arXiv subjects

Xuefeng Liu

Publications and source records attributed to Xuefeng Liu.

At least 73 records · Page 4Linked to original sources

arXiv2022

Projection error-based guaranteed L2 error bounds for finite element approximations of Laplace eigenfunctions

For conforming finite element approximations of the Laplacian eigenfunctions, a fully computable guaranteed error bound in the $L^2$ norm sense is proposed. The bound is based on the a priori error estimate for the Galerkin projection of the conforming finite element method, and has an optimal speed of convergence for the eigenfunctions with the worst regularity. The resulting error estimate bounds the distance of spaces of exact and approximate eigenfunctions and, hence, is robust even in the case of multiple and tightly clustered eigenvalues. The accuracy of the proposed bound is illustrated by numerical examples. The demonstration code is available at https://ganjin.online/xfliu/EigenfunctionEstimation4FEM .

math.NA↗

arXiv2022

Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error constant estimation

A computer-assisted proof is proposed for the Laplacian eigenvalue minimization problems over triangular domains under diameter constraints. The proof utilizes recently developed guaranteed computation methods for both eigenvalues and eigenfunctions of differential operators. The paper also provides an elementary and concise proof of the Hadamard shape derivative, which helps to validate the monotonicity of eigenvalue with respect to shape parameters. Besides the model homogeneous Dirichlet eigenvalue problem, the eigenvalue problem associated with a non-homogeneous Neumann boundary condition, which is related to the Crouzeix--Raviart interpolation error constant, is considered. The computer-assisted proof tells that among the triangles with the unit diameter, the equilateral triangle minimizes the first eigenvalue for each concerned eigenvalue problem.

math.NA↗

arXiv2022

Fully computable a posteriori error bounds for eigenfunctions

For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple eigenvalues, under the settings of target eigenvalue problems. Algorithm I is based on the Rayleigh quotient and the min-max principle that characterizes the eigenvalue problems. The formula for the error estimate provided by Algorithm I is easy to compute and applies to problems with limited information of Rayleigh quotients. Algorithm II, as an extension of the Davis--Kahan method, takes advantage of the dual formulation of differential operators along with the Prager--Synge technique and provides greatly improved accuracy of the estimate, especially for the finite element approximations of eigenfunctions. Numerical examples of eigenvalue problems of matrices and the Laplace operators over convex and non-convex domains illustrate the efficiency of the proposed algorithms.

math.NA↗

arXiv2021

Visformer: The Vision-friendly Transformer

The past year has witnessed the rapid development of applying the Transformer module to vision problems. While some researchers have demonstrated that Transformer-based models enjoy a favorable ability of fitting data, there are still growing number of evidences showing that these models suffer over-fitting especially when the training data is limited. This paper offers an empirical study by performing step-by-step operations to gradually transit a Transformer-based model to a convolution-based model. The results we obtain during the transition process deliver useful messages for improving visual recognition. Based on these observations, we propose a new architecture named Visformer, which is abbreviated from the `Vision-friendly Transformer'. With the same computational complexity, Visformer outperforms both the Transformer-based and convolution-based models in terms of ImageNet classification accuracy, and the advantage becomes more significant when the model complexity is lower or the training set is smaller. The code is available at https://github.com/danczs/Visformer.

cs.CV↗

arXiv2021

Guaranteed a posteriori local error estimation for finite element solutions of boundary value problems

This paper considers the finite element solution of the boundary value problem of Poisson's equation and proposes a guaranteed em a posteriori local error estimation based on the hypercircle method. Compared to the existing literature on qualitative error estimation, the proposed error estimation provides an explicit and sharp bound for the approximation error in the subdomain of interest, and its efficiency can be enhanced by further utilizing a non-uniform mesh. Such a result is applicable to problems without $H^2$-regularity, since it only utilizes the first order derivative of the solution. The efficiency of the proposed method is demonstrated by numerical experiments for both convex and non-convex 2D domains with uniform or non-uniform meshes.

math.NA↗

arXiv2021

Error constant estimation under the maximum norm for linear Lagrange interpolation

For the Lagrange interpolation over a triangular domain, we propose an efficient algorithm to rigorously evaluate the interpolation error constant under the maximum norm by using the finite element method (FEM). In solving the optimization problem corresponding to the interpolation error constant, the maximum norm in the constraint condition is the most difficult part to process. To handle this difficulty, a novel method is proposed by combining the orthogonality of the interpolation associated to the Fujino--Morley FEM space and the convex-hull property of the Bernstein representation of functions in the FEM space. Numerical results for the lower and upper bounds of the interpolation error constant for triangles of various types are presented to verify the efficiency of the proposed method.

math.NA↗

arXiv2021

Computer-assisted proof for the stationary solution existence of the Navier-Stokes equation over 3D domains

