arXiv subjects

Mohammad Monir Uddin

Publications and source records attributed to Mohammad Monir Uddin.

4 recordsLinked to original sources

arXiv2025

$\mathcal{H}_2$-optimal Model Reduction of Linear Quadratic Output Systems in Finite Frequency Range

In frequency-limited model order reduction, the objective is to maintain the frequency response of the original system within a specified frequency range in the reduced-order model. In this paper, a mathematical expression for the frequency-limited $\mathcal{H}_2$ norm is derived, which quantifies the error within the desired frequency interval. Subsequently, the necessary conditions for a local optimum of the frequency-limited $\mathcal{H}_2$ norm of the error are derived. The inherent difficulty in satisfying these conditions within a Petrov-Galerkin projection framework is also discussed. Using the optimality conditions and the Petrov-Galerkin projection, a stationary point iteration algorithm is proposed, which approximately satisfies these optimality conditions upon convergence. The main computational effort in the proposed algorithm involves solving sparse-dense Sylvester equations. These equations are frequently encountered in $\mathcal{H}_2$ model order reduction algorithms and can be solved efficiently. Moreover, the algorithm bypasses the requirement of matrix logarithm computation, which is typically necessary for most frequency-limited reduction methods and can be computationally demanding for high-order systems. An illustrative example is provided to numerically validate the developed theory. The proposed algorithm's effectiveness in accurately approximating the original high-order model within the specified frequency range is demonstrated through the reduction of an advection-diffusion equation-based model, commonly used in model reduction literature for testing algorithms. Additionally, the algorithm's computational efficiency is highlighted by successfully reducing a flexible space structure model of order one million.

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arXiv2024

Time-limited H2-optimal Model Order Reduction of Linear Systems with Quadratic Outputs

An important class of dynamical systems with several practical applications is linear systems with quadratic outputs. These models have the same state equation as standard linear time-invariant systems but differ in their output equations, which are nonlinear quadratic functions of the system states. When dealing with models of exceptionally high order, the computational demands for simulation and analysis can become overwhelming. In such cases, model order reduction proves to be a useful technique, as it allows for constructing a reduced-order model that accurately represents the essential characteristics of the original high-order system while significantly simplifying its complexity. In time-limited model order reduction, the main goal is to maintain the output response of the original system within a specific time range in the reduced-order model. To assess the error within this time interval, a mathematical expression for the time-limited $\mathcal{H}_2$-norm is derived in this paper. This norm acts as a measure of the accuracy of the reduced-order model within the specified time range. Subsequently, the necessary conditions for achieving a local optimum of the time-limited $\mathcal{H}_2$ norm error are derived. The inherent inability to satisfy these optimality conditions within the Petrov-Galerkin projection framework is also discussed. After that, a stationary point iteration algorithm based on the optimality conditions and Petrov-Galerkin projection is proposed. Upon convergence, this algorithm fulfills three of the four optimality conditions. To demonstrate the effectiveness of the proposed algorithm, a numerical example is provided that showcases its ability to effectively approximate the original high-order model within the desired time interval.

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arXiv2024

Integrating Mamba Sequence Model and Hierarchical Upsampling Network for Accurate Semantic Segmentation of Multiple Sclerosis Legion

Integrating components from convolutional neural networks and state space models in medical image segmentation presents a compelling approach to enhance accuracy and efficiency. We introduce Mamba HUNet, a novel architecture tailored for robust and efficient segmentation tasks. Leveraging strengths from Mamba UNet and the lighter version of Hierarchical Upsampling Network (HUNet), Mamba HUNet combines convolutional neural networks local feature extraction power with state space models long range dependency modeling capabilities. We first converted HUNet into a lighter version, maintaining performance parity and then integrated this lighter HUNet into Mamba HUNet, further enhancing its efficiency. The architecture partitions input grayscale images into patches, transforming them into 1D sequences for processing efficiency akin to Vision Transformers and Mamba models. Through Visual State Space blocks and patch merging layers, hierarchical features are extracted while preserving spatial information. Experimental results on publicly available Magnetic Resonance Imaging scans, notably in Multiple Sclerosis lesion segmentation, demonstrate Mamba HUNet's effectiveness across diverse segmentation tasks. The model's robustness and flexibility underscore its potential in handling complex anatomical structures. These findings establish Mamba HUNet as a promising solution in advancing medical image segmentation, with implications for improving clinical decision making processes.

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arXiv2024

Balanced Truncation of Linear Systems with Quadratic Outputs in Limited Time and Frequency Intervals

Model order reduction involves constructing a reduced-order approximation of a high-order model while retaining its essential characteristics. This reduced-order model serves as a substitute for the original one in various applications such as simulation, analysis, and design. Often, there's a need to maintain high accuracy within a specific time or frequency interval, while errors beyond this limit can be tolerated. This paper addresses time-limited and frequency-limited model order reduction scenarios for linear systems with quadratic outputs, by generalizing the recently introduced structure-preserving balanced truncation algorithm. To that end, limited interval system Gramians are defined, and the corresponding generalized Lyapunov equations governing their computation are derived. Additionally, low-rank solutions for these equations are investigated. Next, balanced truncation algorithms are proposed for time-limited and frequency-limited scenarios, each utilizing its corresponding limited-interval system Gramians. The proposed algorithms ensure accurate results within specified time and frequency intervals while preserving the quadratic-output structure. Two benchmark numerical examples are presented to demonstrate the effectiveness of the algorithms, showcasing their ability to achieve superior accuracy within the desired time or frequency interval.

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