Long Chen

Luxi Li, Qin Zou, Fan Zhang et al.

Mural image inpainting is far less explored compared to its natural image counterpart and remains largely unsolved. Most existing image-inpainting methods tend to take the target image as the only input and directly repair the damage to generate a visually plausible result. These methods obtain high performance in restoration or completion of some pre-defined objects, e.g., human face, fabric texture, and printed texts, etc., however, are not suitable for repairing murals with varying subjects and large damaged areas. Moreover, due to discrete colors in paints, mural inpainting may suffer from apparent color bias. To this end, in this paper, we propose a line drawing guided progressive mural inpainting method. It divides the inpainting process into two steps: structure reconstruction and color correction, implemented by a structure reconstruction network (SRN) and a color correction network (CCN), respectively. In structure reconstruction, SRN utilizes the line drawing as an assistant to achieve large-scale content authenticity and structural stability. In color correction, CCN operates a local color adjustment for missing pixels which reduces the negative effects of color bias and edge jumping. The proposed approach is evaluated against the current state-of-the-art image inpainting methods. Qualitative and quantitative results demonstrate the superiority of the proposed method in mural image inpainting. The codes and data are available at https://github.com/qinnzou/mural-image-inpainting.

Long Chen, Michael Holst, Jinchao Xu et al.

The goal of this paper is to design optimal multilevel solvers for the finite element approximation of second order linear elliptic problems with piecewise constant coefficients on bisection grids. Local multigrid and BPX preconditioners are constructed based on local smoothing only at the newest vertices and their immediate neighbors. The analysis of eigenvalue distributions for these local multilevel preconditioned systems shows that there are only a fixed number of eigenvalues which are deteriorated by the large jump. The remaining eigenvalues are bounded uniformly with respect to the coefficients and the meshsize. Therefore, the resulting preconditioned conjugate gradient algorithm will converge with an asymptotic rate independent of the coefficients and logarithmically with respect to the meshsize. As a result, the overall computational complexity is nearly optimal.

Long Chen, Junping Wang, Yanqiu Wang et al.

In this paper, the authors constructed an auxiliary space multigrid preconditioner for the weak Galerkin finite element method for second-order diffusion equations, discretized on simplicial 2D or 3D meshes. The idea of the auxiliary space multigrid preconditioner is to use an auxiliary space as a "coarse" space in the multigrid algorithm, where the discrete problem in the auxiliary space can be easily solved by an existing solver. In this construction, the authors conveniently use the $H^1$ conforming piecewise linear finite element space as an auxiliary space. The main technical difficulty is to build the connection between the weak Galerkin discrete space and the $H^1$ conforming piecewise linear finite element space. The authors successfully constructed such an auxiliary space multigrid preconditioner for the weak Galerkin method, as well as a reduced system of the weak Galerkin method involving only the degrees of freedom on edges/faces. The preconditioned systems are proved to have condition numbers independent of the mesh size. Numerical experiments are conducted to support the theoretical results.

Long Chen, Jun Hu, Xuehai Huang

A block diagonal preconditioner with the minimal residual method and a block triangular preconditioner with the generalized minimal residual method are developed for Hu-Zhang mixed finite element methods of linear elasticity. They are based on a new stability result of the saddle point system in mesh-dependent norms. The mesh-dependent norm for the stress corresponds to the mass matrix which is easy to invert while the displacement it is spectral equivalent to Schur complement. A fast auxiliary space preconditioner based on the $H^1$ conforming linear element of the linear elasticity problem is then designed for solving the Schur complement. For both diagonal and triangular preconditioners, it is proved that the conditioning numbers of the preconditioned systems are bounded above by a constant independent of both the crucial Lamé constant and the mesh-size. Numerical examples are presented to support theoretical results. As byproducts, a new stabilized low order mixed finite element method is proposed and analyzed and superconvergence results of Hu-Zhang element are obtained.

Long Chen, Xuehai Huang

A framework to systematically decouple high order elliptic equations into combination of Poisson-type and Stokes-type equations is developed. The key is to systematically construct the underling commutative diagrams involving the complexes and Helmholtz decompositions in a general way. Discretizing the decoupled formulation leads to a natural superconvergence between the Galerkin projection and the decoupled approximation. Examples include but not limit to: the primal formulations and mixed formulations of biharmonic equation, fourth order curl equation, and triharmonic equation etc. As a by-product, Helmholtz decompositions for many dual spaces are obtained.

