrated fluid bed dryer (VFBD) is presented. The ROM exploits the H2-norm projection-based model order reduction method after a special solution formulation of the model's infinite-dimensional part. This is mainly by introducing a vector field mapping between the model parts casting the semi- discretized PDE into a bilinear form. The ROM produced is then integrated into an nonlinear Kalman Filtering-based observer design also handling the estimation of the model's algebraic variables. Evaluations of the FOM, ROM, ROM-based observer variants, and the FOM-based observer are performed using Monte-Carlo simulations as well as simulations based on experimental data of the real system. It is shown that the ROM could reproduce the FOM states accurately with a relative mean square error below 0.3 % for the experimental data simulation. This is while reaching a computational-time reduction up to a factor of 40. The ROM-based observer with algebraic states correction is shown (using Monte-Carlo simulations) to be able to converge to the true values for all cases regardless of initialization. Moreover, it is also shown that the performance degradation of the observer due to reduction is practically insignificant. This is while the computational speedup of the observer due to reduction reached a factor of more than third order of magnitude.

Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation

We propose a probabilistic way for reducing the cost of classical projection-based model order reduction methods for parameter-dependent linear equations. A reduced order model is here approximated from its random sketch, which is a set of low-dimensional random projections of the reduced approximation space and the spaces of associated residuals. This approach exploits the fact that the residuals associated with approximations in low-dimensional spaces are also contained in low-dimensional spaces. We provide conditions on the dimension of the random sketch for the resulting reduced order model to be quasi-optimal with high probability. Our approach can be used for reducing both complexity and memory requirements. The provided algorithms are well suited for any modern computational environment. Major operations, except solving linear systems of equations, are embarrassingly parallel. Our version of proper orthogonal decomposition can be computed on multiple workstations with a communication cost independent of the dimension of the full order model. The reduced order model can even be constructed in a so-called streaming environment, i.e., under extreme memory constraints. In addition, we provide an efficient way for estimating the error of the reduced order model, which is not only more efficient than the classical approach but is also less sensitive to round-off errors. Finally, the methodology is validated on benchmark problems.

Dynamical Model Reduction Method for Solving Parameter-Dependent Dynamical Systems

We propose a projection-based model order reduction method for the solution of parameter-dependent dynamical systems. The proposed method relies on the construction of time-dependent reduced spaces generated from evaluations of the solution of the full-order model at some selected parameters values. The approximation obtained by Galerkin projection is the solution of a reduced dynamical system with a modified flux which takes into account the time dependency of the reduced spaces. An a posteriori error estimate is derived and a greedy algorithm using this error estimate is proposed for the adaptive selection of parameters values. The resulting method can be interpreted as a dynamical low-rank approximation method with a subspace point of view and a uniform control of the error over the parameter set.

Projection-Based Model Order Reduction Methods for the Estimation of Vector-Valued Variables of Interest

We propose and compare goal-oriented projection based model order reduction methods for the estimation of vector-valued functionals of the solution of parameter-dependent equations. The first projection method is a generalization of the classical primal-dual method to the case of vector-valued variables of interest. We highlight the role played by three reduced spaces: the approximation space and the test space associated to the primal variable, and the approximation space associated to the dual variable. Then we propose a Petrov-Galerkin projection method based on a saddle point problem involving an approximation space for the primal variable and an approximation space for an auxiliary variable. A goal-oriented choice of the latter space, defined as the sum of two spaces, allows us to improve the approximation of the variable of interest compared to a primal-dual method using the same reduced spaces. Then, for both approaches, we derive computable error estimates for the approximations of the variable of interest and we propose greedy algorithms for the goal-oriented construction of reduced spaces. The performance of the algorithms are illustrated on numerical examples and compared to standard (non goal-oriented) algorithms.

The Minimum Norm Least-Squares Solution in Reduction by Krylov Subspace Methods

In recent years, model order reduction (MOR) of interconnect system has become an important technique to reduce the computation complexity and improve the verification efficiency in the nanometer VLSI design. The Krylov subspaces techniques in existing MOR methods are efficient, and have become the methods of choice for generating small-scale macro-models of the large-scale multi-port RCL networks that arise in VLSI interconnect analysis. Although the Krylov subspace projection-based MOR methods have been widely studied over the past decade in the electrical computer-aided design community, all of them do not provide a best optimal solution in a given order. In this paper, a minimum norm least-squares solution for MOR by Krylov subspace methods is proposed. The method is based on generalized inverse (or pseudo-inverse) theory. This enables a new criterion for MOR-based Krylov subspace projection methods. Two numerical examples are used to test the PRIMA method based on the method proposed in this paper as a standard model.

Interpolation of Inverse Operators for Preconditioning Parameter-Dependent Equations

We propose a method for the construction of preconditioners of parameter-dependent matrices for the solution of large systems of parameter-dependent equations. The proposed method is an interpolation of the matrix inverse based on a projection of the identity matrix with respect to the Frobenius norm. Approximations of the Frobenius norm using random matrices are introduced in order to handle large matrices. The resulting statistical estimators of the Frobenius norm yield quasi-optimal projections that are controlled with high probability. Strategies for the adaptive selection of interpolation points are then proposed for different objectives in the context of projection-based model order reduction methods: the improvement of residual-based error estimators, the improvement of the projection on a given reduced approximation space, or the recycling of computations for sampling based model order reduction methods.

Hierarchical Krylov subspace based reduction of large interconnects

In this paper, we propose a new model order reduction approach for large interconnect circuits using hierarchical decomposition and the Krylov subspace projection-based model order reduction methods. The new approach, called hiePrimor, first partitions a large interconnect circuit into a number of smaller subcircuits and then performs the projection-based model order reduction on each of subcircuits in isolation and on the top-level circuit thereafter. The new approach is very amenable for exploiting the multi-core based parallel computing platforms to significantly speed up the reduction process. Theoretically we show that hiePrimor can deliver the same accuracy as the flat reduction method given the same reduction order and it can also preserve the passivity of the reduced models as well. We also show that partitioning has large impacts on the performance of hierarchical reduction and the minimum-span objective should be required to attain the best performance for hierarchical reduction. The proposed method is suitable for reducing large global interconnects like coupled bus, transmission lines, large clock nets in the post-layout stage. Experimental results demonstrate that hiePrimor can be significantly faster and more scalable than the flat projection methods like PRIMA and be order of magnitude faster than PRIMA with parallel computing without loss of accuracy. Interconnect circuits with up to 4 million nodes can be analyzed in a few minutes even in Matlab by the new method.