High Computational Efficiency Research Articles

DF-BETA: An FPGA-based Memory Locality Aware Decision Forest Accelerator via Bit-Level Early Termination

Decision forests, particularly Gradient Boosted Decision Trees (GBDT), are popular due to their high prediction performance and computational efficiency, making them suitable for embedded systems with circuit size and available energy constraints. In this study, we propose a new lightweight GBDT inference acceleration mechanism through the hardware and algorithm co-design. First, we present LoADPack, a hardware-friendly GBDT algorithm that enhances memory access locality. LoADPack obtains trees where the features and thresholds used across the entire ensemble are regular regardless of a branching direction by unifying some nodes and aligning the memory access patterns. Furthermore, we present DF-BETA, a resource-efficient accelerator for the LoADPack algorithm. DF-BETA utilizes MSB-first bit-serial computation to enable early determination of comparison calculations of 32-bit floating-point numbers, optimizing the operation for determining a branch direction. The hardware complexity and computation termination speed vary with the granularity of bit-serial computation. Therefore, we conduct design space exploration of DF-BETA to identify the optimal configuration. Our findings reveal that using 4-bit-serial comparators minimizes circuit size while achieving the leading throughput. Compared to running unconstrained GBDT on a typical accelerator with 32-bit bit-parallel comparators, our accelerator achieves 1.6 times higher throughput on average while maintaining comparable accuracy.

An improved vector method for simultaneous analysis of removability and kinematics in block theory

Block theory is an important and commonly used method for addressing stability problems in rock engineering, and it is very meaningful to simplify its analysis procedure and improve its computational efficiency. In this paper, an improved vector method capable of simultaneously analyzing block removability and kinematics that works well for both convex and concave blocks is proposed. This improved approach is based on rigorously proven theorems. Compared to the original vector method in block theory, it is simpler and more efficient because it skips the removability analysis and simplifies some of the judgment conditions in kinematic analysis. With the results of kinematic analysis, the improved vector method can also determine the removability of the block. The validity of the improved vector method is verified by analyzing a series of individual convex or concave blocks with different removability and failure modes. Finally, the improved vector method is applied to the block progressive failure analysis of an engineering slope, which again validates the effectiveness of the improved vector method. The advantages of the improved vector method are highlighted through a comparison with existing methods, especially its high computational efficiency, ease of understanding and implementation.

A local radial basis function-compact finite difference method for Sobolev equation arising from fluid dynamics

In this article, a new high-order, local meshless technique is presented for numerically solving multi-dimensional Sobolev equation arising from fluid dynamics. In the proposed method, Hermite radial basis function (RBF) interpolation technique is applied to approximate the operators of the model over local stencils. This leads to compact RBF generated finite difference (RBF-FD) formula, which provides a significant improvement in the accuracy and computational efficiency. In the first stage of the proposed method, the time discretization is performed by Crank–Nicolson finite difference scheme along with temporal Richardson extrapolation technique. In the second stage, the space dimension is discretized by applying the local radial basis function-compact finite difference (RBF-CFD) method. By performing some numerical simulations and comparing the results with existing methods, the high accuracy and computational efficiency of the proposed method are clearly demonstrated. The numerical results show that the presented method has fourth-order accuracy in both space and time dimensions. Finally, it can be concluded that the proposed method is a suitable alternative to the existing numerical techniques for the Sobolev model.

