arXiv subjects

Fanyi Yang

Publications and source records attributed to Fanyi Yang.

At least 19 recordsLinked to original sources

arXiv2026

Preconditioned Reconstructed Discontinuous Approximation For Elliptic Interface Problem on Unfitted Mesh

In this paper, we develop an efficient preconditioned unfitted finite element method for the elliptic interface problem, based on the reconstructed discontinuous approximation. The key idea is to impose suitable constraints on the local least squares reconstruction. These constraints ensure the stability near cut interface elements and, more importantly, establish a norm equivalence between the high-order space and the lowest-order piecewise constant space. This result allows us to construct an optimal preconditioner directly from the lowest-order system on the same unfitted mesh for any high-order scheme. The resulting method combines a cut discontinuous Galerkin formulation with Nitsche's penalty technique. The approximation space achieves arbitrarily high order accuracy with only one degree of freedom per element. We prove optimal error estimates and show that the condition number of the preconditioned system is uniformly bounded independently of the mesh size, coefficient contrast, and the location of the interface relative to the mesh. Multigrid algorithms are further designed to efficiently approximate the inverse of the lowest-order system matrix. Numerical experiments in two and three dimensions confirm the optimal convergence rates and demonstrate the robustness and efficiency of the proposed preconditioning method.

math.NA↗

arXiv2026

An unfitted finite element method for elliptic interface problem with low regularity estimates

In this paper, we present and analyze an unfitted finite element method for the elliptic interface problem. We consider the case that the interface is $C^2$-smooth or polygonal, and the exact solution $u \in H^{1+s}(\Omega_0 \cup \Omega_1)$ for any $s > 0$. The stability near the interface is guaranteed by a local polynomial extension technique combined with ghost penalty bilinear forms, from which the robust condition number estimates and the error estimates are derived. Furthermore, the jump penalty term for weakly enforcing the jump condition in our method is also defined based on the local polynomial extension, which enables us to establish the error estimation particularly for solutions with low regularity. We perform a series of numerical tests in two and three dimensions to illustrate the accuracy of the proposed method.

math.NA↗

arXiv2025

REAL-Prover: Retrieval Augmented Lean Prover for Mathematical Reasoning

Nowadays, formal theorem provers have made monumental progress on high-school and competition-level mathematics, but few of them generalize to more advanced mathematics. In this paper, we present REAL-Prover, a new open-source stepwise theorem prover for Lean 4 to push this boundary. This prover, based on our fine-tuned large language model (REAL-Prover-v1) and integrated with a retrieval system (Leansearch-PS), notably boosts performance on solving college-level mathematics problems. To train REAL-Prover-v1, we developed HERALD-AF, a data extraction pipeline that converts natural language math problems into formal statements, and a new open-source Lean 4 interactive environment (Jixia-interactive) to facilitate synthesis data collection. In our experiments, our prover using only supervised fine-tune achieves competitive results with a 23.7% success rate (Pass@64) on the ProofNet dataset-comparable to state-of-the-art (SOTA) models. To further evaluate our approach, we introduce FATE-M, a new benchmark focused on algebraic problems, where our prover achieves a SOTA success rate of 56.7% (Pass@64).

cs.CL↗

arXiv2025

MAIN: Mutual Alignment Is Necessary for instruction tuning

Instruction tuning has empowered large language models (LLMs) to achieve remarkable performance, yet its success heavily depends on the availability of large-scale, high-quality instruction-response pairs. To meet this demand, various methods have been developed to synthesize data at scale. However, current methods for scaling up data generation often overlook a crucial aspect: the alignment between instructions and responses. We hypothesize that the quality of instruction-response pairs is determined not by the individual quality of each component, but by the degree of mutual alignment. To address this, we propose a Mutual Alignment Framework (MAIN) which enforces coherence between instructions and responses through mutual constraints. We demonstrate that MAIN generalizes well across model architectures and sizes, achieving state-of-the-art performance on LLaMA, Mistral, and Qwen models across diverse benchmarks. This work underscores the critical role of instruction-response alignment in enabling generalizable and high-quality instruction tuning for LLMs. All code is available from our repository.

