Boundary Problem Research Articles

NeighborNet: Learning Intra- and Inter-Image Pixel Neighbor Representation for Breast Lesion Segmentation.

Breast lesion segmentation from ultrasound images is essential in computer-aided breast cancer diagnosis. To alleviate the problems of blurry lesion boundaries and irregular morphologies, common practices combine CNN and attention to integrate global and local information. However, previous methods use two independent modules to extract global and local features separately, such feature-wise inflexible integration ignores the semantic gap between them, resulting in representation redundancy/insufficiency and undesirable restrictions in clinic practices. Moreover, medical images are highly similar to each other due to the imaging methods and human tissues, but the captured global information by transformer-based methods in the medical domain is limited within images, the semantic relations and common knowledge across images are largely ignored. To alleviate the above problems, in the neighbor view, this paper develops a pixel neighbor representation learning method (NeighborNet) to flexibly integrate global and local context within and across images for lesion morphology and boundary modeling. Concretely, we design two neighbor layers to investigate two properties (i.e., number and distribution) of neighbors. The neighbor number for each pixel is not fixed but determined by itself. The neighbor distribution is extended from one image to all images in the datasets. With the two properties, for each pixel at each feature level, the proposed NeighborNet can evolve into the transformer or degenerate into the CNN for adaptive context representation learning to cope with the irregular lesion morphologies and blurry boundaries. The state-of-the-art performances on three ultrasound datasets prove the effectiveness of the proposed NeighborNet. The code is available at: https://github.com/fjcaoww/NeighborNet.

Regularity of flat free boundaries for two-phase p(x)-Laplacian problems with right hand side

We consider viscosity solutions to two-phase free boundary problems for the p(x)-Laplacian with non-zero right hand side. We prove that flat free boundaries are C1,γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\gamma }$$\\end{document}. No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the p(x)-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when p(x)≡p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(x)\\equiv p$$\\end{document}, i.e., for the p-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability.

A Generalized 3D-DDM for both Fracture-Type and Asymmetric Closed-Contour Problems: Modified Fully Analytical Approach and Applications

The displacement discontinuity method (DDM) has been widely used for large-scale engineering problems, e.g., hydraulic fracturing related to metal mining and hydrocarbon extraction. However, there is a lack of detailed explanation for the implementation of 3D-DDM in closed-contour boundary problem, and applications to address asymmetric closed-contour boundary problem remains limited. In this paper, we have theoretically derived and elaborated the numerical procedure, identified the caveats of DDM in solving fracture-type and closed-contour boundary problems. Subsequently, we modified the analytical integral and differential expressions for the basic solution on 3D-DDM based on triangular elements and provided a more general suite of expressions. Additionally, implementation details including element traversal criteria and precautions against rigid body movement are introduced to address the asymmetric closed-contour boundary problem. The accuracy of the generalized 3D-DDM were verified through simulations involving a penny-shaped fracture and a spherical cavity. Finally, the generalized 3D-DDM was employed to elucidate the stress redistribution and interference with respect to the tunnel excavation. This study demonstrates that: (1) the modified analytical integral and differential expressions combined with the element traversal criterion and precautions against rigid body movement provide a detailed explanation of how 3D-DDM can be applied to asymmetric closed-contour boundary problem; (2) during tunnel excavation, the arch flanks and floor would suffer from the biggest damage risk; and (3) the interaction effect between adjacent tunnels plays a role when the tunnel spacing is less than eight times the radius of the arch.

Space‐time stochastic Galerkin boundary elements for acoustic scattering problems

SummaryAcoustic emission or scattering problems naturally involve uncertainties about the sound sources or boundary conditions. This article initiates the study of time domain boundary elements for such stochastic boundary problems for the acoustic wave equation. We present a space‐time stochastic Galerkin boundary element method which is applied to sound‐hard, sound‐soft and absorbing scatterers. Uncertainties in both the sources and the boundary conditions are considered using a polynomial chaos expansion. The numerical experiments illustrate the performance and convergence of the proposed method in model problems and present an application to a problem from traffic noise.

