Hui Guo

Hui Guo, Shu Hu, Xin Wang et al.

AI-synthesized faces are visually challenging to discern from real ones. They have been used as profile images for fake social media accounts, which leads to high negative social impacts. Although progress has been made in developing automatic methods to detect AI-synthesized faces, there is no open platform to study the human performance of AI-synthesized faces detection. In this work, we develop an online platform called Open-eye to study the human performance of AI-synthesized face detection. We describe the design and workflow of the Open-eye in this paper.

Shanmin Yang, Hui Guo, Shu Hu et al.

Deepfake technology poses a significant threat to security and social trust. Although existing detection methods have shown high performance in identifying forgeries within datasets that use the same deepfake techniques for both training and testing, they suffer from sharp performance degradation when faced with cross-dataset scenarios where unseen deepfake techniques are tested. To address this challenge, we propose a Deep Information Decomposition (DID) framework to enhance the performance of Cross-dataset Deepfake Detection (CrossDF). Unlike most existing deepfake detection methods, our framework prioritizes high-level semantic features over specific visual artifacts. Specifically, it adaptively decomposes facial features into deepfake-related and irrelevant information, only using the intrinsic deepfake-related information for real/fake discrimination. Moreover, it optimizes these two kinds of information to be independent with a de-correlation learning module, thereby enhancing the model's robustness against various irrelevant information changes and generalization ability to unseen forgery methods. Our extensive experimental evaluation and comparison with existing state-of-the-art detection methods validate the effectiveness and superiority of the DID framework on cross-dataset deepfake detection.

Hui Guo, Xin Wang, Siwei Lyu

The COVID pandemic has led to the wide adoption of online video calls in recent years. However, the increasing reliance on video calls provides opportunities for new impersonation attacks by fraudsters using the advanced real-time DeepFakes. Real-time DeepFakes pose new challenges to detection methods, which have to run in real-time as a video call is ongoing. In this paper, we describe a new active forensic method to detect real-time DeepFakes. Specifically, we authenticate video calls by displaying a distinct pattern on the screen and using the corneal reflection extracted from the images of the call participant's face. This pattern can be induced by a call participant displaying on a shared screen or directly integrated into the video-call client. In either case, no specialized imaging or lighting hardware is required. Through large-scale simulations, we evaluate the reliability of this approach under a range in a variety of real-world imaging scenarios.

Hui Guo, Qihang Zheng, Chenghai Huo et al.

The efficient deployment of large language models (LLMs) is hindered by memory architecture heterogeneity, where traditional compilers suffer from fragmented workflows and high adaptation costs. We present nncase, an open-source, end-to-end compilation framework designed to unify optimization across diverse targets. Central to nncase is an e-graph-based term rewriting engine that mitigates the phase ordering problem, enabling global exploration of computation and data movement strategies. The framework integrates three key modules: Auto Vectorize for adapting to heterogeneous computing units, Auto Distribution for searching parallel strategies with cost-aware communication optimization, and Auto Schedule for maximizing on-chip cache locality. Furthermore, a buffer-aware Codegen phase ensures efficient kernel instantiation. Evaluations show that nncase outperforms mainstream frameworks like MLC LLM and Intel IPEX on Qwen3 series models and achieves performance comparable to the hand-optimized llama.cpp on CPUs, demonstrating the viability of automated compilation for high-performance LLM deployment. The source code is available at https://github.com/kendryte/nncase.

Chenghong Sun, Weidong Ji, Guohui Zhou et al.

The main purpose of relation extraction is to extract the semantic relationships between tagged pairs of entities in a sentence, which plays an important role in the semantic understanding of sentences and the construction of knowledge graphs. In this paper, we propose that the key semantic information within a sentence plays a key role in the relationship extraction of entities. We propose the hypothesis that the key semantic information inside the sentence plays a key role in entity relationship extraction. And based on this hypothesis, we split the sentence into three segments according to the location of the entity from the inside of the sentence, and find the fine-grained semantic features inside the sentence through the intra-sentence attention mechanism to reduce the interference of irrelevant noise information. The proposed relational extraction model can make full use of the available positive semantic information. The experimental results show that the proposed relation extraction model improves the accuracy-recall curves and P@N values compared with existing methods, which proves the effectiveness of this model.

