BANBEIS

We consider the problem of frequency limited H2 optimal model order reduction for large-scale sparse linear systems. A set of first-order H2 optimality conditions are derived for the frequency limited model order reduction problem. These conditions involve the solution of two frequency limited Sylvester equations that are known to be computationally complex. We discuss a framework for solving these matrix equations efficiently. The idea is also extended to the frequency limited H2 optimal model order reduction of index-1 descriptor systems. Numerical experiments are carried out using Python programming language and the results are presented to demonstrate the approximation accuracy and computational efficiency of the proposed technique.

This paper studies the model order reduction of second-order index-1 descriptor systems using a tangential interpolation projection method based on the Iterative Rational Krylov Algorithm (IRKA). Our primary focus is to reduce the system into a second-order form so that the structure of the original system can be preserved. For this purpose, the IRKA based tangential interpolatory method is modified to deal with the second-order structure of the underlying descriptor system efficiently in an implicit way. The paper also shows that by exploiting the symmetric properties of the system the implementing computational costs can be reduced significantly. Theoretical results are verified for the model reduction of the piezo actuator based adaptive spindle support which is second-order index-1 differential-algebraic form. The efficiency and accuracy of the method are demonstrated by analyzing the numerical results.

This paper discusses on the stability preservation technique of the frequency limited balanced reduced order model of a class of large-scale sparse descriptor system. For this purpose, firstly we modify standard rational Krylov subsapce method (RKSM) for solving frequency-limited Lyapunov equation pair to find out low-rank Gramian factors what is necessary to apply in balanced truncation technique. Next, we construct two projector matrices, and project the balanced unstable reduced system matrices found after implementing balanced truncation technique to ensure stability. Several data of index-1 descriptor system models are nominated for numerical experiments to demonstrate the efficiency of the proposed technique.

This paper discusses frequency limited balanced truncation of a class of large-scale sparse descriptor system by preserving the sparsity of the system. For this purpose we compute the low-rank Gramian factors by solving the frequency limited Lyapunov equations. We modify the standard rational Krylov subspace method (RKSM) for solving the Lyapunov equations efficiently and implicitly. Several data of index-l descriptor system models are nominated for numerical experiments to demonstrate the efficiency of the proposed techniques.

The usage of mathematical models of physical systems are increasing day by day in various disciplines of science and engineering for simulation, optimization, or control. Descriptor systems is a special kind of formation of physical systems arisen in many practical oriented fields whose dynamics maintain Differential- Algebraic Equations. Such type of systems are originated by finite elements or difference methods which becomes huge and complex to analyze along with the increment of the fineness of the grid resolution. As a result, the necessity of model order reduction comes up in order to minimize the complexity of the models during controlling by preserving the input-output relation of the original large-scale models. However, although reducing the dimension of large-scale models on infinite time and frequency domains has a great theoretical significance, the reduced order models of original large-scale models on restricted time and frequency intervals are more demandable to the analyzers and engineers for practical investigation. This dissertation elaborately discusses the model generations for data extraction to form the large-scale state-space systems and the projection based techniques to calculate the approximate low-rank solutions of the original state-space systems on definite time and frequency intervals. We impose relevant governing equations to create physical as well as data models. Balanced truncation is one of the most notable methods for the reduction of the model dimensions of linear time-invariant systems which requires computing the numerical solutions of two Lyapunov equations, commonly known as Gramians. Among some widely used approaches, Rational Krylov Subspace Method is one of the most effective procedures for finding the Gramians of the Lyapunov equations of the large-scale sparse dynamical systems which has already been developed to compute the low-rank time and frequency indefinite solutions. Besides, it has been also reformed to compute the low-rank approximation of the standard Lyapunov equations on limited time and frequency intervals for small-dense state-space systems.

This thesis mainly focuses on computational techniques applied to stabilize unstable Navier-Stokes models. The models arising from the Navier stokes equation are an essential aspect in engineering applications and applied mathematics in fluid mechanics, which is significantly depends on Reynolds number (Re), and if Re?300, the corresponding model will be unstable. The computation steps are designed to approximate the full models with the ROMs, find the reduced-order feedback matrices, and attain the optimal feedback matrices for stabilizing the desired Navier-Stokes models. The prime concern is exploring the Riccati-based boundary feedback stabilization of incompressible Navier-Stokes flow via Krylov subspace techniques. Since the volume of data derived from the original models is large, the feedback stabilization process through the Riccati equation is always infeasible. Therefore, a H_2 optimal model-order reduction scheme for reduced-order modeling, preserving the sparsity of the system, is required. Some conventional methods exist, but they have some adversities, such as the requirement of high computation time and memory allocation, complex matrix algebra, and uncertainty of the stability of the reduced-order models. To overcome these drawbacks, an extended form of Krylov subspace-based Two-Sided Iterative Algorithm (TSIA) is implemented to stabilize non-symmetric index-2 descriptor systems explored from unstable Navier-Stokes models. The proposed techniques are sparsity-preserving and utilize the Wilson condition to efficiently satisfy the reduced-order modeling approach through the sparse-dense Sylvester equations. To solve the desired Sylvester equations, sparsity-preserving Krylov subspaces are structured via the system of linear equations with a compact form of matrix-vector operations. Inverse projections approaches are applied to get the optimal feedback matrix from reduced-order models. To validate the efficiency of the proposed techniques, transient behaviors of the target systems are observed, incorporating the tabular and figurative comparisons with MATLAB simulations. Finally, to reveal the advancement of the proposed techniques, we compare our work with some existing results. From the tabular and graphical comparisons of the results of numerical computations, it is observed that RKSM is not applicable for the target models due to the non-symmetric structure. In contrast, TSIA can be suitably applied to solve Sparse-dense Sylvester equations for reduced-order modeling. Furthermore, by the TSIA, full models can be efficiently approximated by the corresponding ROMs with minimized H_2 error norm, and the inverse projection scheme is effective in computing the optimal feedback matrices from the reduced-order feedback matrices to stabilize the target models more efficiently than existing methods. Thus, it can be concluded that by utilizing TSIA, unstable Navier-Stokes models can be stabilized with better accuracy and less computing time.

Street surface weakening, for example, potholes, has caused drivers substantial money-related harm each year. Notwithstanding, viable street condition observing has been a proceeding with a challenge to street proprietors. Profundity cameras have a small field of view and can be effectively influenced by vehicle bobbing. Customary picture handling strategies are dependent on calculations. For example, the division can't adjust to shifting ecological and camera situations. In this paper, the object detection API for pothole detection is used to test the set of images and videos and give the output results of the tested images and videos. By evaluating the R-CNN algorithm and SSD mobile net algorithm, the results of the test showed successful results in getting potholes from test images with a maximum confidence level of 93%.