arXiv subjects

Haijun Yu

Publications and source records attributed to Haijun Yu.

At least 19 recordsLinked to original sources

arXiv2026

Quantum Compressed Sensing CT Reconstruction Algorithm Based on Penalized Weighted Least Squares and Guided Total Variation

Objective. Existing quadratic unconstrained binary optimization (QUBO)-based sparse-view computed tomography (CT) reconstruction neglects photon-counting statistics and anatomical heterogeneity. We address both limitations within the QUBO framework.Approach. We propose a quantum compressed-sensing CT method combining penalized weighted least squares (PWLS) and guided total variation (GTV). PWLS weights projection residuals by photon-count reliability, whereas GTV uses gradients from a prior image reconstructed by the simultaneous algebraic reconstruction technique (SART) to preserve edges and suppress noise in homogeneous regions. After binary encoding, both terms form a unified QUBO model. Experiments used four 40 times 40 CT images under a 10-view fan-beam geometry with Poisson noise. Comparisons included conventional reconstruction methods, QUBO variants, gradient descent, simulated annealing, and a D-Wave hybrid quantum-classical solver.Main results. PWLS-GTV achieved the best reconstruction quality across all cases. In the representative chest case, it reached a peak signal-to-noise ratio (PSNR) of 36.64 dB, compared with 22.48 dB for SART, the best conventional baseline. GTV consistently outperformed conventional total variation. Simulated annealing and the D-Wave hybrid solver produced similar reconstructions, whereas gradient descent was ineffective. Repeated hybrid-solver runs showed stable performance.Significance. The framework incorporates photon-statistical weighting and structure-guided regularization into QUBO-based CT reconstruction without changing its quadratic form, providing a proof of concept for quantum-assisted sparse-view CT reconstruction.

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arXiv2026

Penalty-Free Natural Deep Ritz Method Based on de Rham Complex for High-Dimensional Dirichlet Boundary Value Problems

Deep neural networks show great promise for high-dimensional PDEs, yet enforcing essential boundary conditions remains challenging, especially as penalty parameters require problem-specific retuning with increasing dimensionality. In this work, we extend the Natural Deep Ritz Method (NatDRM) [H. Yu and S. Zhang, J. Comput. Phys., 537 (2025)] to a unified framework for all dimensions $d \geq 2$ based on the de Rham complex and its penalty-free boundary decomposition: curl-type operators act on scalar potentials in 2D, vector potentials in 3D, and antisymmetric second-order tensor potentials in $d \geq 4$, respectively. This method converts Dirichlet constraints into three coupled natural (Neumann-type) subproblems with corresponding Ritz-type losses, eliminating the need for a boundary penalty parameter $\beta$. We derive dimension-unified discrete losses, lightweight boundary-based gauge-fixing regularizations to resolve curl-kernel non-uniqueness, and a joint training procedure; extensions to variable-coefficient elliptic and semilinear Poisson problems are formulated at the first subproblem level. Numerical experiments on smooth benchmarks up to 6D show that NatDRM, without any penalty tuning, matches or exceeds the accuracy of optimally tuned DRM and PINN in most cases. It converges stably in 6D where penalized DRM fails for most penalty values, and exhibits synchronous decay of interior and boundary errors, resolving the inherent imbalance of penalty-based methods.

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arXiv2026

Quantum CT via Dynamic Interval Encoding and Prior-Balanced QUBO Reconstruction

Quadratic unconstrained binary optimization (QUBO)-based quantum computed tomography (CT) casts reconstruction as a binary quadratic problem for quantum annealing and hybrid quantum--classical solvers. For grayscale CT, however, image encoding is constrained by the binary-variable budget: fixed global bit-plane encodings increase QUBO size and coupling complexity as gray-level precision improves, whereas low-bit encodings introduce quantization error. We propose a QUBO-based grayscale CT reconstruction framework that combines dynamic interval encoding with prior-balanced optimization. Each refinement round encodes active pixels only within local gray-level intervals around the current estimate, and a boundary-hit-guided update rule adaptively switches between search expansion and local refinement. To improve optimization stability, the method balances projection-domain data consistency and an edge-preserving quadratic prior before forming the final QUBO. Sparse-view and limited-angle fan-beam CT experiments show that the proposed method recovers structures and gray-level distributions more faithfully than the evaluated analytic, iterative, variational, and representation-based baselines. Expressivity analysis and ablation studies further indicate that the improvement mainly arises from effective gray-level representation through dynamic local encoding and more stable data-fidelity--prior coupling. Experiments on the D-Wave hybrid binary quadratic model (BQM) solver further demonstrate that the formulation is executable on a hardware-backed hybrid quantum--classical backend.