Nonlinear subspace clustering by functional link neural networks

Nonlinear subspace clustering based on a feed-forward neural network has been demonstrated to provide better clustering accuracy than some advanced subspace clustering algorithms. While this approach demonstrates impressive outcomes, it involves a balance between effectiveness and computational cost. In this study, we employ a functional link neural network to transform data samples into a nonlinear domain. Subsequently, we acquire a self-representation matrix through a learning mechanism that builds upon the mapped samples. As the functional link neural network is a single-layer neural network, our proposed method achieves high computational efficiency while ensuring desirable clustering performance. By incorporating the local similarity regularization to enhance the grouping effect, our proposed method further improves the quality of the clustering results. We name our method as Functional Link Neural Network Subspace Clustering (FLNNSC). Furthermore, we propose a convex combination subspace clustering scheme that combines a linear subspace clustering method with the functional link neural network subspace clustering approach. This combination method is named as Convex Combination Subspace Clustering (CCSC), which allows for a dynamic balance between linear and nonlinear representations. Extensive experiments conducted on four widely used datasets, including Extended Yale B, USPS, COIL20, and ORL, demonstrate that both FLNNSC and CCSC outperform several state-of-art subspace clustering methods in terms of clustering accuracy. Our affinity graph experiments reveal that FLNNSC exhibits clear block diagonal structures. We provide recommendations for hyperparameters in FLNNSC by performing a parameter sensitivity analysis, and empirically verify the convergence of FLNNSC. Additionally, we show that FLNNSC has a lower computational cost compared to two high-performing methods.

A novel nonlinear prestack inversion method based on nutcracker optimization algorithm with high convergence speed

Predicting reservoir parameters with high accuracy is still a crucial work of oil reservoir exploration and development. Due to the limitation of computational efficiency, deterministic methods are primarily used in practical production applications for predicting reservoir parameters. When the nonlinear forward equations are exceptionally complex and the initial model constructed deviates significantly from the true reservoir parameters, deterministic methods may have difficulty obtaining reasonable predictions of reservoir parameters. Compared to deterministic methods, intelligent optimization methods based on nature-inspired metaheuristic algorithms have unique advantages because they do not require derivative information, can achieve global optimization, and have less reliance on initial model. Therefore, they perform better in solving complex nonlinear optimization problems. In this paper, a new intelligent optimization algorithm called Nutcracker Optimization Algorithm (NOA) with a high convergence speed is introduced. By utilizing this optimization algorithm to solve the nonlinear inversion problem constructed by the highly nonlinear exact Zoeppritz equations, we analyze the potential of nonlinear reservoir parameters prediction methods based on intelligent optimization algorithms in practical production applications. The synthetic data test shows that, compared to the classical quantum particle swarm optimization (QPSO) algorithm and the highly-cited whale optimization algorithm (WOA), the prestack nonlinear inversion method based on NOA proposed in this paper ensures high convergence accuracy and exhibits high computational efficiency. It significantly reduces computation time and holds great potential for practical production applications. The field data test shows that the proposed method can rapidly and accurately estimates reservoir parameters, validating the feasibility and effectiveness of the proposed method. This has important theoretical value and practical significance for advancing the application of intelligent optimization algorithms in field reservoir exploration and development.

A novel conditional generative model for efficient ensemble forecasts of state variables in large-scale geological carbon storage

Integrating monitoring data to efficiently update reservoir pressure and CO2 plume distribution forecasts presents a significant challenge in geological carbon storage (GCS) applications. Inverse modeling techniques are commonly used to fuse observational data and refine reservoir model parameters, thereby improving state variable forecasts. However, these techniques often rely on linear or Gaussian assumptions, which can limit their effectiveness in accurately predicting state variables. Moreover, simulating large-scale three-dimensional (3D) GCS problems is computationally expensive, making iterative runs in inverse problems prohibitive. To address these challenges, we propose a conditional generative model utilizing the score-based diffusion method for real-time 3D pressure and saturation field distribution predictions. Our approach involves solving the score function with a mini-batch-based Monte Carlo estimator to generate labeled data. This data is subsequently employed to train a fully connected neural network, enabling it to learn the conditional sample generator within a supervised learning framework. This method enables the rapid generation of a large ensemble of predictions, facilitating comprehensive uncertainty quantification of state variables. We applied our method to forecast the dynamic 3D distributions of pressure and saturation fields over a 30-year injection period. The statistical assessment with low root mean square error (RMSE) values demonstrates that our method can accurately predict the spatiotemporal distributions of both pressure and saturation fields. Moreover, the developed conditional generative model shows high computational efficiency by generating 100 ensemble forecasts of 3D state variables in less than 10 min. The consistency between ensemble averages and ground truth values further illustrates the model’s capability to capture state variable dynamics during the CO2 plume injection process. Notably, the ground truth values fall within the ensemble forecasts, indicating that our uncertainty quantification effectively captures variability and potential noise in the observations. Thus, the developed conditional generative model proves to be a more efficient, accurate, and practical tool for GCS applications, facilitating timely risk analysis and informed decision-making.