Long Chen, Jun Hu, Xuehai Huang

A V-cycle multigrid method for the Hellan-Herrmann-Johnson (HHJ) discretization of the Kirchhoff plate bending problems is developed in this paper. It is shown that the contraction number of the V-cycle multigrid HHJ mixed method is bounded away from one uniformly with respect to the mesh size. The uniform convergence is achieved for the V-cycle multigrid method with only one smoothing step and without full elliptic regularity. The key is a stable decomposition of the kernel space which is derived from an exact sequence of the HHJ mixed method, and the strengthened Cauchy Schwarz inequality. Some numerical experiments are provided to confirm the proposed V-cycle multigrid method. The exact sequences of the HHJ mixed method and the corresponding commutative diagram is of some interest independent of the current context.

Jian Huang, Long Chen, Hongxing Rui

An efficient nonlinear multigrid method for a mixed finite element method of the Darcy-Forchheimer model is constructed in this paper. A Peaceman-Rachford type iteration is used as a smoother to decouple the nonlinearity from the divergence constraint. The nonlinear equation can be solved element-wise with a closed formulae. The linear saddle point system for the constraint is reduced into a symmetric positive definite system of Poisson type. Furthermore an empirical choice of the parameter used in the splitting is proposed and the resulting multigrid method is robust to the so-called Forchheimer number which controls the strength of the nonlinearity. By comparing the number of iterations and CPU time of different solvers in several numerical experiments, our multigrid method is shown to convergent with a rate independent of the mesh size and the Forchheimer number and with a nearly linear computational cost.

Yuhang He, Lin Chen, Junkun Xie et al.

We propose a novel framework to learn 3D point cloud semantics from 2D multi-view image observations containing pose error. On the one hand, directly learning from the massive, unstructured and unordered 3D point cloud is computationally and algorithmically more difficult than learning from compactly-organized and context-rich 2D RGB images. On the other hand, both LiDAR point cloud and RGB images are captured in standard automated-driving datasets. This motivates us to conduct a "task transfer" paradigm so that 3D semantic segmentation benefits from aggregating 2D semantic cues, albeit pose noises are contained in 2D image observations. Among all difficulties, pose noise and erroneous prediction from 2D semantic segmentation approaches are the main challenges for the task transfer. To alleviate the influence of those factor, we perceive each 3D point using multi-view images and for each single image a patch observation is associated. Moreover, the semantic labels of a block of neighboring 3D points are predicted simultaneously, enabling us to exploit the point structure prior to further improve the performance. A hierarchical full attention network~(HiFANet) is designed to sequentially aggregates patch, bag-of-frames and inter-point semantic cues, with hierarchical attention mechanism tailored for different level of semantic cues. Also, each preceding attention block largely reduces the feature size before feeding to the next attention block, making our framework slim. Experiment results on Semantic-KITTI show that the proposed framework outperforms existing 3D point cloud based methods significantly, it requires much less training data and exhibits tolerance to pose noise. The code is available at https://github.com/yuhanghe01/HiFANet.

Long Chen, Jun Hu, Xuehai Huang et al.

A posteriori error estimators for the symmetric mixed finite element methods for linear elasticity problems of Dirichlet and mixed boundary conditions are proposed. Stability and efficiency of the estimators are proved. Finally, we provide numerical examples to verify the theoretical results.

Bin Zheng, Luoping Chen, Xiaozhe Hu et al.

In this paper, we study fast iterative solvers for the solution of fourth order parabolic equations discretized by mixed finite element methods. We propose to use consistent mass matrix in the discretization and use lumped mass matrix to construct efficient preconditioners. We provide eigenvalue analysis for the preconditioned system and estimate the convergence rate of the preconditioned GMRes method. Furthermore, we show that these preconditioners only need to be solved inexactly by optimal multigrid algorithms. Our numerical examples indicate that the proposed preconditioners are very efficient and robust with respect to both discretization parameters and diffusion coefficients. We also investigate the performance of multigrid algorithms with either collective smoothers or distributive smoothers when solving the preconditioner systems.