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arXiv2026

An Efficient Laguerre Minimum Action Method for Computing Quasi-Potentials

Minimum action methods provide a powerful framework for analyzing rare transitions in small-noise-driven dynamical systems, but their practical performance is often limited by time truncation and parameter sensitivity in infinite-horizon problems. In this paper, we develop an efficient Laguerre spectral minimum action method (LMAM) for computing quasi-potentials associated with fixed points of dynamical systems. Based on the large deviation framework, the method computes minimum action paths by formulating the problem on a semi-infinite time interval and discretize the temporal direction using Laguerre functions. An appropriate time rescaling strategy is proposed to enhance accuracy and convergence of the Laguerre spectral approximation. To efficiently handle nonlinear terms, we employ an improved procedure for evaluating Laguerre--Gauss--Radau quadrature, which enables stable and accurate double-precision computations with a large number of Laguerre modes. Precise numerical analysis for the linear problem and a local result for the nonlinear case are developed. Numerical experiments including both ordinary and partial differential equations (Allen-Cahn and Navier-Stokes) are presented to illustrate the accuracy and efficiency of the proposed method.

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arXiv2026

Predictive Moving Sample Method for Physics-Informed Neural Solvers of Time-Dependent PDEs

Time-dependent partial differential equations (PDEs) often develop sharp fronts, localized peaks, and other moving structures that occupy only a small portion of the space--time domain but dominate the approximation error. This makes fixed or uniformly sampled collocation strategies inefficient for physics-informed neural networks (PINNs), especially in high dimensions and over long-time prediction intervals. We propose the predictive moving sample method (PMSM), which builds on the moving sample method (MSM) in \cite{xu2026moving} by replacing its full time domain iterative training with a progressive time-stepping strategy and simplifying the velocity-field loss to further reduce the per-step cost. To improve practicality for long-time prediction, we further introduce the windowed-reset predictive moving sample method (WR-PMSM), which restricts extension training to an active time window and periodically resets the reference state, thereby reducing the growth of optimization cost while preserving global consistency through a final refinement stage. Across four representative benchmarks, PMSM consistently outperforms both standard PINNs and the original MSM under matched collocation budgets. These results suggest that transporting samples according to residual dynamics provides an effective and practical route to neural network solvers for time-dependent PDEs.

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arXiv2026

Improving Generalization of Deep Learning for Brain Metastases Segmentation Across Institutions

Background: Deep learning has demonstrated significant potential for automated brain metastases (BM) segmentation; however, models trained at a singular institution often exhibit suboptimal performance at various sites due to disparities in scanner hardware, imaging protocols, and patient demographics. The goal of this work is to create a domain adaptation framework that will allow for BM segmentation to be used across multiple institutions. Methods: We propose a VAE-MMD preprocessing pipeline that combines variational autoencoders (VAE) with maximum mean discrepancy (MMD) loss, incorporating skip connections and self-attention mechanisms alongside nnU-Net segmentation. The method was tested on 740 patients from four public databases: Stanford, UCSF, UCLM, and PKG, evaluated by domain classifier's accuracy, sensitivity, precision, F1/F2 scores, surface Dice (sDice), and 95th percentile Hausdorff distance (HD95). Results: VAE-MMD reduced domain classifier accuracy from 0.91 to 0.50, indicating successful feature alignment across institutions. Reconstructed volumes attained a PSNR greater than 36 dB, maintaining anatomical accuracy. The combined method raised the mean F1 by 11.1% (0.700 to 0.778), the mean sDice by 7.93% (0.7121 to 0.7686), and reduced the mean HD95 by 65.5% (11.33 to 3.91 mm) across all four centers compared to the baseline nnU-Net. Conclusions: VAE-MMD effectively diminishes cross-institutional data heterogeneity and enhances BM segmentation generalization across volumetric, detection, and boundary-level metrics without necessitating target-domain labels, thereby overcoming a significant obstacle to the clinical implementation of AI-assisted segmentation.