Full waveform inversion based on deep learning and the phase-preserving symplectic partitioned Runge-Kutta method

To obtain more accurate full waveform inversion results, we present a forward modeling method with minimal phase error, low numerical dispersion, and high computational efficiency. To solve the 2D acoustic wave equation, we utilize a finite-difference (FD) scheme with optimized coefficients for spatial discretization, combined with a phase-preserving symplectic partitioned Runge-Kutta method for temporal discretization. This results in the development of the optimized symplectic partitioned Runge-Kutta (OSPRK) forward modeling method. We further apply the OSPRK method in conjunction with a recurrent neural network (RNN) for full waveform inversion (FWI). Our study explores the impact of various loss functions, Nadam optimizer parameters, and the incorporation of physical information operators on inversion performance. Numerical experiments demonstrate that the OSPRK method significantly reduces numerical dispersion compared to traditional FD methods. The Log-Cosh loss function offers superior stability across different learning rates, while the Nesterov-accelerated Adaptive Moment Estimation (Nadam) optimizer with optimized parameters greatly enhances convergence speed and inversion accuracy. Furthermore, the inclusion of physical information operators markedly improves inversion outcomes.

Single-Scene SAR Image Data Augmentation Based on SBR and GAN for Target Recognition

High-performance neural networks for synthetic aperture radar (SAR) automatic target recognition (ATR) often encounter the challenge of data scarcity. The lack of sufficient labeled SAR image datasets leads to the consideration of using simulated data to supplement the dataset. On the one hand, electromagnetic computation simulations provide high amplitude accuracy but are inefficient for large-scale datasets due to their complex computations and physical models. On the other hand, ray tracing simulations offer high geometric accuracy and computational efficiency but struggle with low amplitude correctness, hindering accurate numerical feature extraction. Furthermore, the emergence of generative adversarial networks (GANs) provides a way to generate simulated datasets, trying to balance computational efficiency with image quality. Nevertheless, the simulated SAR images generated based on random noise lack constraints, and it is also difficult to generate images that exceed the parameter conditions of the real image’s training set. Hence, it is essential to integrate physics-based simulation techniques into GANs to enhance the generalization ability of the imaging parameters. In this paper, we present the SingleScene-SAR Simulator, an efficient framework for SAR image simulation that operates under limited real SAR data. This simulator integrates rasterized shooting and bouncing rays (SBR) with cycle GAN, effectively achieving both amplitude correctness and geometric accuracy. The simulated images are appropriate for augmenting datasets in target recognition networks. Firstly, the SingleScene-SAR Simulator employs a rasterized SBR algorithm to generate radar cross section (RCS) images of target models. Secondly, a specific training pattern for cycle GAN is established to translate noisy RCS images into simulated SAR images that closely resemble real ones. Finally, these simulated images are utilized for data augmentation. Experimental results based on the constructed dataset show that with only one scene SAR image containing 30 target chips, the SingleScene-SAR Simulator can efficiently produce simulated SAR images that exhibit high similarity in both spatial and statistical distributions compared with real images. By employing simulated SAR images for data augmentation, the accuracy of target recognition networks can be consistently and significantly enhanced.