Peng Huang, Yiming Wang, Yineng Chen et al.

Echocardiography plays an important role in the screening and diagnosis of cardiovascular diseases. However, automated intelligent analysis of echocardiographic data remains challenging due to complex cardiac dynamics and strong view heterogeneity. In recent years, visual language models (VLM) have opened a new avenue for building ultrasound understanding systems for clinical decision support. Nevertheless, most existing methods formulate this task as a direct mapping from video and question to answer, making them vulnerable to template shortcuts and spurious explanations. To address these issues, we propose EchoTrust, an evidence-driven Actor-Verifier framework for trustworthy reasoning in echocardiography VLM-based agents. EchoTrust produces a structured intermediate representation that is subsequently analyzed by distinct roles, enabling more reliable and interpretable decision-making for high-stakes clinical applications.

Jianxiang Peng, Junhao Li, Hongxiang Wang et al.

With the rapid development of Large Language Models (LLMs), a large number of benchmarks have been proposed. However, most benchmarks lack unified evaluation standard and require the manual implementation of custom scripts, making results hard to ensure consistency and reproducibility. Furthermore, mainstream evaluation frameworks are centralized, with datasets and answers, which increases the risk of benchmark leakage. To address these issues, we propose a Decentralized Evaluation Protocol (DEP), a decentralized yet unified and standardized evaluation framework through a matching server without constraining benchmarks. The server can be mounted locally or deployed remotely, and once adapted, it can be reused over the long term. By decoupling users, LLMs, and benchmarks, DEP enables modular, plug-and-play evaluation: benchmark files and evaluation logic stay exclusively on the server side. In remote setting, users cannot access the ground truth, thereby achieving data isolation and leak-proof evaluation. To facilitate practical adoption, we develop DEP Toolkit, a protocol-compatible toolkit that supports features such as breakpoint resume, concurrent requests, and congestion control. We also provide detailed documentation for adapting new benchmarks to DEP. Using DEP toolkit, we evaluate multiple LLMs across benchmarks. Experimental results verify the effectiveness of DEP and show that it reduces the cost of deploying benchmark evaluations. As of February 2026, we have adapted over 60 benchmarks and continue to promote community co-construction to support unified evaluation across various tasks and domains.

Hang Yang, Hao Chen, Hui Guo et al.

Accurate and efficient question-answering systems are essential for delivering high-quality patient care in the medical field. While Large Language Models (LLMs) have made remarkable strides across various domains, they continue to face significant challenges in medical question answering, particularly in understanding domain-specific terminologies and performing complex reasoning. These limitations undermine their effectiveness in critical medical applications. To address these issues, we propose a novel approach incorporating similar case generation within a multi-agent medical question-answering (MedQA) system. Specifically, we leverage the Llama3.1:70B model, a state-of-the-art LLM, in a multi-agent architecture to enhance performance on the MedQA dataset using zero-shot learning. Our method capitalizes on the model's inherent medical knowledge and reasoning capabilities, eliminating the need for additional training data. Experimental results show substantial performance gains over existing benchmark models, with improvements of 7% in both accuracy and F1-score across various medical QA tasks. Furthermore, we examine the model's interpretability and reliability in addressing complex medical queries. This research not only offers a robust solution for medical question answering but also establishes a foundation for broader applications of LLMs in the medical domain.

Yicui Peng, Hao Chen, Chingsheng Lin et al.

Providing explanations within the recommendation system would boost user satisfaction and foster trust, especially by elaborating on the reasons for selecting recommended items tailored to the user. The predominant approach in this domain revolves around generating text-based explanations, with a notable emphasis on applying large language models (LLMs). However, refining LLMs for explainable recommendations proves impractical due to time constraints and computing resource limitations. As an alternative, the current approach involves training the prompt rather than the LLM. In this study, we developed a model that utilizes the ID vectors of user and item inputs as prompts for GPT-2. We employed a joint training mechanism within a multi-task learning framework to optimize both the recommendation task and explanation task. This strategy enables a more effective exploration of users' interests, improving recommendation effectiveness and user satisfaction. Through the experiments, our method achieving 1.59 DIV, 0.57 USR and 0.41 FCR on the Yelp, TripAdvisor and Amazon dataset respectively, demonstrates superior performance over four SOTA methods in terms of explainability evaluation metric. In addition, we identified that the proposed model is able to ensure stable textual quality on the three public datasets.