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arXiv2026

Fast Jacobi Spectral Methods and Closure Approximations for the Homogeneous FENE Model of Complex Fluids

The Finitely Extensible Nonlinear Elastic (FENE) dumbbell model is a widely used mathematical model for complex fluids. Direct simulation of the FENE Fokker--Planck equation is computationally challenging due to high dimensionality and singularity of its potential. In this paper, we develop two fast Jacobi-Spherical Harmonic spectral methods for the spatially homogeneous FENE Fokker--Planck equation. These methods effectively resolve the singularity near the boundary by combining properly designed Jacobi polynomials with a weighted variational formulation. A semi-implicit backward differentiation formula of second-order (BDF2) is employed for time marching, and its energy stability is rigorously proved. The resulting linear algebraic system possesses a sparse structure and can be efficiently solved. Numerical results verify the spectral convergence and efficiency of the direct spectral solvers, establishing them as a reliable tool for generating reference solutions for challenging benchmark problems. Furthermore, to achieve an optimal trade-off between accuracy and efficiency, we compare several closure approximation models, including the industry workhorse Peterlin approximation (FENE-P), the quasi-equilibrium approximation (FENE-QE), and a novel neural network implementation for FENE-QE proposed in this paper (FENE-QE-NN). Numerical experiments in extensional and shear flows demonstrate the superior accuracy and efficiency of the proposed methods compared to traditional approaches.

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arXiv2026

Scaling Optimized Spectral Approximations on Unbounded Domains: The Generalized Hermite and Laguerre Methods

We propose a novel error analysis framework for scaled generalized Laguerre and generalized Hermite approximations.This framework can be regarded as an analogue of the Nyquist-Shannon sampling theorem: It characterizes the spatial and frequency bandwidths that can be effectively captured by Laguerre or Hermite sampling points. Provided a function satisfies the corresponding bandwidth constraints, it can be accurately approximated within this framework. The proposed framework is notably more powerful than classical theory -- it not only provides systematic guidance for choosing the optimal scaling factor, but also predicts root-exponential and other intricate convergence behaviors that classical approaches fail to capture. Leveraging this framework, we conducted a detailed comparative study of Hermite and Laguerre approximations. We find that functions with similar decay and oscillation characteristics may nonetheless display markedly different convergence rates. Furthermore, approximations based on two concatenated sets of Laguerre functions may offer significant advantages over those using a single set of Hermite functions.

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arXiv2026

Structure-constrained Language-informed Diffusion Model for Unpaired Low-dose Computed Tomography Angiography Reconstruction

The application of iodinated contrast media (ICM) improves the sensitivity and specificity of computed tomography (CT) for a wide range of clinical indications. However, overdose of ICM can cause problems such as kidney damage and life-threatening allergic reactions. Deep learning methods can generate CT images of normal-dose ICM from low-dose ICM, reducing the required dose while maintaining diagnostic power. However, existing methods are difficult to realize accurate enhancement with incompletely paired images, mainly because of the limited ability of the model to recognize specific structures. To overcome this limitation, we propose a Structure-constrained Language-informed Diffusion Model (SLDM), a unified medical generation model that integrates structural synergy and spatial intelligence. First, the structural prior information of the image is effectively extracted to constrain the model inference process, thus ensuring structural consistency in the enhancement process. Subsequently, semantic supervision strategy with spatial intelligence is introduced, which integrates the functions of visual perception and spatial reasoning, thus prompting the model to achieve accurate enhancement. Finally, the subtraction angiography enhancement module is applied, which serves to improve the contrast of the ICM agent region to suitable interval for observation. Qualitative analysis of visual comparison and quantitative results of several metrics demonstrate the effectiveness of our method in angiographic reconstruction for low-dose contrast medium CT angiography.

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arXiv2026

Moving sample method for solving time-dependent partial differential equations

Solving time-dependent partial differential equations (PDEs) that exhibit sharp gradients or local singularities is computationally demanding, as traditional physics-informed neural networks (PINNs) often suffer from inefficient point allocation that wastes resources on regions already well-resolved. This paper presents an adaptive sampling framework for PINNs aimed at efficiently solving time-dependent partial differential equations with pronounced local singularities. The method employs a residual-driven strategy, where the spatial-temporal distribution of training points is iteratively updated according to the error field from the previous iteration. This targeted allocation enables the network to concentrate computational effort on regions with significant residuals, achieving higher accuracy with fewer sampling points compared to uniform sampling. Numerical experiments on representative PDE benchmarks demonstrate that the proposed approach improves solution quality.

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arXiv2025

A Morphology-Adaptive Random Feature Method for Inverse Source Problem of the Helmholtz Equation

The inverse source problem for the Helmholtz equation poses significant challenges, particularly when sources exhibit complex or discontinuous geometries. Traditional numerical methods suffer from prohibitive computational costs, while machine learning-based approaches such as Physics-Informed Neural Networks (PINNs) and the Random Feature Method (RFM) -- though computationally efficient for inverse problems -- lack the intrinsic machinery to handle the sharp morphological features in such singular problems, leading to inaccurate solutions. To address this issue, we propose the Morphology-Adaptive Random Feature Method (MA-RFM), a novel two-phase framework that adaptively locates critical regions and adds morphology activation functions for tackling the multi-frequency inverse source problem with complex geometry. Our framework recasts the ill-posed inverse problem into a well-posed, strictly convex optimization problem by reformulating the governing Helmholtz equation as a Tikhonov-regularized integral equation via its fundamental solution. In the first stage, the Integral Adaptive RFM (IA-RFM), employs an adaptive algorithm to rapidly localize the source support, thereby reducing computational overhead and accelerating convergence. In the second stage, posterior geometric information is progressively integrated into the solver via hybrid basis functions, enabling a precise reconstruction of complex morphologies. The MA-RFM extends the capabilities of RFM to handle PDEs with singular solutions while preserving its mesh-free efficiency. We demonstrate the superior performance of our approach through ample challenging 2D and 3D benchmark problems, even under limited and noisy measurement conditions, highlighting its robustness and accuracy in reconstructing complex and disjoint sources.

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arXiv2025

Efficient Image-to-Image Schr\"odinger Bridge for CT Field of View Extension

Computed tomography (CT) is a cornerstone imaging modality for non-invasive, high-resolution visualization of internal anatomical structures. However, when the scanned object exceeds the scanner's field of view (FOV), projection data are truncated, resulting in incomplete reconstructions and pronounced artifacts near FOV boundaries. Conventional reconstruction algorithms struggle to recover accurate anatomy from such data, limiting clinical reliability. Deep learning approaches have been explored for FOV extension, with diffusion generative models representing the latest advances in image synthesis. Yet, conventional diffusion models are computationally demanding and slow at inference due to their iterative sampling process. To address these limitations, we propose an efficient CT FOV extension framework based on the image-to-image Schr\"odinger Bridge (I$^2$SB) diffusion model. Unlike traditional diffusion models that synthesize images from pure Gaussian noise, I$^2$SB learns a direct stochastic mapping between paired limited-FOV and extended-FOV images. This direct correspondence yields a more interpretable and traceable generative process, enhancing anatomical consistency and structural fidelity in reconstructions. I$^2$SB achieves superior quantitative performance, with root-mean-square error (RMSE) values of 49.8 HU on simulated noisy data and 152.0 HU on real data, outperforming state-of-the-art diffusion models such as conditional denoising diffusion probabilistic models (cDDPM) and patch-based diffusion methods. Moreover, its one-step inference enables reconstruction in just 0.19 s per 2D slice, representing over a 700-fold speedup compared to cDDPM (135 s) and surpassing DiffusionGAN (0.58 s), the second fastest. This combination of accuracy and efficiency indicates that I$^2$SB has potential for real-time or clinical deployment.

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arXiv2025

An Improved Adaptive Orthogonal Basis Deflation Method for Multiple Solutions with Applications to Nonlinear Elliptic Equations in Varying Domains

Multiple solutions are common in various non-convex problems arising from industrial and scientific computing. Nonetheless, understanding the nontrivial solutions' qualitative properties seems limited, partially due to the lack of efficient and reliable numerical methods. In this paper, we design a dedicated numerical method to explore these nontrivial solutions further. We first design an improved adaptive orthogonal basis deflation method by combining the adaptive orthogonal basis method with a bisection-deflation algorithm. We then apply the proposed new method to study the impact of domain changes on multiple solutions of certain nonlinear elliptic equations. When the domain varies from a circular disk to an elliptical disk, the corresponding functional value changes dramatically for some particular solutions, which indicates that these nontrivial solutions in the circular domain may become unstable in the elliptical domain. Moreover, several theoretical results on multiple solutions in existing literature are verified. For the nonlinear Sine-Gordon equation with parameter $\lambda$, nontrivial solutions are found for $\lambda > \lambda_2$, here $\lambda_2$ is the second eigenvalue of the corresponding linear eigenvalue problem. For the singularly perturbed Ginzburg-Landau equation, highly concentrated solutions are numerically found which suggests that their convergent limit is a delta function when the perturbation parameter goes to zero

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arXiv2024

An Efficient Stochastic Subgradient Method for the Global Placement Problem in Very Large-Scale Integration Circuits