Efficient and Accurate Brain Tumor Classification Using Hybrid MobileNetV2–Support Vector Machine for Magnetic Resonance Imaging Diagnostics in Neoplasms

Background/Objectives: Magnetic Resonance Imaging (MRI) plays a vital role in brain tumor diagnosis by providing clear visualization of soft tissues without the use of ionizing radiation. Given the increasing incidence of brain tumors, there is an urgent need for reliable diagnostic tools, as misdiagnoses can lead to harmful treatment decisions and poor outcomes. While machine learning has significantly advanced medical diagnostics, achieving both high accuracy and computational efficiency remains a critical challenge. Methods: This study proposes a hybrid model that integrates MobileNetV2 for feature extraction with a Support Vector Machine (SVM) classifier for the classification of brain tumors. The model was trained and validated using the Kaggle MRI brain tumor dataset, which includes 7023 images categorized into four types: glioma, meningioma, pituitary tumor, and no tumor. MobileNetV2’s efficient architecture was leveraged for feature extraction, and SVM was used to enhance classification accuracy. Results: The proposed hybrid model showed excellent results, achieving Area Under the Curve (AUC) scores of 0.99 for glioma, 0.97 for meningioma, and 1.0 for both pituitary tumors and the no tumor class. These findings highlight that the MobileNetV2-SVM hybrid not only improves classification accuracy but also reduces computational overhead, making it suitable for broader clinical use. Conclusions: The MobileNetV2-SVM hybrid model demonstrates substantial potential for enhancing brain tumor diagnostics by offering a balance of precision and computational efficiency. Its ability to maintain high accuracy while operating efficiently could lead to better outcomes in medical practice, particularly in resource limited settings.

TMS: Ensemble Deep Learning Model for Accurate Classification of Monkeypox Lesions Based on Transformer Models with SVM

Background/Objectives:The emergence of monkeypox outside its endemic region in Africa has raised significant concerns within the public health community due to its rapid global dissemination. Early clinical differentiation of monkeypox from similar diseases, such as chickenpox and measles, presents a challenge. The Monkeypox Skin Lesion Dataset (MSLD) used in this study comprises monkeypox skin lesions, which were collected primarily from publicly accessible sources. The dataset contains 770 original images captured from 162 unique patients. The MSLD includes four distinct class labels: monkeypox, measles, chickenpox, and normal. Methods: This paper presents an ensemble model for classifying the monkeypox dataset, which includes transformer models and support vector machine (SVM). The model development process begins with an evaluation of seven convolutional neural network (CNN) architectures. The proposed model is developed by selecting the top four models based on evaluation metrics for performance. The top four CNN architectures, namely EfficientNetB0, ResNet50, MobileNet, and Xception, are used for feature extraction. The high-dimensional feature vectors extracted from each network are then concatenated and optimized before being inputted into the SVM classifier. Results: The proposed ensemble model, in conjunction with the SVM classifier, achieves an accuracy of 95.45b%. Furthermore, the model demonstrates high precision (95.51%), recall (95.45%), and F1 score (95.46%), indicating its effectiveness in identifying monkeypox lesions. Conclusions: The results of the study show that the proposed hybrid framework achieves robust diagnostic performance in monkeypox detection, offering potential utility for enhanced disease monitoring and outbreak management. The model’s high diagnostic accuracy and computational efficiency indicate that it can be used as an additional tool for clinical decision support.