Long Chen, Yuli Wu, Dorit Merhof

Instance segmentation aims to delineate each individual object of interest in an image. State-of-the-art approaches achieve this goal by either partitioning semantic segmentations or refining coarse representations of detected objects. In this work, we propose a novel approach to solve the problem via object layering, i.e. by distributing crowded, even overlapping objects into different layers. By grouping spatially separated objects in the same layer, instances can be effortlessly isolated by extracting connected components in each layer. In comparison to previous methods, our approach is not affected by complex object shapes or object overlaps. With minimal post-processing, our method yields very competitive results on a diverse line of datasets: C. elegans (BBBC), Overlapping Cervical Cells (OCC) and cultured neuroblastoma cells (CCDB). The source code is publicly available.

Long Chen

The first order condition of the constrained minimization problem leads to a saddle point problem. A multigrid method using a multiplicative Schwarz smoother for saddle point problems can thus be interpreted as a successive subspace optimization method based on a multilevel decomposition of the constraint space. Convergence theory is developed for successive subspace optimization methods based on two assumptions on the space decomposition: stable decomposition and strengthened Cauchy-Schwarz inequality, and successfully applied to the saddle point systems arising from mixed finite element methods for Poisson and Stokes equations. Uniform convergence is obtained without the full regularity assumption of the underlying partial differential equations. As a byproduct, a V-cycle multigrid method for non-conforming finite elements is developed and proved to be uniform convergent with even one smoothing step.

Ruchi Guo, Shuhao Cao, Long Chen

A Transformer-based deep direct sampling method is proposed for electrical impedance tomography, a well-known severely ill-posed nonlinear boundary value inverse problem. A real-time reconstruction is achieved by evaluating the learned inverse operator between carefully designed data and the reconstructed images. An effort is made to give a specific example to a fundamental question: whether and how one can benefit from the theoretical structure of a mathematical problem to develop task-oriented and structure-conforming deep neural networks? Specifically, inspired by direct sampling methods for inverse problems, the 1D boundary data in different frequencies are preprocessed by a partial differential equation-based feature map to yield 2D harmonic extensions as different input channels. Then, by introducing learnable non-local kernels, the direct sampling is recast to a modified attention mechanism. The new method achieves superior accuracy over its predecessors and contemporary operator learners and shows robustness to noises in benchmarks. This research shall strengthen the insights that, despite being invented for natural language processing tasks, the attention mechanism offers great flexibility to be modified in conformity with the a priori mathematical knowledge, which ultimately leads to the design of more physics-compatible neural architectures.

Long Chen

We propose a modification of the weak Galerkin methods and show its equivalence to a new version of virtual element methods. We also show the original weak Galerkin method is equivalent to the non-conforming virtual element method. As a consequence, ideas and techniques used for one method can be transferred to another. The key of the connection is the degree of freedoms.

Qian Zhang, Long Chen, Yifeng Xu

In this paper an iterative minimization method is proposed to approximate the minimizer to the double-well energy functional arising in the phase-field theory. The method is based on a quadratic functional posed over a nonempty closed convex set and is shown to be unconditionally energy stable. By the minimization approach, we also derive an variant of the first-order scheme for the Allen-Cahn equation, which has been constructed in the context of Invariant Energy Quadratization, and prove its unconditional energy stability.

Xiaoshuang Huang, Hongxiang Li, Meng Cao et al.

Recent developments underscore the potential of textual information in enhancing learning models for a deeper understanding of medical visual semantics. However, language-guided medical image segmentation still faces a challenging issue. Previous works employ implicit and ambiguous architectures to embed textual information. This leads to segmentation results that are inconsistent with the semantics represented by the language, sometimes even diverging significantly. To this end, we propose a novel cross-modal conditioned Reconstruction for Language-guided Medical Image Segmentation (RecLMIS) to explicitly capture cross-modal interactions, which assumes that well-aligned medical visual features and medical notes can effectively reconstruct each other. We introduce conditioned interaction to adaptively predict patches and words of interest. Subsequently, they are utilized as conditioning factors for mutual reconstruction to align with regions described in the medical notes. Extensive experiments demonstrate the superiority of our RecLMIS, surpassing LViT by 3.74% mIoU on the publicly available MosMedData+ dataset and achieving an average increase of 1.89% mIoU for cross-domain tests on our QATA-CoV19 dataset. Simultaneously, we achieve a relative reduction of 20.2% in parameter count and a 55.5% decrease in computational load. The code will be available at https://github.com/ShashankHuang/RecLMIS.