Jianxiang Lu, Cong Xie, Hui Guo

As large-scale text-to-image generation models have made remarkable progress in the field of text-to-image generation, many fine-tuning methods have been proposed. However, these models often struggle with novel objects, especially with one-shot scenarios. Our proposed method aims to address the challenges of generalizability and fidelity in an object-driven way, using only a single input image and the object-specific regions of interest. To improve generalizability and mitigate overfitting, in our paradigm, a prototypical embedding is initialized based on the object's appearance and its class, before fine-tuning the diffusion model. And during fine-tuning, we propose a class-characterizing regularization to preserve prior knowledge of object classes. To further improve fidelity, we introduce object-specific loss, which can also use to implant multiple objects. Overall, our proposed object-driven method for implanting new objects can integrate seamlessly with existing concepts as well as with high fidelity and generalization. Our method outperforms several existing works. The code will be released.

Hao Chen, Hui Guo, Baochen Hu et al.

The rapid growth of social media has resulted in an explosion of online news content, leading to a significant increase in the spread of misleading or false information. While machine learning techniques have been widely applied to detect fake news, the scarcity of labeled datasets remains a critical challenge. Misinformation frequently appears as paired text and images, where a news article or headline is accompanied by a related visuals. In this paper, we introduce a self-learning multimodal model for fake news classification. The model leverages contrastive learning, a robust method for feature extraction that operates without requiring labeled data, and integrates the strengths of Large Language Models (LLMs) to jointly analyze both text and image features. LLMs are excel at this task due to their ability to process diverse linguistic data drawn from extensive training corpora. Our experimental results on a public dataset demonstrate that the proposed model outperforms several state-of-the-art classification approaches, achieving over 85% accuracy, precision, recall, and F1-score. These findings highlight the model's effectiveness in tackling the challenges of multimodal fake news detection.

Yineng Chen, Peng Huang, Aozhong Zhang et al.

Foundation models have achieved remarkable results in medical image analysis. However, its large network architecture and high computational complexity significantly impact inference speed, limiting its application on terminal medical devices. Quantization, a technique that compresses models into low-bit versions, is a solution to this challenge. In this paper, we propose a post-training quantization algorithm, Permutation-COMQ. It eliminates the need for backpropagation by using simple dot products and rounding operations, thereby removing hyperparameter tuning and simplifying the process. Additionally, we introduce a weight-aware strategy that reorders the weight within each layer to address the accuracy degradation induced by channel-wise scaling during quantization, while preserving channel structure. Experiments demonstrate that our method achieves the best results in 2-bit, 4-bit, and 8-bit quantization.

Hui Guo, Grace Y. Yi, Boyu Wang

While crowdsourcing has emerged as a practical solution for labeling large datasets, it presents a significant challenge in learning accurate models due to noisy labels from annotators with varying levels of expertise. Existing methods typically estimate the true label posterior, conditioned on the instance and noisy annotations, to infer true labels or adjust loss functions. These estimates, however, often overlook potential misspecification in the true label posterior, which can degrade model performances, especially in high-noise scenarios. To address this issue, we investigate learning from noisy annotations with an estimated true label posterior through the framework of conditional distributionally robust optimization (CDRO). We propose formulating the problem as minimizing the worst-case risk within a distance-based ambiguity set centered around a reference distribution. By examining the strong duality of the formulation, we derive upper bounds for the worst-case risk and develop an analytical solution for the dual robust risk for each data point. This leads to a novel robust pseudo-labeling algorithm that leverages the likelihood ratio test to construct a pseudo-empirical distribution, providing a robust reference probability distribution in CDRO. Moreover, to devise an efficient algorithm for CDRO, we derive a closed-form expression for the empirical robust risk and the optimal Lagrange multiplier of the dual problem, facilitating a principled balance between robustness and model fitting. Our experimental results on both synthetic and real-world datasets demonstrate the superiority of our method.

Yingsong Huang, Hui Guo, Jing Huang et al.