On nonminimizing solutions of elliptic free boundary problems

On nonminimizing solutions of elliptic free boundary problems

New bandgap analysis method for metamaterial structures using variational principle

This paper presents an efficient bandgap analysis method for metamaterial structures based on variational principle and linear expression technique. This is in view of the challenges in constructing a displacement function that satisfies the Bloch boundary when applying the traditional energy method to analyze the dispersion relations of metamaterial structures. Also, the inclusion of the wave numbers in the displacement function leads to inefficient computation. While there have been methods, such as virtual springs, have been proposed to deal with such problems, issues remain to be resolved. The main idea of the current paper is the solution of linear uncorrelated coefficients by processing the constraints. In this way, the unknown coefficients in the equations of motion can be linearly expressed, thereby allowing the equations to become variational. Both Gaussian elimination and null-space techniques are used to implement the method. The solution procedures of the two methods are detailed and demonstrated by using one-dimensional and two-dimensional metamaterial structures. The accuracy of the method is verified by comparing the results with those from using finite element method. The calculation efficiency is compared with that of the traditional energy method in terms of both the matrix dimension and the number of the points of wave numbers. The results show that the method has a much higher calculation efficiency than that of the traditional energy method. The method proposed in this paper is free of convergence problems in dealing with boundary problems. Furthermore, in the discussion of combined periodic structures, it is shown that the method still has good geometrical applicability in structures containing multiple boundary conditions, showing great potential in solving complex engineering problems.

An alternative two-phase formulation and analytical solution for a Stefan problem in one-dimension accounting for volume change and sensible heat

Stefan problems, or free boundary problems, in the phase change process occur in many engineering problems. This paper develops a compact formula for a two-phase melting process in a one-dimensional semi-infinite slab with a novel approach. Unlike the classic Stefan problem, the novel approach considers volume change (or density difference) and the net-sensible-heat of the two phases. Furthermore, an analytical solution is proposed. To illustrate, the melting process of ice-water in a pipe is investigated. The results have been validated through an exact solution and comparisons with other relevant models. A special case of a one-phase problem is also proposed. The proposed model can be used as an alternative tool for engineers in estimating the position of the melt-front, depending on the dominant effect on the phase change process.

Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

In the modern world, where efficiency, stability, and precision play a crucial role, the development and application of optimal control strategies in oscillatory systems hold significant importance. The issues related to the numerical solution of control problems associated with damping oscillatory systems consisting of two objects are considered. To numerically solve the discussed problem, the gradient projection method, based on the formula for the first variation of the functional, and the method of successive approximations, associated with the linearity of boundary problems describing oscillatory processes, are applied. The oscillations of one object are described by a wave equation with first-order boundary conditions, while a second-order ordinary differential equation models the oscillations of the other object. Furthermore, the original and the adjoint boundary value problems are solved using direct methods at each iteration step. An algorithm for the numerical solution of the problem is proposed, and based on this algorithm, a software code for implementation is developed. The numerical results obtained in the study demonstrate that there is convergence in terms of functionality, and the approximately optimal controls found in this process are minimizing sequences in the control space. The mechanism of controlling and regulating the operation of the system according to its input constraints is provided by observed feedback, allowing systems with limited excitation to maintain stability and optimal functioning in conditions of changing external or internal circumstances. The obtained results can also be used to forecast the system's behavior in the future, resource planning, prevention of emergencies, or optimization of production processes

Partial data inverse problems for magnetic Schrödinger operators on conformally transversally anisotropic manifolds

We study inverse boundary problems for the magnetic Schrödinger operator with Hölder continuous magnetic potentials and continuous electric potentials on a conformally transversally anisotropic Riemannian manifold of dimension n ⩾ 3 with connected boundary. A global uniqueness result is established for magnetic fields and electric potentials from the partial Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective.

Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge

AbstractWe consider an autonomous, indefinite Lagrangian L admitting an infinitesimal symmetry K whose associated Noether charge is linear in each tangent space. Our focus lies in investigating solutions to the Euler-Lagrange equations having fixed energy and that connect a given point p to a flow line $$\gamma =\gamma (t)$$ γ = γ ( t ) of K that does not cross p. By utilizing the invariance of L under the flow of K, we simplify the problem into a two-point boundary problem. Consequently, we derive an equation that involves the differential of the “arrival time” t, seen as a functional on the infinite dimensional manifold of connecting paths satisfying the semi-holonomic constraint defined by the Noether charge. When L is positively homogeneous of degree 2 in the velocities, the resulting equation establishes a variational principle that extends the Fermat’s principle in a stationary spacetime. Furthermore, we also analyze the scenario where the Noether charge is affine.

A(nother) democratic case for federalism

This work offers a new democratic case for federalism, understood as a form of governance in which multiple entities in a country possess final decision-making authority (viz., can make decisions free from others substituting their decisions, issuing fines, etc.) over at least one subject (e.g., immigration, defense). It argues that leading solutions to the democratic boundary problem provide overlapping arguments for federalism. The underlying logic and many details of the most commonly cited solutions focused on those relevantly affected by and subject to decisions each support three distinct arguments (focused on voteshares, other forms of democratic influence, and persistent minorities) for multiple demoi possessing authority in a polity. Federalism is the best available method for recognizing the distinct demoi. This not only supports federal governance as opposed to unitary governance and subsidiarity. It also suggests that democracy and federalism are importantly related and have several implications for institutional design.

Modeling the barotropic tidal dynamics of the Kuril region

   A 3D boundary-value problem for computation barotropic tidal dynamics is solved at Kuril region, including southern part of the Okhotsk Sea, the Kuril Straits and its continental slope. The model is realised in statement for the contravariant components of velocity vector of the Programmed Complex “Cardinal”. Modeling the tidal dynamics of the straits of the Kuril Сhain and its continental slope is of particular importance due to the high geostrategic prestige of the region. The exceptional complexity of the relief of the area containing dozens of underwater volcanoes requires solving the problem in a complete non-hydrostatic formulation; this necessitates a multiprocessor implementation of the model with a high grid resolution for model representativity. An efficient methods for tidal dynamic reproduction at the 2D vertical cross-cuts domain is described. A statement of boundary problem at vertical cross-cuts is considered both in non-hydrostatic formulation and hydrostatic approximation. The results of computation contain the field of level tidal currents generated by the dominant wave K1 and summary wave in the synodic period; results of comparison the bottom vertical velocity at submarine mountain and on sharp depth changes in the hydrostatic and non-hydrostatic statements, result related to structure vertical velocity at cross-cut, the frequency spectrum and energy of tidal currents.

Measuring absorbed energy in the human auditory system using finite element models: A comparison with experimental results.

There are different ways to analyze energy absorbance (EA) in the human auditory system. In previous research, we developed a complete finite element model (FEM) of the human auditory system. In this current work, the external auditory canal (EAC), middle ear, and inner ear (spiral cochlea, vestibule, and semi-circular canals) were modelled based on human temporal bone histological sections. Multiple acoustic, structure, and fluid-coupled analyses were conducted using the FEM to perform harmonic analyses in the 0.1-10 kHz range. Once the FEM had been validated with published experimental data, its numerical results were used to calculate the EA or energy reflected (ER) by the tympanic membrane. This EA was also measured in clinical audiology tests which were used as a diagnostic parameter. A mathematical approach was developed to calculate the EA and ER, with numerical and experimental results showing adequate correlation up to 1 kHz. Another published FEM had adapted its boundary conditions to replicate experimental results. Here, we recalculated those numerical results by applying the natural boundary conditions of human hearing and found that the results almost totally agreed with our FEM. This boundary problem is frequent and problematic in experimental hearing test protocols: the more invasive they are, the more the results are affected. One of the main objectives of using FEMs is to explore how the experimental test conditions influence the results. Further work will still be required to uncover the relationship between middle ear structures and EA to clarify how to best use FEMs. Moreover, the FEM boundary conditions must be more representative in future work to ensure their adequate interpretation.

Single-stage oriented object detection via Corona Heatmap and Multi-stage Angle Prediction

Oriented-object detection is a pivotal research subject within computer vision. Nevertheless, extant-oriented detection methodologies frequently grapple with boundary problems when handling targets with large aspect ratios and arbitrary orientations. This study introduces a Single-stage oriented-object detection via Corona Heatmap and Multi-stage Angle Prediction (SOD) to address the above challenges. Specifically, the angle is conceptualised as a multi-stage classification prediction; the initial stage forecasts the approximate rotational direction of the target. Based on the outcome of the first stage, the subsequent stage uses Gaussian smoothing encoding to predict the precise angle, mitigating discrepancies caused by floating-point representation. Moreover, this study proposes a spatially constrained corona heatmap to rectify the issue of central position prediction drift in the Gaussian heatmap when dealing with targets with large aspect ratios. The post-processing methodology is refined based on the above approach, effectively mitigating the deviation of the predicted target’s central position of the predicted target. Visual results and experimental analyses conducted on multiple challenging datasets corroborate the efficacy and competitiveness of the proposed method.

Oscillatory properties of selfadjoint boundary problems of the fourth order

A series of results and methods is presented, which make it possible to trace the relationship between the number of inner zeros of nontrivial solutions of fourth order selfadjoint boundary problems with separated boundary conditions and the negative inertia index.

Subsonic flows with a contact discontinuity in a finitely long axisymmetric cylinder

This paper concerns the structural stability of subsonic flows with a contact discontinuity in a finitely long axisymmetric cylinder. We establish the existence and uniqueness of axisymmetric subsonic flows with a contact discontinuity by prescribing the horizontal mass flux distribution, the swirl velocity, the entropy and the Bernoulli’s quantity at the entrance and the radial velocity at the exit. It can be formulated as a free boundary problem with the contact discontinuity to be determined simultaneously with the flows. Compared with the two-dimensional case, a new difficulty arises due to the singularity near the axis. An invertible modified Lagrangian transformation is introduced to overcome this difficulty and straighten the contact discontinuity. The key elements in our analysis are to utilize the deformation-curl decomposition introduced in [S. Weng and Z. Xin, A deformation-curl decomposition for three dimensional steady Euler equations, Sci. Sin. Math. 49 (2019) 307–320 (in Chinese): doi:10.1360/N012018-00125] to effectively decouple the hyperbolic and elliptic modes in the steady axisymmetric Euler system and to use the implicit function theorem to locate the contact discontinuity.

AFDet: alignment and focusing for aerial object detection

ABSTRACT Object detection in aerial images has become a focus in recent years due to the expansion of its application fields. Aerial objects have the characteristics of multi-scale, arbitrary angle and dense arrangement, which brings considerable challenges to the task. Based on the efficient one-stage detectors, most methods design modules to adaptively align the feature receptive field with the anchors to improve the accuracy of the bounding box regression and calculate loss according to parameter bias. Current methods are not adequately data-driven in feature optimization and the angle parameter regression faces boundary problems. To handle these problems, in this letter, we propose an Align-Focus detector (AFDet) for aerial image object detection. We introduce the ideology of deformable multi-head self-attention to feature optimization and design a new Align-Focus Module (AFM) which can sensitively focus feature encoding response to the real texture area of the objects. In addition, we apply KF-IOU Loss to solve the boundary problem. We conduct necessary experiments on DOTA and HRSC2016 datasets, and AFDet achieves the highest mAP performance compared to the existing methods.

ViT-Based Image Regression Model for Shear-Strength Prediction of Transparent Soil

The direct-shear test is the primary method used to test the shear strength of transparent soil, but this experiment is complex and easily influenced by experimental conditions. In order to simplify the process of obtaining the shear strength of transparent soil, an image regression model based on a vision transformer (ViT) is proposed in this paper; this is used to recognize the shear strength of the soil based on images of transparent-soil patches. This model uses a convolutional neural network (CNN) to decompose the transparent-soil images into multiple image patches containing high-order features, utilizes a ViT for feature extraction, and designs a regression network to facilitate the transfer of information between the abstract image features and shear strength. This model solves the problem of boundary blurring and difficult-to-identify features in speckle images. To demonstrate the effectiveness of the proposed model, different parameters related to transparent soil were obtained by controlling the particle size of fused quartz sand and the content of aerosol; in addition, the friction angle and cohesive force of the transparent soil under different proportions were measured using direct-shear tests, serving as two datasets. The results show that the proposed method achieves correlations of 0.93 and 0.94 in the two prediction tasks, thus outperforming existing deep learning models.

Higher-order compact finite difference method for a free boundary problem of ductal carcinoma in situ

A higher-order compact finite difference scheme is established to investigate the model of ductal carcinoma in situ, a free boundary value problem. For the first time, to determine the growth of a tumor numerically, a compact finite difference method is applied in the spatial direction, and the Crank-Nicolson scheme is used in the temporal direction. Through stability analysis, it has been demonstrated that the proposed numerical scheme is unconditionally stable. The convergence of the numerical scheme is analyzed theoretically by von Neumann's method. It has been proved that the proposed numerical method is second-order convergent in time and fourth-order convergent in space in the L2-norm. Two numerical examples are provided to verify the scheme's efficiency and validate the theoretical results. The tabular and graphical representations confirm the high accuracy and adaptability of the scheme.

On the physical vacuum free boundary problem of the 1D shallow water equations coupled with the Poisson equation

On the physical vacuum free boundary problem of the 1D shallow water equations coupled with the Poisson equation