MugenNet: A Novel Combined Convolution Neural Network and Transformer Network with Application in Colonic Polyp Image Segmentation

Accurate polyp image segmentation is of great significance, because it can help in the detection of polyps. Convolutional neural network (CNN) is a common automatic segmentation method, but its main disadvantage is the long training time. Transformer is another method that can be adapted to the automatic segmentation method by employing a self-attention mechanism, which essentially assigns different importance weights to each piece of information, thus achieving high computational efficiency during segmentation. However, a potential drawback with Transformer is the risk of information loss. The study reported in this paper employed the well-known hybridization principle to propose a method to combine CNN and Transformer to retain the strengths of both. Specifically, this study applied this method to the early detection of colonic polyps and to implement a model called MugenNet for colonic polyp image segmentation. We conducted a comprehensive experiment to compare MugenNet with other CNN models on five publicly available datasets. An ablation experiment on MugenNet was conducted as well. The experimental results showed that MugenNet can achieve a mean Dice of 0.714 on the ETIS dataset, which is the optimal performance on this dataset compared to other models, with an inference speed of 56 FPS. The overall outcome of this study is a method to optimally combine two methods of machine learning which are complementary to each other.

The Mixed Variable Transfer Matrix Method and Its Application in Predicting the Frequency Domain Vibration Characteristics of Hydraulic Pipelines

The fluid–structure interaction effect should not be disregarded when examining the vibration characteristics of hydraulic pipeline systems. The transfer matrix method (TMM) is an efficacious method for analyzing the vibration characteristics of hydraulic pipelines in the frequency domain, offering advantages such as simplicity and efficiency. However, the TMM suffers the problem of high frequency instability when dealing with long-span hydraulic pipelines, which restricts its practical application. Currently, several modified transfer matrix methods face challenges such as low computational efficiency and difficulties in handling complex boundaries. In response to these issues, this paper proposes a novel modified transfer matrix method known as the mixed variable transfer matrix method. This innovative method possesses clear physical significance and effectively prevents the transfer matrix from becoming singular without necessitating the subdivision of the pipeline length. Consequently, it addresses high-frequency instability while maintaining high computational efficiency. Moreover, this method is capable of addressing complex boundary problems by integrating boundary matrices, thereby demonstrating enhanced applicability compared to existing methods. The performance of the proposed method was validated through the utilization of classic Dubee pipeline impact test data, and the result shows maximum errors of 3.03% relative to the public data. Subsequently, an experiment was conducted on a section of hydraulic piping within a ship’s steering system. A hydraulic fluid noise generator was established to induce fluid pulsation excitation to the pipeline, thereby simulating the actual boundary conditions encountered in a ship’s hydraulic pipeline system so as to corroborate the efficacy of the proposed method in predicting the frequency domain vibration characteristics of a real hydraulic pipeline system. The experimental results indicate that the proposed method offers significant advantages in terms of high precision, efficiency, and stability, shows maximum errors of 4.35% relative to experimental data, and demonstrates promising prospects for engineering applications.

Improved fast deconvolution algorithms based on functional beamforming for gas leakage sound source imaging

Fast deconvolution algorithms have high computational efficiency, but they often overlook spatial variations in the point spread function (PSF) and wrap-around errors, resulting in the degradation of sound source imaging performance. Accordingly, this paper proposes two improved fast deconvolution algorithms based on functional beamforming (FB). The proposed algorithms establish a linear equation involving the FB output, sound source distribution, and raised-power spatial shift-invariant PSF, specially with irregular focus grid point. Furthermore, iterative solutions are obtained based on fast Fourier transform, enhancing imaging performance while maintaining computational efficiency. Simulations and experimental results demonstrate that compared to the fast deconvolution algorithms, the proposed algorithms expand the effective imaging area to 36.3°, and the Rényi entropy is reduced by approximately 45 %, which has higher imaging spatial resolution and dynamic range. Finally, the compressed gas leakage source imaging results validate the effectiveness of the proposed algorithms, indicating that they have promising engineering application prospects.