The rapid progress of diffusion models highlights the growing need for detecting generated images. Previous research demonstrates that incorporating diffusion-based measurements, such as reconstruction error, can enhance the generalizability of detectors. However, ignoring the differing impacts of aleatoric and epistemic uncertainty on reconstruction error can undermine detection performance. Aleatoric uncertainty, arising from inherent data noise, creates ambiguity that impedes accurate detection of generated images. As it reflects random variations within the data (e.g., noise in natural textures), it does not help distinguish generated images. In contrast, epistemic uncertainty, which represents the model's lack of knowledge about unfamiliar patterns, supports detection. In this paper, we propose a novel framework, Diffusion Epistemic Uncertainty with Asymmetric Learning~(DEUA), for detecting diffusion-generated images. We introduce Diffusion Epistemic Uncertainty~(DEU) estimation via the Laplace approximation to assess the proximity of data to the manifold of diffusion-generated samples. Additionally, an asymmetric loss function is introduced to train a balanced classifier with larger margins, further enhancing generalizability. Extensive experiments on large-scale benchmarks validate the state-of-the-art performance of our method.

Donglian Li, Hui Guo, Minglang Chen et al.

Accurate segmentation of anatomical structures in the apical four-chamber (A4C) view of fetal echocardiography is essential for early diagnosis and prenatal evaluation of congenital heart disease (CHD). However, precise segmentation remains challenging due to ultrasound artifacts, speckle noise, anatomical variability, and boundary ambiguity across different gestational stages. To reduce the workload of sonographers and enhance segmentation accuracy, we propose DCD, an advanced deep learning-based model for automatic segmentation of key anatomical structures in the fetal A4C view. Our model incorporates a Dense Atrous Spatial Pyramid Pooling (Dense ASPP) module, enabling superior multi-scale feature extraction, and a Convolutional Block Attention Module (CBAM) to enhance adaptive feature representation. By effectively capturing both local and global contextual information, DCD achieves precise and robust segmentation, contributing to improved prenatal cardiac assessment.

Xiaoyi Feng, Kaifeng Zou, Caichun Cen et al.

Existing optical flow datasets focus primarily on real-world simulation or synthetic human motion, but few are tailored to Celluloid(cel) anime character motion: a domain with unique visual and motion characteristics. To bridge this gap and facilitate research in optical flow estimation and downstream tasks such as anime video generation and line drawing colorization, we introduce LinkTo-Anime, the first high-quality dataset specifically designed for cel anime character motion generated with 3D model rendering. LinkTo-Anime provides rich annotations including forward and backward optical flow, occlusion masks, and Mixamo Skeleton. The dataset comprises 395 video sequences, totally 24,230 training frames, 720 validation frames, and 4,320 test frames. Furthermore, a comprehensive benchmark is constructed with various optical flow estimation methods to analyze the shortcomings and limitations across multiple datasets.

Xin Wang, Hui Guo, Shu Hu et al.

Generative Adversarial Networks (GAN) have led to the generation of very realistic face images, which have been used in fake social media accounts and other disinformation matters that can generate profound impacts. Therefore, the corresponding GAN-face detection techniques are under active development that can examine and expose such fake faces. In this work, we aim to provide a comprehensive review of recent progress in GAN-face detection. We focus on methods that can detect face images that are generated or synthesized from GAN models. We classify the existing detection works into four categories: (1) deep learning-based, (2) physical-based, (3) physiological-based methods, and (4) evaluation and comparison against human visual performance. For each category, we summarize the key ideas and connect them with method implementations. We also discuss open problems and suggest future research directions.

Hui Guo, Shu Hu, Xin Wang et al.

GAN-based techniques that generate and synthesize realistic faces have caused severe social concerns and security problems. Existing methods for detecting GAN-generated faces can perform well on limited public datasets. However, images from existing public datasets do not represent real-world scenarios well enough in terms of view variations and data distributions (where real faces largely outnumber synthetic faces). The state-of-the-art methods do not generalize well in real-world problems and lack the interpretability of detection results. Performance of existing GAN-face detection models degrades significantly when facing imbalanced data distributions. To address these shortcomings, we propose a robust, attentive, end-to-end network that can spot GAN-generated faces by analyzing their eye inconsistencies. Specifically, our model learns to identify inconsistent eye components by localizing and comparing the iris artifacts between the two eyes automatically. Our deep network addresses the imbalance learning issues by considering the AUC loss and the traditional cross-entropy loss jointly. Comprehensive evaluations of the FFHQ dataset in terms of both balanced and imbalanced scenarios demonstrate the superiority of the proposed method.