cs.CL↗

arXiv2024

An efficient particle locating method on unstructured meshes in two and three dimensions based on patch searching

We present a particle locating method for unstructured meshes in two and three dimensions. Our algorithm is based on a patch searching process, and includes two steps. We first locate the given point to a patch near a vertex, and then the host element is determined within the patch domain. Here, the patch near a vertex is the domain of elements around this vertex. We prove that in the first step the patch can be rapidly identified by constructing an auxiliary Cartesian grid with a prescribed resolution. Then, the second step can be converted into a searching problem, which can be easily solved by searching algorithms. Only coordinates to particles are required in our method. We conduct a series of numerical tests in two and three dimensions to illustrate the robustness and efficiency of our method.

math.NA↗

arXiv2024

A Preconditioned Discontinuous Galerkin Method for Biharmonic Equation with $C^0$-Reconstructed Approximation

In this paper, we present a high-order finite element method based on a reconstructed approximation to the biharmonic equation. In our construction, the space is reconstructed from nodal values by solving a local least squares fitting problem per element. It is shown that the space can achieve an arbitrarily high-order accuracy and share the same nodal degrees of freedom with the $C^0$ linear space. The interior penalty discontinuous Galerkin scheme can be directly applied to the reconstructed space for solving the biharmonic equation. We prove that the numerical solution converges with optimal orders under error measurements. More importantly, we establish a norm equivalence between the reconstructed space and the continuous linear space. This property allows us to precondition the linear system arising from the high-order space by the linear space on the same mesh. This preconditioner is shown to be optimal in the sense that the condition number of the preconditioned system admits a uniform upper bound independent of the mesh size. Numerical examples in two and three dimensions are provided to illustrate the accuracy of the scheme and the efficiency of the preconditioning method.

math.NA↗

arXiv2024

PKU-AIGIQA-4K: A Perceptual Quality Assessment Database for Both Text-to-Image and Image-to-Image AI-Generated Images

In recent years, image generation technology has rapidly advanced, resulting in the creation of a vast array of AI-generated images (AIGIs). However, the quality of these AIGIs is highly inconsistent, with low-quality AIGIs severely impairing the visual experience of users. Due to the widespread application of AIGIs, the AI-generated image quality assessment (AIGIQA), aimed at evaluating the quality of AIGIs from the perspective of human perception, has garnered increasing interest among scholars. Nonetheless, current research has not yet fully explored this field. We have observed that existing databases are limited to images generated from single scenario settings. Databases such as AGIQA-1K, AGIQA-3K, and AIGCIQA2023, for example, only include images generated by text-to-image generative models. This oversight highlights a critical gap in the current research landscape, underscoring the need for dedicated databases catering to image-to-image scenarios, as well as more comprehensive databases that encompass a broader range of AI-generated image scenarios. Addressing these issues, we have established a large scale perceptual quality assessment database for both text-to-image and image-to-image AIGIs, named PKU-AIGIQA-4K. We then conduct a well-organized subjective experiment to collect quality labels for AIGIs and perform a comprehensive analysis of the PKU-AIGIQA-4K database. Regarding the use of image prompts during the training process, we propose three image quality assessment (IQA) methods based on pre-trained models that include a no-reference method NR-AIGCIQA, a full-reference method FR-AIGCIQA, and a partial-reference method PR-AIGCIQA. Finally, leveraging the PKU-AIGIQA-4K database, we conduct extensive benchmark experiments and compare the performance of the proposed methods and the current IQA methods.

cs.CV↗

arXiv2024

Superclosenes error estimates for the div least-squares finite element method on elliptic problems

In this paper we provide some error estimates for the div least-squares finite element method on elliptic problems. The main contribution is presenting a complete error analysis, which improves the current \emph{state-of-the-art} results. The error estimates for both the scalar and the flux variables are established by specially designed dual arguments with the help of two projections: elliptic projection and H(div) projection, which are crucial to supercloseness estimates. In most cases, $H^3$ regularity is omitted to get the optimal convergence rate for vector and scalar unknowns, and most of our results require a lower regularity for the vector variable than the scalar. Moreover, a series of supercloseness results are proved, which are \emph{never seen} in the previous work of least-squares finite element methods.