FPGA-based low-light image enhancement using Retinex algorithm and coarse-grained reconfigurable architecture

Advancements in digital imaging and video processing are often challenged by low-light environments, leading to degraded visual quality. This affects critical sectors such as medical imaging, aerospace, and underwater exploration, where uneven lighting can compromise safety and clarity. To enhance image quality in low-light conditions using a computationally efficient system. This paper introduces an FPGA-based system utilizing the Retinex algorithm for low-light image enhancement, implemented on a Coarse-Grained Reconfigurable Architecture (CGRA). The system is designed using Verilog HDL on a Xilinx FPGA, prioritizing hardware optimization to achieve high-quality outputs with minimal latency. The system achieves a processing rate of 60 frames per second (fps) for images with a resolution of 720 × 576. Quantitative evaluations show a Peak Signal-to-Noise Ratio (PSNR) improvement to 43.18 dB, a Structural Similarity Index (SSIM) of 0.92, and a Mean Squared Error (MSE) reduction, demonstrating significant enhancements in image quality. The design also achieves a low power consumption of 0.186 W and efficient resource utilization, with only 2.2% of Slice LUTs and Slice Registers used. The FPGA-based system demonstrates significant improvements in image quality with high computational efficiency, proving beneficial for critical applications in various sectors.

End-to-End Deployment of Winograd-Based DNNs on Edge GPU

The Winograd algorithm reduces the computational complexity of convolutional neural networks (CNNs) by minimizing the number of multiplications required for convolutions, making it particularly suitable for resource-constrained edge devices. Concurrently, most edge hardware accelerators utilize 8-bit integer arithmetic to enhance energy efficiency and reduce inference latency, requiring the quantization of CNNs before deployment. Combining Winograd-based convolution with quantization offers the potential for both performance acceleration and reduced energy consumption. However, prior research has identified significant challenges in this combination, particularly due to numerical instability and substantial accuracy degradation caused by the transformations required in the Winograd domain, making the two techniques incompatible on edge hardware. In this work, we describe our latest training scheme, which addresses these challenges, enabling the successful integration of Winograd-accelerated convolution with low-precision quantization while maintaining high task-related accuracy. Our approach mitigates the numerical instability typically introduced during the transformation, ensuring compatibility between the two techniques. Additionally, we extend our work by presenting a custom-optimized CUDA implementation of quantized Winograd convolution for NVIDIA edge GPUs. This implementation takes full advantage of the proposed training scheme, achieving both high computational efficiency and accuracy, making it a compelling solution for edge-based AI applications. Our training approach enables significant MAC reduction with minimal impact on prediction quality. Furthermore, our hardware results demonstrate up to a 3.4× latency reduction for specific layers, and a 1.44× overall reduction in latency for the entire DeepLabV3 model, compared to the standard implementation.

An integer arithmetic-based AMG preconditioned FGMRES solver

We consider solving a sparse linear system using integer (fixed-point) arithmetic. Integer arithmetic has attracted attention in scientific computing because of its high computational efficiency. Furthermore, considering the current circumstances of hardware development, integer arithmetic is expected to become increasingly important. Nevertheless, integer arithmetic has not been widely used for solving linear systems because it lacks robustness against overflow and underflow, making it hard to solve practical problems. Thus, we propose a new integer-based implementation framework for the flexible GMRES (FGMRES) method, which enables integer-based solvers to solve linear systems with the same accuracy as conventional floating-point solvers. In addition, we propose an integer-only algebraic multigrid preconditioner. Combining it with the integer-based FGMRES framework, we develop an integer-based solver. Numerical experiments on CPUs showed that the developed integer-based solver has a comparable convergence rate to floating-point solvers. We also found the test cases where the integer-based solver runs faster than the floating-point solvers.

Automatic Feature Selection for Atom-Centered Neural Network Potentials Using a Gradient Boosting Decision Algorithm.