Hui Guo, Shu Hu, Xin Wang et al.

Generative adversary network (GAN) generated high-realistic human faces have been used as profile images for fake social media accounts and are visually challenging to discern from real ones. In this work, we show that GAN-generated faces can be exposed via irregular pupil shapes. This phenomenon is caused by the lack of physiological constraints in the GAN models. We demonstrate that such artifacts exist widely in high-quality GAN-generated faces and further describe an automatic method to extract the pupils from two eyes and analysis their shapes for exposing the GAN-generated faces. Qualitative and quantitative evaluations of our method suggest its simplicity and effectiveness in distinguishing GAN-generated faces.

Shiyi Yang, Hui Guo, Nour Moustafa

Machine learning (ML)-based intrusion detection systems (IDSs) play a critical role in discovering unknown threats in a large-scale cyberspace. They have been adopted as a mainstream hunting method in many organizations, such as financial institutes, manufacturing companies and government agencies. However, existing designs achieve a high threat detection performance at the cost of a large number of false alarms, leading to alert fatigue. To tackle this issue, in this paper, we propose a neural-network-based defense mechanism named DarkHunter. DarkHunter incorporates both supervised learning and unsupervised learning in the design. It uses a deep ensemble network (trained through supervised learning) to detect anomalous network activities and exploits an unsupervised learning-based scheme to trim off mis-detection results. For each detected threat, DarkHunter can trace to its source and present the threat in its original traffic format. Our evaluations, based on the UNSW-NB15 dataset, show that DarkHunter outperforms the existing ML-based IDSs and is able to achieve a high detection accuracy while keeping a low false positive rate.

Peilun Wu, Hui Guo

Email threat is a serious issue for enterprise security. The threat can be in various malicious forms, such as phishing, fraud, blackmail and malvertisement. The traditional anti-spam gateway often maintains a greylist to filter out unexpected emails based on suspicious vocabularies present in the email's subject and contents. However, this type of signature-based approach cannot effectively discover novel and unknown suspicious emails that utilize various evolving malicious payloads. To address the problem, in this paper, we present Holmes, an efficient and lightweight semantic based engine for anomalous email detection. Holmes can convert each email event log into a sentence through word embedding and then identify abnormalities that deviate from a historical baseline based on those translated sentences. We have evaluated the performance of Holmes in a real-world enterprise environment, where around 5,000 emails are sent/received each day. In our experiments, Holmes shows a high capability to detect email threats, especially those that cannot be handled by the enterprise anti-spam gateway. It is also demonstrated through our experiment that Holmes can discover more concealed malicious emails that are immune from several commercial detection tools.

Shiyi Yang, Peilun Wu, Hui Guo

Network intrusion detection (NID) is an essential defense strategy that is used to discover the trace of suspicious user behaviour in large-scale cyberspace, and machine learning (ML), due to its capability of automation and intelligence, has been gradually adopted as a mainstream hunting method in recent years. However, traditional ML based network intrusion detection systems (NIDSs) are not effective to recognize unknown threats and their high detection rate often comes with the cost of high false alarms, which leads to the problem of alarm fatigue. To address the above problems, in this paper, we propose a novel neural network based detection system, DualNet, which is constructed with a general feature extraction stage and a crucial feature learning stage. DualNet can rapidly reuse the spatial-temporal features in accordance with their importance to facilitate the entire learning process and simultaneously mitigate several optimization problems occurred in deep learning (DL). We evaluate the DualNet on two benchmark cyber attack datasets, NSL-KDD and UNSW-NB15. Our experiment shows that DualNet outperforms classical ML based NIDSs and is more effective than existing DL methods for NID in terms of accuracy, detection rate and false alarm rate.

Peilun Wu, Nour Moustafa, Shiyi Yang et al.

High false alarm rate and low detection rate are the major sticking points for unknown threat perception. To address the problems, in the paper, we present a densely connected residual network (Densely-ResNet) for attack recognition. Densely-ResNet is built with several basic residual units, where each of them consists of a series of Conv-GRU subnets by wide connections. Our evaluation shows that Densely-ResNet can accurately discover various unknown threats that appear in edge, fog and cloud layers and simultaneously maintain a much lower false alarm rate than existing algorithms.