math.NA↗

arXiv2023

PKU-I2IQA: An Image-to-Image Quality Assessment Database for AI Generated Images

As image generation technology advances, AI-based image generation has been applied in various fields and Artificial Intelligence Generated Content (AIGC) has garnered widespread attention. However, the development of AI-based image generative models also brings new problems and challenges. A significant challenge is that AI-generated images (AIGI) may exhibit unique distortions compared to natural images, and not all generated images meet the requirements of the real world. Therefore, it is of great significance to evaluate AIGIs more comprehensively. Although previous work has established several human perception-based AIGC image quality assessment (AIGCIQA) databases for text-generated images, the AI image generation technology includes scenarios like text-to-image and image-to-image, and assessing only the images generated by text-to-image models is insufficient. To address this issue, we establish a human perception-based image-to-image AIGCIQA database, named PKU-I2IQA. We conduct a well-organized subjective experiment to collect quality labels for AIGIs and then conduct a comprehensive analysis of the PKU-I2IQA database. Furthermore, we have proposed two benchmark models: NR-AIGCIQA based on the no-reference image quality assessment method and FR-AIGCIQA based on the full-reference image quality assessment method. Finally, leveraging this database, we conduct benchmark experiments and compare the performance of the proposed benchmark models. The PKU-I2IQA database and benchmarks will be released to facilitate future research on \url{https://github.com/jiquan123/I2IQA}.

cs.CV↗

arXiv2023

Preconditioned NonSymmetric/Symmetric Discontinuous Galerkin Method for Elliptic Problem with Reconstructed Discontinuous Approximation

In this paper, we propose and analyze an efficient preconditioning method for the elliptic problem based on the reconstructed discontinuous approximation method. We reconstruct a high-order piecewise polynomial space that arbitrary order can be achieved with one degree of freedom per element. This space can be directly used with the symmetric/nonsymmetric interior penalty discontinuous Galerkin method. Compared with the standard DG method, we can enjoy the advantage on the efficiency of the approximation. Besides, we establish an norm equivalence result between the reconstructed high-order space and the piecewise constant space. This property further allows us to construct an optimal preconditioner from the piecewise constant space. The upper bound of the condition number to the preconditioned symmetric/nonsymmetric system is shown to be independent of the mesh size. Numerical experiments are provided to demonstrate the validity of the theory and the efficiency of the proposed method.

math.NA↗

arXiv2023

The Least Squares Finite Element Method for Elasticity Interface Problem on Unfitted Mesh

In this paper, we propose and analyze the least squares finite element methods for the linear elasticity interface problem in the stress-displacement system on unfitted meshes. We consider the cases that the interface is $C^2$ or polygonal, and the exact solution $(\sigma,u)$ belongs to $H^s(div; \Omega_0 \cup \Omega_1) \times $H^{1+s}(\Omega_0 \cup \Omega_1)$ with $s > 1/2$. Two types of least squares functionals are defined to seek the numerical solution. The first is defined by simply applying the $L^2$ norm least squares principle, and requires the condition $s \geq 1$. The second is defined with a discrete minus norm, which is related to the inner product in $H^{-1/2}(\Gamma)$. The use of this discrete minus norm results in a method of optimal convergence rates and allows the exact solution has the regularity of any $s > 1/2$. The stability near the interface for both methods is guaranteed by the ghost penalty bilinear forms and we can derive the robust condition number estimates. The convergence rates under $L^2$ norm and the energy norm are derived for both methods. We illustrate the accuracy and the robustness of the proposed methods by a series of numerical experiments for test problems in two and three dimensions.

math.NA↗

arXiv2022

An unfitted finite element method with direct extension stabilization for time-harmonic Maxwell problems on smooth domains

We propose an unfitted finite element method for numerically solving the time-harmonic Maxwell equations on a smooth domain. The model problem involves a Lagrangian multiplier to relax the divergence constraint of the vector unknown. The embedded boundary of the domain is allowed to cut through the background mesh arbitrarily. The unfitted scheme is based on a mixed interior penalty formulation, where Nitsche penalty method is applied to enforce the boundary condition in a weak sense, and a penalty stabilization technique is adopted based on a local direct extension operator to ensure the stability for cut elements. We prove the inf-sup stability and obtain optimal convergence rates under the energy norm and the $L^2$ norm for both the vector unknown and the Lagrangian multiplier. Numerical examples in both two and three dimensions are presented to illustrate the accuracy of the method.

math.NA↗

arXiv2021

An unfitted finite element method by direct extension for elliptic problems on domains with curved boundaries and interfaces

We propose and analyze an unfitted finite element method for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element space defined on interior elements, in the sense that there is no degree of freedom locating in boundary/interface elements. The boundary/jump conditions are imposed in a weak sense in the scheme. The method is shown to be stable without any mesh adjustment or any special stabilization. Optimal convergence rates under the $L^2$ norm and the energy norm are derived. Numerical results in both two and three dimensions are presented to illustrate the accuracy and the robustness of the method.

math.NA↗

arXiv2021

A discontinuous least squares finite element method for Helmholtz equations

We propose a discontinuous least squares finite element method for solving the Helmholtz equation. The method is based on the L2 norm least squares functional with the weak imposition of the continuity across the interior faces as well as the boundary conditions. We minimize the functional over the discontinuous polynomial spaces to seek numerical solutions. The wavenumber explicit error estimates to our method are established. The optimal convergence rate in the energy norm with respect to a fixed wavenumber is attained. The least squares functional can naturally serve as a posteriori estimator in the h-adaptive procedure. It is convenient to implement the code due to the usage of discontinuous elements. Numerical results in two and three dimensions are presented to verify the error estimates.

math.NA↗

arXiv2020

A Discontinuous Galerkin Method by Patch Reconstruction for Elliptic Interface Problem on Unfitted Mesh

We propose a discontinuous Galerkin(DG) method to approximate the elliptic interface problem on unfitted mesh using a new approximation space. The approximation space is constructed by patch reconstruction with one degree of freedom per element. The optimal error estimates in both L2 norm and DG energy norm are obtained, without the typical constraints for DG method on how the interface intersects to the elements in the mesh. Other than enjoying the advantages of DG method, our method may achieve even better efficiency than the conforming finite element method, as illustrated by numerical examples.

math.NA↗

arXiv2020

A discontinuous least squares finite element method for time-harmonic Maxwell equations

We propose and analyze a discontinuous least squares finite element method for solving the indefinite time-harmonic Maxwell equations. The scheme is based on the $L^2$ norm least squares functional with the weak imposition of the continuity across the interior faces. We minimize the functional over the piecewise polynomial spaces to seek numerical solutions. The method is shown to be stable without any constraint on the mesh size. We prove the convergence orders under both the energy norm and the $L^2$ norm. Numerical results in two and three dimensions are presented to verify the error estimates.

math.NA↗

arXiv2020

A sequential least squares method for elliptic equations in non-divergence form

We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equations in two steps. We first obtain a numerical approximation to the gradient in a piecewisely irrotational polynomial space. Then together with the numerical gradient, we seek a numerical solution of the primitive variable in continuous finite element space. The variational setting naturally provides a posteriori error which could be used in an adaptive refinement algorithm. The error estimates in $L^2$ norm and energy norms for both two unknowns are derived. By a series of numerical experiments, we verify the convergence rates and show the efficiency of the adaptive algorithm.

math.NA↗

arXiv2020

Reconstructed Discontinuous Approximation to Stokes Equation in A Sequential Least Squares Formulation

We propose a new least squares finite element method to solve the Stokes problem with two sequential steps. The approximation spaces are constructed by patch reconstruction with one unknown per element. For the first step, we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace. By this space, we minimize a least squaresfunctional toobtain thenumericalapproximationstothe gradientof thevelocityand the pressure. In the second step, we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergence-free space. We derive error estimates for all unknowns under L2 norms and energy norms. Numerical results in two dimensions and three dimensions verify the convergence rates and demonstrate the great flexibility of our method.

math.NA↗