Atom-centered neural network (ANN) potentials have shown high accuracy and computational efficiency in modeling atomic systems. A crucial step in developing reliable ANN potentials is the proper selection of atom-centered symmetry functions (ACSFs), also known as atomic features, to describe atomic environments. Inappropriate selection of ACSFs can lead to poor-quality ANN potentials. Here, we propose a gradient boosting decision tree (GBDT)-based framework for the automatic selection of optimal ACSFs. This framework takes uniformly distributed sets of ACSFs as input and evaluates their relative importance. The ACSFs with high average importance scores are selected and used to train an ANN potential. We applied this method to the Ge system, resulting in an ANN potential with root-mean-square errors (RMSE) of 10.2 meV/atom for energy and 84.8 meV/Å for force predictions, utilizing only 18 ACSFs to achieve a balance between accuracy and computational efficiency. The framework is validated using the grid searching method, demonstrating that ACSFs selected with our framework are in the optimal region. Furthermore, we also compared our method with commonly used feature selection algorithms. The results show that our algorithm outperforms the others in terms of effectiveness and accuracy. This study highlights the significance of the ACSF parameter effect on the ANN performance and presents a promising method for automatic ACSF selection, facilitating the development of machine learning potentials.

A Time‐Space Dual Adaptive Uncoupled Method for Supersonic Combustion

ABSTRACTHigh computational complexity due to rapidly increasing numerical stiffness is a difficult problem for simulating a supersonic reactive flow by using the uncoupled method. On the basis of our previous work, this paper proposes a dual adaptive method to ensure high calculation efficiency and good robustness in simulating stiff cases. The principle of this method is to realize adaptive coordination for the advection and reaction time steps in accordance with the non‐uniform feature of stiffness in the space and time dimensions. The proposed method can advance by a small time step in strong stiffness while with a large one in weak stiffness through the “prediction‐correction‐recovery” strategy. Some classical problems are chosen to verify the performance of the proposed method. The proposed method improved the computation efficiency by at least comparing with the previous method [1] and widened the error tolerance of the initial time step.

Full 2D optimized window coefficients for improved range velocity estimation in 5G joint communication and sensing

Abstract This work presents an approach for optimization of window coefficients for 5G user equipment side sensing, using orthogonal frequency division multiplexing radar-based range and velocity estimation, based on the sounding reference signal (SRS) from the 5G New Radio (NR) standard. The signal configuration and the corresponding waveform are generated in compliance with the 3rd Generation Partnership Project (3GPP) standard for 5G. The limitations of conventional signal processing for resources available for sensing with the SRS are highlighted. The proposed approach, which optimizes the window coefficients to improve the sensing capabilities, is implemented through two methods. The first method employs a decoupled optimization strategy for range and velocity, showing high computational efficiency. Our results demonstrate that this method significantly improves the peak sidelobe level (PSL) of the velocity profile by over $\mathrm{15}\,\mathrm{dB}$ , although it does not address the issue of diagonally located sidelobes, which occur due to non-uniform resource distribution. The second method adopts a comprehensive full 2D optimization technique. While it requires more computational resources and does not improve the PSL beyond the first method’s achievements, it mitigates the diagonally located sidelobes issue. The level of these have been improved by more than $\mathrm{3}\,\mathrm{dB}$ .

Damped Multistep Formulas for Solving Stiff Dynamic Problems

A novel series of damped multistep formulas of matrix coefficients are developed to find the solutions of dynamic problems with negligible stiff components or eigenmodes. They are derived from the concept of eigenmode and the development details are presented. A total of six multistep formulas is developed and they have an order of accuracy from 1 to 6 corresponding to the 1-step to 6-step multistep formula. Each formula has damping to suppress or remove the stiff components or eigenmodes that are of no importance or of no interest while the slow eigenmode solutions can be still accurately calculated. The 1-step and 2-step formulas of the series are L-stable. Besides, each formula can have a non-iterative solution procedure and hence it is of high computational efficiency in contrast to the corresponding conventional backward differentiation method. This series can give a better quality of numerical solutions for dynamic problems with negligible stiff components or eigenmodes when compared to the methods with no damping. Numerical tests reveal that this series performs well like the backward differentiation method and it can save many efforts in computing if a non-iterative solution procedure is used when compared to the backward differentiation method.