Peilun Wu, Hui Guo, Nour Moustafa

One challenge for building a secure network communication environment is how to effectively detect and prevent malicious network behaviours. The abnormal network activities threaten users' privacy and potentially damage the function and infrastructure of the whole network. To address this problem, the network intrusion detection system (NIDS) has been used. By continuously monitoring network activities, the system can timely identify attacks and prompt counter-attack actions. NIDS has been evolving over years. The current-generation NIDS incorporates machine learning (ML) as the core technology in order to improve the detection performance on novel attacks. However, the high detection rate achieved by a traditional ML-based detection method is often accompanied by large false-alarms, which greatly affects its overall performance. In this paper, we propose a deep neural network, Pelican, that is built upon specially-designed residual blocks. We evaluated Pelican on two network traffic datasets, NSL-KDD and UNSW-NB15. Our experiments show that Pelican can achieve a high attack detection performance while keeping a much low false alarm rate when compared with a set of up-to-date machine learning based designs.

Peilun Wu, Hui Guo

Network attack is a significant security issue for modern society. From small mobile devices to large cloud platforms, almost all computing products, used in our daily life, are networked and potentially under the threat of network intrusion. With the fast-growing network users, network intrusions become more and more frequent, volatile and advanced. Being able to capture intrusions in time for such a large scale network is critical and very challenging. To this end, the machine learning (or AI) based network intrusion detection (NID), due to its intelligent capability, has drawn increasing attention in recent years. Compared to the traditional signature-based approaches, the AI-based solutions are more capable of detecting variants of advanced network attacks. However, the high detection rate achieved by the existing designs is usually accompanied by a high rate of false alarms, which may significantly discount the overall effectiveness of the intrusion detection system. In this paper, we consider the existence of spatial and temporal features in the network traffic data and propose a hierarchical CNN+RNN neural network, LuNet. In LuNet, the convolutional neural network (CNN) and the recurrent neural network (RNN) learn input traffic data in sync with a gradually increasing granularity such that both spatial and temporal features of the data can be effectively extracted. Our experiments on two network traffic datasets show that compared to the state-of-the-art network intrusion detection techniques, LuNet not only offers a high level of detection capability but also has a much low rate of false positive-alarm.

Peilun Wu, Hui Guo, Richard Buckland

Convolution Neural Network (ConvNet) offers a high potential to generalize input data. It has been widely used in many application areas, such as visual imagery, where comprehensive learning datasets are available and a ConvNet model can be well trained and perform the required function effectively. ConvNet can also be applied to network intrusion detection. However, the currently available datasets related to the network intrusion are often inadequate, which makes the ConvNet learning deficient, hence the trained model is not competent in detecting unknown intrusions. In this paper, we propose a ConvNet model using transfer learning for network intrusion detection. The model consists of two concatenated ConvNets and is built on a two-stage learning process: learning a base dataset and transferring the learned knowledge to the learning of the target dataset. Our experiments on the NSL-KDD dataset show that the proposed model can improve the detection accuracy not only on the test dataset containing mostly known attacks (KDDTest+) but also on the test dataset featuring many novel attacks (KDDTest-21) -- about 2.68\% improvement on KDDTest+ and 22.02\% on KDDTest-21 can be achieved, as compared to the traditional ConvNet model.

Hui Guo, Yang Yang

In this paper, we develop bound-preserving discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacement problems. We consider the problem with two components and the (volumetric) concentration of the $i$th component of the fluid mixture, $c_i$, should be between $0$ and $1$. However, $c_i$ does not satisfy the maximum principle. Therefore, the numerical techniques introduced in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied directly. The main idea is to apply the positivity-preserving techniques to both $c_1$ and $c_2$, respectively and enforce $c_1+c_2=1$ simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure $dp/dt$ as a source in the concentration equation. Moreover, $c_i's$ are not the conservative variables, as a result, the classical bound-preserving limiter in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied. Therefore, another limiter will be introduced. Numerical experiments will be given to demonstrate the accuracy in $L^\infty$-norm and good performance of the numerical technique.