This paper proposes a computer-assisted solution existence verification method for the stationary Navier-Stokes equation over general 3D domains. The proposed method verifies that the exact solution as the fixed point of the Newton iteration exists around the approximate solution through rigorous computation and error estimation. The explicit values of quantities required by applying the fixed point theorem are obtained by utilizing newly developed quantitative error estimation for finite element solutions to boundary value problems and eigenvalue problems of the Stokes equation.

math.NA↗

arXiv2021

A Survey on Incorporating Domain Knowledge into Deep Learning for Medical Image Analysis

Although deep learning models like CNNs have achieved great success in medical image analysis, the small size of medical datasets remains a major bottleneck in this area. To address this problem, researchers have started looking for external information beyond current available medical datasets. Traditional approaches generally leverage the information from natural images via transfer learning. More recent works utilize the domain knowledge from medical doctors, to create networks that resemble how medical doctors are trained, mimic their diagnostic patterns, or focus on the features or areas they pay particular attention to. In this survey, we summarize the current progress on integrating medical domain knowledge into deep learning models for various tasks, such as disease diagnosis, lesion, organ and abnormality detection, lesion and organ segmentation. For each task, we systematically categorize different kinds of medical domain knowledge that have been utilized and their corresponding integrating methods. We also provide current challenges and directions for future research.

eess.IV↗

arXiv2020

MangaGAN: Unpaired Photo-to-Manga Translation Based on The Methodology of Manga Drawing

Manga is a world popular comic form originated in Japan, which typically employs black-and-white stroke lines and geometric exaggeration to describe humans' appearances, poses, and actions. In this paper, we propose MangaGAN, the first method based on Generative Adversarial Network (GAN) for unpaired photo-to-manga translation. Inspired by how experienced manga artists draw manga, MangaGAN generates the geometric features of manga face by a designed GAN model and delicately translates each facial region into the manga domain by a tailored multi-GANs architecture. For training MangaGAN, we construct a new dataset collected from a popular manga work, containing manga facial features, landmarks, bodies, and so on. Moreover, to produce high-quality manga faces, we further propose a structural smoothing loss to smooth stroke-lines and avoid noisy pixels, and a similarity preserving module to improve the similarity between domains of photo and manga. Extensive experiments show that MangaGAN can produce high-quality manga faces which preserve both the facial similarity and a popular manga style, and outperforms other related state-of-the-art methods.

cs.CV↗

arXiv2020

SelectScale: Mining More Patterns from Images via Selective and Soft Dropout

Convolutional neural networks (CNNs) have achieved remarkable success in image recognition. Although the internal patterns of the input images are effectively learned by the CNNs, these patterns only constitute a small proportion of useful patterns contained in the input images. This can be attributed to the fact that the CNNs will stop learning if the learned patterns are enough to make a correct classification. Network regularization methods like dropout and SpatialDropout can ease this problem. During training, they randomly drop the features. These dropout methods, in essence, change the patterns learned by the networks, and in turn, forces the networks to learn other patterns to make the correct classification. However, the above methods have an important drawback. Randomly dropping features is generally inefficient and can introduce unnecessary noise. To tackle this problem, we propose SelectScale. Instead of randomly dropping units, SelectScale selects the important features in networks and adjusts them during training. Using SelectScale, we improve the performance of CNNs on CIFAR and ImageNet.

cs.CV↗

arXiv2020

An End-to-end Method for Producing Scanning-robust Stylized QR Codes

Quick Response (QR) code is one of the most worldwide used two-dimensional codes.~Traditional QR codes appear as random collections of black-and-white modules that lack visual semantics and aesthetic elements, which inspires the recent works to beautify the appearances of QR codes. However, these works adopt fixed generation algorithms and therefore can only generate QR codes with a pre-defined style. In this paper, combining the Neural Style Transfer technique, we propose a novel end-to-end method, named ArtCoder, to generate the stylized QR codes that are personalized, diverse, attractive, and scanning-robust.~To guarantee that the generated stylized QR codes are still scanning-robust, we propose a Sampling-Simulation layer, a module-based code loss, and a competition mechanism. The experimental results show that our stylized QR codes have high-quality in both the visual effect and the scanning-robustness, and they are able to support the real-world application.

cs.CV↗

arXiv2020

Explicit a posteriori and a priori error estimation for the finite element solution of Stokes equations

For the Stokes equation over 2D and 3D domains, explicit a posteriori and a priori error estimation are novelly developed for the finite element solution. The difficulty in handling the divergence-free condition of the Stokes equation is solved by utilizing the extended hypercircle method along with the Scott-Vogelius finite element scheme. Since all terms in the error estimation have explicit values, by further applying the interval arithmetic and verified computing algorithms, the computed results provide rigorous estimation for the approximation error. As an application of the proposed error estimation, the eigenvalue problem of the Stokes operator is considered and rigorous bounds for the eigenvalues are obtained. The efficiency of proposed error estimation is demonstrated by solving the Stokes equation on both convex and non-convex 3D domains.

math.NA↗

arXiv2020

Targeting SARS-CoV-2 with AI- and HPC-enabled Lead Generation: A First Data Release