Long Chen, Xiaotian Song, Andy Song et al.

Spiking Large Language Models have been shown as a good alternative to LLMs in various scenarios. Existing methods for creating Spiking LLMs, i.e., direct training and ANN-SNN conversion, often suffer from performance degradation and relatively high computational costs. To address these issues, we propose a novel Fast ANN-SNN conversion strategy (FAS) that transforms LLMs into spiking LLMs in two stages. The first stage employs a full-parameter fine-tuning of pre-trained models, so it does not need any direct training from scratch. The second stage introduces a coarse-to-fine calibration method to reduce conversion errors and improve accuracy. Experiments on both language and vision-language tasks across four different scales of LLMs demonstrate that FAS can achieve state-of-the-art performance yet with significantly reduced inference latency and computational costs. Notably, FAS only takes eight timesteps to achieve an accuracy of 3\% higher than that of the OPT-7B model, while reducing energy consumption by 96.63\%. The source code is available at https://github.com/lc783/FAS

Xianda Guo, Chenming Zhang, Youmin Zhang et al.

We present LightStereo, a cutting-edge stereo-matching network crafted to accelerate the matching process. Departing from conventional methodologies that rely on aggregating computationally intensive 4D costs, LightStereo adopts the 3D cost volume as a lightweight alternative. While similar approaches have been explored previously, our breakthrough lies in enhancing performance through a dedicated focus on the channel dimension of the 3D cost volume, where the distribution of matching costs is encapsulated. Our exhaustive exploration has yielded plenty of strategies to amplify the capacity of the pivotal dimension, ensuring both precision and efficiency. We compare the proposed LightStereo with existing state-of-the-art methods across various benchmarks, which demonstrate its superior performance in speed, accuracy, and resource utilization. LightStereo achieves a competitive EPE metric in the SceneFlow datasets while demanding a minimum of only 22 GFLOPs and 17 ms of runtime, and ranks 1st on KITTI 2015 among real-time models. Our comprehensive analysis reveals the effect of 2D cost aggregation for stereo matching, paving the way for real-world applications of efficient stereo systems. Code will be available at https://github.com/XiandaGuo/OpenStereo.

Xuehui Wang, Qing Wang, Yuzhi Zhao et al.

Despite convolutional network-based methods have boosted the performance of single image super-resolution (SISR), the huge computation costs restrict their practical applicability. In this paper, we develop a computation efficient yet accurate network based on the proposed attentive auxiliary features (A$^2$F) for SISR. Firstly, to explore the features from the bottom layers, the auxiliary feature from all the previous layers are projected into a common space. Then, to better utilize these projected auxiliary features and filter the redundant information, the channel attention is employed to select the most important common feature based on current layer feature. We incorporate these two modules into a block and implement it with a lightweight network. Experimental results on large-scale dataset demonstrate the effectiveness of the proposed model against the state-of-the-art (SOTA) SR methods. Notably, when parameters are less than 320k, A$^2$F outperforms SOTA methods for all scales, which proves its ability to better utilize the auxiliary features. Codes are available at https://github.com/wxxxxxxh/A2F-SR.

Boyi Sun, Yuhang Liu, Xingxia Wang et al.

Point cloud data labeling is considered a time-consuming and expensive task in autonomous driving, whereas annotation-free learning training can avoid it by learning point cloud representations from unannotated data. In this paper, we propose AFOV, a novel 3D \textbf{A}nnotation-\textbf{F}ree framework assisted by 2D \textbf{O}pen-\textbf{V}ocabulary segmentation models. It consists of two stages: In the first stage, we innovatively integrate high-quality textual and image features of 2D open-vocabulary models and propose the Tri-Modal contrastive Pre-training (TMP). In the second stage, spatial mapping between point clouds and images is utilized to generate pseudo-labels, enabling cross-modal knowledge distillation. Besides, we introduce the Approximate Flat Interaction (AFI) to address the noise during alignment and label confusion. To validate the superiority of AFOV, extensive experiments are conducted on multiple related datasets. We achieved a record-breaking 47.73\% mIoU on the annotation-free 3D segmentation task in nuScenes, surpassing the previous best model by 3.13\% mIoU. Meanwhile, the performance of fine-tuning with 1\% data on nuScenes and SemanticKITTI reached a remarkable 51.75\% mIoU and 48.14\% mIoU, outperforming all previous pre-trained models

Jinhan Wang, Long Chen, Aparna Khare et al.

