Shahkar Ahmad Nahvi

Trajectory based methods for nonlinear MOR: Review and perspectives

Trajectory based methods constitute an important class amongst the techniques for Model Order Reduction (MOR) of nonlinear dynamical systems. This is an attempt to trace their development, right from the original formulation to the current state of the art. Outstanding issues that pose challenges and possible directions in which solutions can be found are discussed. Relevant insights from the authors experience in using these methods for nonlinear MOR are given.

Original article: Nonlinearity-aware sub-model combination in trajectory based methods for nonlinear Mor

Trajectory based methods approximate nonlinear dynamical systems by superposition of dimensionally reduced linear systems. The linear systems are obtained by linearisations at multiple points along a state-trajectory. They are combined in a weighted sum and the combinations are switched appropriately to approximate the dynamic behaviour of the nonlinear system. Weights assigned at a specimen point on the trajectory generally depend on the euclidean distance to the linearisation points. In this work, limitations of the conventional weight-assignment scheme are pointed out. It is shown that the procedure is similar across all nonlinearities, and hence ignores the nonlinear vector field curvature for superposition. Additionally, it results in an inadequate assessment of the linear systems when they are equidistant from the specimen point. An improved method for weight-assignment, which uses state-velocities in addition to state-positions is proposed. The method naturally takes into account the system nonlinearity and is hence called Nonlinearity-aware Trajectory Piece-wise Linear (Ntpwl) method. Further, a computationally efficient procedure for estimating the state-velocity is introduced. The new strategy is illustrated and assessed with the help of case studies and it is shown that the Ntpwl model substantially improves the approximation of the nonlinear systems considered. Increased robustness to training and negligible stretching of the computational resources is also obtained.

Piece-wise Quasi-linear Approximation for Nonlinear Model Reduction

The trajectory piece-wise linear (TPWL) method is a popular technique for nonlinear model order reduction (MOR). Though widely studied, it has primarily been restricted to applications modeled by nonlinear systems with linear input operators. This paper is an effort to bridge this gap. We illustrate problems in the TPWL method in creating reduced order models for nonlinear systems with nonlinear input operators. We also propose a solution based on a quasi-linear formulation of the nonlinear system at approximation points. This results in a method for nonlinear MOR, called the trajectory piece-wise quasi-linear (TPWQ) method. TPWQ is formulated, numerically validated and a new technique to reduce the computational costs associated with simulating the quasi-linear systems is also demonstrated.

Optimal control of a heat conduction problem using its low order approximation

Optimal control of a large dynamical system is accomplished by designing the control strategy on its low order approximation. The Large system is the Finite Element (FE) model of a heat conduction problem and its low order approximation is obtained using Krylov Subspace Projection. It is seen that this approach provides good dividends as the desired cost functional is minimized reasonably at substantially reduced computational cost. A study of the sub-optimality caused is however required and is pointed out as a subject of possible exploration.

A scheme for comprehensive computational cost reduction in proper orthogonal decomposition

Abstract This paper addresses the issue of offline and online computational cost reduction of the proper orthogonal decomposition (POD) which is a popular nonlinear model order reduction (MOR) technique. Online computational cost is reduced by using the discrete empirical interpolation method (DEIM), which reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD: this is the POD-DEIM approach. Offline computational cost is reduced by generating an approximate snapshot-ensemble of the nonlinear dynamical system, consequently, completely avoiding the need to simulate the full-order system. Two snapshot ensembles: one of the states and the other of the nonlinear function are obtained by simulating the successive linearization of the original nonlinear system. The proposed technique is applied to two benchmark large-scale nonlinear dynamical systems and clearly demonstrates comprehensive savings in computational cost and time with insignificant or no deterioration in performance.

Fast simulation of nonlinear dynamical systems for application in reduced order modelling

The Trajectory piecewise linear (TPWL) representation of nonlinear dynamical systems requires an a-priori solution of the nonlinear system trajectory. This paper proposes a new algorithm for finding an approximate nonlinear system trajectory to reduce the computational burden of the TPWL process. Additionally, the new algorithm has an error assessment feature that provides a less heuristic alternative to the conventional methods. It is shown that the TPWL model can be obtained with lesser user intervention using the new algorithm and also comprises of a smaller number of constituent linear systems.

Approximate Snapshot-ensemble Generation for Basis Extraction in Proper Orthogonal Decomposition

Abstract This paper tries to address the problem of high computational burden posed by the basis extraction procedure in Proper Orthogonal Decomposition (POD). A proposal for generating approximate snapshots of the nonlinear system, to reduce this burden is presented. Simulation results are shown which demonstrate that a substantial reduction in computational time with no or insignificant deterioration in performance is easily obtained.

AFAS - adaptive fast approximate simulation for non-linear model reduction

In this work, we examine the problem of selecting linearisation points for trajectory piecewise linear (TPWL) approximation of non-linear dynamical systems. Linearisation point selection is a crucial step in the TPWL process, the quality and complexity of the approximation rests on it. In contrast to the popular approaches wherein linearisations are done at constant, pre-selected Euclidean distances in the state-space, we propose a new and simple error measure that helps in assessing the linearisation point requirement at different points on the non-linear system trajectory. Based on this error measure a new scheme to simulate the non-linear system, create linearisations at viable points and obtain a better TPWL approximation is presented. Finally, we substantiate our observations and propositions by detailed numerical tests on two non-linear circuits.

Trajectory piece-wise quasi-linear approximation of large non-linear dynamic systems

In this work we extend the trajectory piece-wise linear (TPWL) approximation of large, non-linear and input-affine dynamical systems to non-linear systems with non-linear input operators. The new technique is called the trajectory piece-wise quasi-linear (TPWQ) approximation. We explain the motivation for this technique in light of previously reported under-performance of the TPWL method and show that its a more general alternative, developing it is formulation and demonstrating its effectiveness in the process.