Researchers across the globe are seeking to rapidly repurpose existing drugs or discover new drugs to counter the the novel coronavirus disease (COVID-19) caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). One promising approach is to train machine learning (ML) and artificial intelligence (AI) tools to screen large numbers of small molecules. As a contribution to that effort, we are aggregating numerous small molecules from a variety of sources, using high-performance computing (HPC) to computer diverse properties of those molecules, using the computed properties to train ML/AI models, and then using the resulting models for screening. In this first data release, we make available 23 datasets collected from community sources representing over 4.2 B molecules enriched with pre-computed: 1) molecular fingerprints to aid similarity searches, 2) 2D images of molecules to enable exploration and application of image-based deep learning methods, and 3) 2D and 3D molecular descriptors to speed development of machine learning models. This data release encompasses structural information on the 4.2 B molecules and 60 TB of pre-computed data. Future releases will expand the data to include more detailed molecular simulations, computed models, and other products.

q-bio.BM↗

arXiv2020

A Deep-learning-based Method for PIR-based Multi-person Localization

Device-free localization (DFL) based on pyroelectric infrared (PIR) sensors has attracted much attention due to its advantages of low cost, low power consumption, and privacy protection. However, most existing PIR-based DFL methods require high deployment density to achieve high localization accuracy. Recently, a few works proposed that the deployment density can be reduced by deeply analyzing the analog output of PIR sensors. However, these methods can not well handle the localization task in multi-person scenarios yet. In this paper, we propose a novel neural network for PIR-based multi-person localization, which appropriately leverages a series of domain knowledge. Specifically, the proposed network consists of two modules: one is for determining the number of persons and another is for determining their locations. Meanwhile, the module of person counting is further designed as a two-stage network: one stage is for signal separation and another is for single-person detection. The module for localization is also designed as a two-stage network: one stage is for signal extraction and another is for single-person localization. Through the above methods, we succeed to remarkably reduce the deployment density of the traditional PIR-based method by about 76\%, while maintaining the localization accuracy.

eess.SP↗

arXiv2020

Network Adjustment: Channel Search Guided by FLOPs Utilization Ratio

Automatic designing computationally efficient neural networks has received much attention in recent years. Existing approaches either utilize network pruning or leverage the network architecture search methods. This paper presents a new framework named network adjustment, which considers network accuracy as a function of FLOPs, so that under each network configuration, one can estimate the FLOPs utilization ratio (FUR) for each layer and use it to determine whether to increase or decrease the number of channels on the layer. Note that FUR, like the gradient of a non-linear function, is accurate only in a small neighborhood of the current network. Hence, we design an iterative mechanism so that the initial network undergoes a number of steps, each of which has a small `adjusting rate' to control the changes to the network. The computational overhead of the entire search process is reasonable, i.e., comparable to that of re-training the final model from scratch. Experiments on standard image classification datasets and a wide range of base networks demonstrate the effectiveness of our approach, which consistently outperforms the pruning counterpart. The code is available at https://github.com/danczs/NetworkAdjustment.

cs.CV↗

arXiv2019

A new PIR-based method for real-time tracking

Pyroelectric infrared (PIR) sensors are considered to be promising devices for device-free localization due to its advantages of low cost, less intrusive, and the immunity from multi-path fading. However, most of the existing PIR-based localization systems only utilize the binary information of PIR sensors and therefore require a large number of PIR sensors and a careful deployment. A few works directly map the raw data of PIR sensors to one's location using machine learning approaches. However, these approaches require to collect abundant training data and suffer from environmental change. In this paper, we propose a PIR-based device-free localization approach based on the raw data of PIR sensors. The key of this approach is to extract a new type of location information called as the azimuth change. The extraction of the azimuth change relies on the physical properties of PIR sensors. Therefore, no abundant training data are needed and the system is robust to environmental change. Through experiments, we demonstrated that a device-free localization system incorporating the information of azimuth change outperforms the state-of-the-art approaches in terms of higher location accuracy. In addition, the information of the azimuth change can be easily integrated with other PIR-based localization systems to improve their localization accuracy.

eess.SP↗

arXiv2019

Explicit a posteriori local error estimation for FEM solutions

For the finite element solution of Poisson's equation, a local a posteriori error estimation based on the Hypercircle method is proposed. Even for the solution of Poisson's equation without the $H^2$ regularity, this method can provide explicit local error estimation. The efficiency of the proposed method is demonstrated by numerical experiments for the boundary value problem of Poisson's equation defined on the 2D and 3D domains.

math.NA↗

arXiv2019

Explicit eigenvalue bounds of differential operators defined by symmetric positive semi-definite bilinear forms

Recently, the eigenvalue problems formulated with symmetric positive definite bilinear forms have been well investigated with the aim of explicit bounds for the eigenvalues. In this paper, the existing theorems for bounding eigenvalues are further extended to deal with the case of eigenvalue problems defined by positive semi-definite bilinear forms. As an application, the eigenvalue estimation theorems are applied to the error constant estimation for polynomial projections.

math.NA↗