We propose an approach for continuous prediction of turn-taking and backchanneling locations in spoken dialogue by fusing a neural acoustic model with a large language model (LLM). Experiments on the Switchboard human-human conversation dataset demonstrate that our approach consistently outperforms the baseline models with single modality. We also develop a novel multi-task instruction fine-tuning strategy to further benefit from LLM-encoded knowledge for understanding the tasks and conversational contexts, leading to additional improvements. Our approach demonstrates the potential of combined LLMs and acoustic models for a more natural and conversational interaction between humans and speech-enabled AI agents.

Tan M. Nguyen, Vai Suliafu, Stanley J. Osher et al.

We propose FMMformers, a class of efficient and flexible transformers inspired by the celebrated fast multipole method (FMM) for accelerating interacting particle simulation. FMM decomposes particle-particle interaction into near-field and far-field components and then performs direct and coarse-grained computation, respectively. Similarly, FMMformers decompose the attention into near-field and far-field attention, modeling the near-field attention by a banded matrix and the far-field attention by a low-rank matrix. Computing the attention matrix for FMMformers requires linear complexity in computational time and memory footprint with respect to the sequence length. In contrast, standard transformers suffer from quadratic complexity. We analyze and validate the advantage of FMMformers over the standard transformer on the Long Range Arena and language modeling benchmarks. FMMformers can even outperform the standard transformer in terms of accuracy by a significant margin. For instance, FMMformers achieve an average classification accuracy of $60.74\%$ over the five Long Range Arena tasks, which is significantly better than the standard transformer's average accuracy of $58.70\%$.

Long Chen, Lukas Platinsky, Stefanie Speichert et al.

We investigate what grade of sensor data is required for training an imitation-learning-based AV planner on human expert demonstration. Machine-learned planners are very hungry for training data, which is usually collected using vehicles equipped with the same sensors used for autonomous operation. This is costly and non-scalable. If cheaper sensors could be used for collection instead, data availability would go up, which is crucial in a field where data volume requirements are large and availability is small. We present experiments using up to 1000 hours worth of expert demonstration and find that training with 10x lower-quality data outperforms 1x AV-grade data in terms of planner performance. The important implication of this is that cheaper sensors can indeed be used. This serves to improve data access and democratize the field of imitation-based motion planning. Alongside this, we perform a sensitivity analysis of planner performance as a function of perception range, field-of-view, accuracy, and data volume, and the reason why lower-quality data still provide good planning results.

Long Chen, Junyu Dong, Huiyu Zhou

Underwater object detection technique is of great significance for various applications in underwater the scenes. However, class imbalance issue is still an unsolved bottleneck for current underwater object detection algorithms. It leads to large precision discrepancies among different classes that the dominant classes with more training data achieve higher detection precisions while the minority classes with fewer training data achieves much lower detection precisions. In this paper, we propose a novel class-wise style augmentation (CWSA) algorithm to generate a class-balanced underwater dataset Balance18 from the public contest underwater dataset URPC2018. CWSA is a new kind of data augmentation technique which augments the training data for the minority classes by generating various colors, textures and contrasts for the minority classes. Compare with previous data augmentation algorithms such flipping, cropping and rotations, CWSA is able to generate a class balanced underwater dataset with diverse color distortions and haze-effects.

Kai Zhang, Martin Danelljan, Yawei Li et al.

This paper reviews the AIM 2020 challenge on efficient single image super-resolution with focus on the proposed solutions and results. The challenge task was to super-resolve an input image with a magnification factor x4 based on a set of prior examples of low and corresponding high resolution images. The goal is to devise a network that reduces one or several aspects such as runtime, parameter count, FLOPs, activations, and memory consumption while at least maintaining PSNR of MSRResNet. The track had 150 registered participants, and 25 teams submitted the final results. They gauge the state-of-the-art in efficient single image super-resolution.

Teng Liu, Xing Yang, Hong Wang et al.