The placement problem in Very Large-Scale Integration (VLSI) circuits is a critical step in chip design. Its primary goal is to optimize the wirelength of circuit components within a confined area while adhering to nonoverlapping constraints. This paper introduces a novel approach that directly optimizes the original nonsmooth wirelength and proposes an innovative penalty model tailored for the global placement problem. Specifically, we transform the nonoverlapping constraints into rectified linear penalty functions, allowing for a more precise formulation of the problem. Notably, we recast the resultant optimization problem into a form analogous to training deep neural network with Rectified Linear Units (ReLU). Leveraging automatic differentiation techniques from deep learning, we efficiently compute the subgradient of the objective function. This facilitates the application of stochastic subgradient methods to solve the model. To enhance the algorithm's performance, several advanced techniques are further introduced, leading to significant improvements in both efficiency and solution quality. Numerical experiments were conducted on Gigascale Systems Research Center (GSRC) benchmark and International Symposium on Physical Design 2005 (ISPD2005) benchmark circuits. The results demonstrate that our proposed model and algorithm achieve significant reductions in wirelength while effectively eliminating overlaps. This highlights the potential of our approach as a transformative advancement for VLSI placement. Furthermore, we establish a rigorous convergence proof for the proposed stochastic subgradient method. To the best of our knowledge, it constitutes the first such result for the ReLU-type nonsmooth and nonconvex optimization problems.

math.OC↗

arXiv2024

Scaling Optimized Hermite Approximation Methods

Hermite polynomials and functions have extensive applications in scientific and engineering problems. Although it is recognized that employing the scaled Hermite functions rather than the standard ones can remarkably enhance the approximation performance, the understanding of the scaling factor remains insufficient. Due to the lack of theoretical analysis, recent publications still cast doubt on whether the Hermite spectral method is inferior to other methods. To dispel this doubt, we show in this article that the inefficiency of the Hermite spectral method comes from the imbalance in the decay speed of the objective function within the spatial and frequency domains. Proper scaling can render the Hermite spectral methods comparable to other methods. To make it solid, we propose a novel error analysis framework for the scaled Hermite approximation. Taking the $L^2$ projection error as an example, our framework illustrates that there are three different components of errors: the spatial truncation error, the frequency truncation error, and the Hermite spectral approximation error. Through this perspective, finding the optimal scaling factor is equivalent to balancing the spatial and frequency truncation errors. As applications, we show that geometric convergence can be recovered by proper scaling for a class of functions. Furthermore, we show that proper scaling can double the convergence order for smooth functions with algebraic decay. The perplexing pre-asymptotic sub-geometric convergence when approximating algebraic decay functions can be perfectly explained by this framework.

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arXiv2024

A Natural Deep Ritz Method for Essential Boundary Value Problems

Deep neural network approaches show promise in solving partial differential equations. However, unlike traditional numerical methods, they face challenges in enforcing essential boundary conditions. The widely adopted penalty-type methods, for example, offer a straightforward implementation but introduces additional complexity due to the need for hyper-parameter tuning; moreover, the use of a large penalty parameter can lead to artificial extra stiffness, complicating the optimization process. In this paper, we propose a novel, intrinsic approach to impose essential boundary conditions through a framework inspired by intrinsic structures. We demonstrate the effectiveness of this approach using the deep Ritz method applied to Poisson problems, with the potential for extension to more general equations and other deep learning techniques. Numerical results are provided to substantiate the efficiency and robustness of the proposed method.

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arXiv2024

Energetic Spectral-Element Time Marching Methods for Phase-Field Nonlinear Gradient Systems

We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen--Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen--Cahn equation, we also apply the method to a conservative Allen--Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen--Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems.

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arXiv2024

On the Influence of Smoothness Constraints in Computed Tomography Motion Compensation

Computed tomography (CT) relies on precise patient immobilization during image acquisition. Nevertheless, motion artifacts in the reconstructed images can persist. Motion compensation methods aim to correct such artifacts post-acquisition, often incorporating temporal smoothness constraints on the estimated motion patterns. This study analyzes the influence of a spline-based motion model within an existing rigid motion compensation algorithm for cone-beam CT on the recoverable motion frequencies. Results demonstrate that the choice of motion model crucially influences recoverable frequencies. The optimization-based motion compensation algorithm is able to accurately fit the spline nodes for frequencies almost up to the node-dependent theoretical limit according to the Nyquist-Shannon theorem. Notably, a higher node count does not compromise reconstruction performance for slow motion patterns, but can extend the range of recoverable high frequencies for the investigated algorithm. Eventually, the optimal motion model is dependent on the imaged anatomy, clinical use case, and scanning protocol and should be tailored carefully to the expected motion frequency spectrum to ensure accurate motion compensation.

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