We transpose an optimal control technique to the image segmentation problem. The idea is to consider image segmentation as a parameter estimation problem. The parameter to estimate is the color of the pixels of the image. We use the adaptive parameterization technique which builds iteratively an optimal representation of the parameter into uniform regions that form a partition of the domain, hence corresponding to a segmentation of the image. We minimize an error function during the iterations, and the partition of the image into regions is optimally driven by the gradient of this error. The resulting segmentation algorithm inherits desirable properties from its optimal control origin: soundness, robustness, and flexibility.
The lack of smoothness is a common feature of weak solutions of nonlinear hyperbolic equations and is a crucial issue in their approximation. This has motivated several efforts to define appropriate indicators, based on the values of the approximate solutions, in order to detect the most troublesome regions of the domain. This information helps to adapt the approximation scheme in order to avoid spurious oscillations when using high-order schemes. In this paper we propose a genuinely multidimensional extension of the WENO procedure in order to overcome the limitations of indicators based on di…
Monash University Malaysia（莫纳什大学马来西亚分校）；Sunway Medical Centre（双威医疗中心）；The University of Oklahoma（俄克拉荷马大学）
Convolutional neural networks (CNNs) have been used quite successfully for semantic segmentation of brain tumors. However, current CNNs and attention mechanisms are stochastic in nature and neglect the morphological indicators used by radiologists to manually annotate regions of interest. In this paper, we introduce a channel and spatial wise asymmetric attention (CASPIAN) by leveraging the inherent structure of tumors to detect regions of saliency. To demonstrate the efficacy of our proposed layer, we integrate this into a well-established convolutional neural network (CNN) architecture to ac…
Goal-Oriented Adaptivity and Multilevel Preconditioning for the Poisson-Boltzmann Equation
TOBB University of Economics and Technology（TOBB经济与技术大学）；Louisiana State University（路易斯安那州立大学）；Sandia National Laboratories（桑迪亚国家实验室）；University of California San Diego（加州大学圣地亚哥分校）
Error indicators and adaptivity for a least-squares method applied to the Monge-Ampère equation
Geneva School of Business Administration (HEG-GENÈVE), University of Applied Sciences and Arts Western Switzerland (HES-SO)（日内瓦商业管理学院（HEG-GENÈVE），瑞士西部应用科学艺术大学（HES-SO））；Institute of Mathematics, Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院数学研究所）
In this paper we introduce an algorithm based on a sparse grid adaptive refinement, for the approximation of the eigensolutions to parametric problems arising from elliptic partial differential equations. In particular, we are interested in detecting the crossing of the hypersurfaces describing the eigenvalues as a function of the parameters. The a priori matching is followed by an a posteriori verification, driven by a suitably defined error indicator. At a given refinement level, a sparse grid approach is adopted for the construction of the grid of the next level, by using the marking given …
In this paper, we design the first residual type a posteriori error estimator for mixed interior penalty discontinuous Galerkin method for the H(curl)-elliptic problems. Then we prove that our residual based a posteriori error indicator is both reliable and efficient. At last, we present some numerical experiments to validate the performance of the indicator within an adaptive mesh refinement procedure.
In this article, a new unified duality theory is developed for Petrov-Galerkin finite element methods. This novel theory is then used to motivate goal-oriented adaptive mesh refinement strategies for use with discontinuous Petrov-Galerkin (DPG) methods. The focus of this article is mainly on broken ultraweak variational formulations of stationary boundary value problems, however, many of the ideas presented within are general enough that they be extended to any such well-posed variational formulation. The proposed goal-oriented adaptive mesh refinement procedures require the construction of re…
In the flow and transport numerical simulation, mesh adaptivity strategy is important in reducing the usage of CPU time and memory. The refinement based on the pressure error estimator is commonly-used approach without considering the flux error which plays important role in coupling flow and transport systems. We derive a posteriori error estimators for Enhanced Velocity Mixed Finite Element Method (EVMFEM) in the incompressible Darcy flow. We show numerically difference of the explicit residual based error estimator and implicit error estimators, where Arbogast and Chen post-processing proce…
Mathematical modeling and numerical multigoal-oriented a posteriori error control and adaptivity for a stationary, nonlinear, coupled flow temperature model with temperature dependent density
In this paper we propose an adaptive scheme for the solution of time-dependent boundary value problems (BVPs). To solve numerically these problems, we consider the kernel-based method of lines that allows us to split the spatial and time derivatives, dealing with each separately. This adaptive algorithm is based on a leave-one-out cross validation (LOOCV) technique, which is employed as an error indicator. By this scheme, we can first detect the domain areas where the error is estimated to be too large -- generally due to steep variations or quick changes in the solution -- and then accordingl…
Adaptive Radial Basis Function-generated Finite Differences method for contact problems
Parallel and Distributed Systems Laboratory, “Jožef Stefan” Institute（约泽夫·斯特凡研究所并行与分布式系统实验室）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）
Lattice-Boltzmann methods are known for their simplicity, efficiency and ease of parallelization, usually relying on uniform Cartesian meshes with a strong bond between spatial and temporal discretization. This fact complicates the crucial issue of reducing the computational cost and the memory impact by automatically coarsening the grid where a fine mesh is unnecessary, still ensuring the overall quality of the numerical solution through error control. This work provides a possible answer to this interesting question, by connecting, for the first time, the field of lattice-Boltzmann Methods (…
The paper is concerned with space-time IgA approximations of parabolic initial-boundary value problems. We deduce guaranteed and fully computable error bounds adapted to special features of IgA approximations and investigate their applicability. The derivation method is based on the analysis of respective integral identities and purely functional arguments. Therefore, the estimates do not contain mesh-dependent constants and are valid for any approximation from the admissible (energy) class. In particular, they provide computable error bounds for norms associated with stabilised space-time IgA…
We propose an adaptive mesh refinement strategy for immersed isogeometric analysis, with application to steady heat conduction and viscous flow problems. The proposed strategy is based on residual-based error estimation, which has been tailored to the immersed setting by the incorporation of appropriately scaled stabilization and boundary terms. Element-wise error indicators are elaborated for the Laplace and Stokes problems, and a THB-spline-based local mesh refinement strategy is proposed. The error estimation .and adaptivity procedure is applied to a series of benchmark problems, demonstrat…