Digital quadruplets aiming to improve road safety, traffic efficiency, and driving cooperation for future connected automated vehicles are proposed with the enlightenment of ACP based parallel driving. The ACP method denotes Artificial societies, Computational experiments, and Parallel execution modules for cyber-physical-social systems. Four agents are designed in the framework of digital quadruplets: descriptive vehicles, predictive vehicles, prescriptive vehicles, and real vehicles. The three virtual vehicles (descriptive, predictive, and prescriptive) dynamically interact with the real one in order to enhance the safety and performance of the real vehicle. The details of the three virtual vehicles in the digital quadruplets are described. Then, the interactions between the virtual and real vehicles are presented. The experimental results of the digital quadruplets demonstrate the effectiveness of the proposed framework.

John Houston, Guido Zuidhof, Luca Bergamini et al.

Motivated by the impact of large-scale datasets on ML systems we present the largest self-driving dataset for motion prediction to date, containing over 1,000 hours of data. This was collected by a fleet of 20 autonomous vehicles along a fixed route in Palo Alto, California, over a four-month period. It consists of 170,000 scenes, where each scene is 25 seconds long and captures the perception output of the self-driving system, which encodes the precise positions and motions of nearby vehicles, cyclists, and pedestrians over time. On top of this, the dataset contains a high-definition semantic map with 15,242 labelled elements and a high-definition aerial view over the area. We show that using a dataset of this size dramatically improves performance for key self-driving problems. Combined with the provided software kit, this collection forms the largest and most detailed dataset to date for the development of self-driving machine learning tasks, such as motion forecasting, motion planning and simulation. The full dataset is available at http://level5.lyft.com/.

Long Chen, Dorit Merhof

Automated brain structure segmentation is important to many clinical quantitative analysis and diagnoses. In this work, we introduce MixNet, a 2D semantic-wise deep convolutional neural network to segment brain structure in multi-modality MRI images. The network is composed of our modified deep residual learning units. In the unit, we replace the traditional convolution layer with the dilated convolutional layer, which avoids the use of pooling layers and deconvolutional layers, reducing the number of network parameters. Final predictions are made by aggregating information from multiple scales and modalities. A pyramid pooling module is used to capture spatial information of the anatomical structures at the output end. In addition, we test three architectures (MixNetv1, MixNetv2 and MixNetv3) which fuse the modalities differently to see the effect on the results. Our network achieves the state-of-the-art performance. MixNetv2 was submitted to the MRBrainS challenge at MICCAI 2018 and won the 3rd place in the 3-label task. On the MRBrainS2018 dataset, which includes subjects with a variety of pathologies, the overall DSC (Dice Coefficient) of 84.7% (gray matter), 87.3% (white matter) and 83.4% (cerebrospinal fluid) were obtained with only 7 subjects as training data.

Shuhao Cao, Long Chen, Frank Lin

This short note reports a new derivation of the optimal order of the a priori error estimates for conforming virtual element methods (VEM) on 3D polyhedral meshes based on an error equation. The geometric assumptions, which are necessary for the optimal order of the conforming VEM error estimate in the $H^1$-seminorm, are relaxed for that in a bilinear form-induced energy norm.

Long Chen, Yongke Wu

Finite element exterior calculus (FEEC) has been developed as a systematical framework for constructing and analyzing stable and accurate numerical method for partial differential equations by employing differential complexes. This paper is devoted to analyze convergence of adaptive mixed finite element methods for Hodge Laplacian equations based on FEEC without considering harmonic forms. More precisely, a residual type posteriori error estimates is obtained by using the Hodge decomposition, the regular decomposition and bounded commuting quasi-interpolants. An additional marking strategy is added to ensure the quasi-orthogonality. Using this quasi-orthogonality, the uniform convergence of adaptive mixed finite element methods is obtained without assuming the initial mesh size is small enough.

Long Chen, Michael Holst, Jinchao Xu

A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson-Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson-Boltzmann equation is introduced as an auxiliary problem, making it possible to study the original nonlinear equation with delta distribution sources. A priori error estimates for the finite element approximation are obtained for the regularized Poisson-Boltzmann equation based on certain quasi-uniform grids in two and three dimensions. Adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge. The Poisson-Boltzmann equation does not appear to have been previously studied in detail theoretically, and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the Poisson-Boltzmann equation. Note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear Poisson-Boltzmann equation with delta distribution sources, and it also introduces the first provably convergent adaptive method for the equation. This last result is currently one of only a handful of existing convergence results of this type for nonlinear problems.

Long Chen, Michael Holst, Jinchao Xu

The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasi-orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace, while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result. This discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation.