Gangjoon Yoon

Analyses on the finite difference method by Gibou et al. for Poisson equation

Gibou et al. in 4 introduced a finite difference method for solving the Poisson equation in irregular domains with the Dirichlet boundary condition. Contrary to its great importance, its properties have not been mathematically analyzed, but have just been numerically observed. In this article, we present two analyses for the method. One proves that its solution is second order accurate, and the other estimates the condition number of its linear system. According to our estimation, the condition number of the unpreconditioned linear system is of size O ( 1 / ( h ? h min ) ) , and each of Jacobi, SGS, and ILU preconditioned systems is of size O ( h - 2 ) . Furthermore, our analysis shows that the condition number of MILU is of size O ( h - 1 ) , the most successful one.

Convergence Analysis of the Standard Central Finite Difference Method for Poisson Equation

We consider the standard central finite difference method for solving the Poisson equation with the Dirichlet boundary condition. This scheme is well known to produce second order accurate solutions. From numerous tests, its numerical gradient was reported to be also second order accurate, but the observation has not been proved yet except for few specific domains. In this work, we first introduce a refined error estimate near the boundary and a discrete version of the divergence theorem. Applying the divergence theorem with the estimate, we prove the second order accuracy of the numerical gradient in arbitrary smooth domains.

User-drawn sketch-based 3D object retrievalusing sparse coding

Abstract3D object retrieval from user-drawn (sketch) queries is one of the important research issues in the areas of pattern recognition and computer graphics for simulation, visualization, and Computer Aided Design. The performance of any content-based 3D object retrieval system crucially depends on the availability of effective descriptors and similarity measures for this kind of data. We present a sketch-based approach for improving 3D object retrieval effectiveness by optimizing the representation of one particular type of features (oriented gradients) using a sparse coding approach. We perform experiments, the results of which show that the retrieval quality improves over alternative features and codings. Based our findings, the coding can be proposed for sketch-based 3D object retrieval systems relying on oriented gradient features.

Structure preserving dimensionality reduction for visual object recognition

Robust object recognition has drawn increasing attention in the field of computer vision and machine learning with fast development in feature extraction and classification techniques, and release of public datasets, such as Caltech datasets, Pascal Visual Object Classes, and ImageNet. Recently, deep learning based object recognition systems have shown significant performance improvements in visual object recognition tasks using innovative learning methodology. However, high dimensional space searching and recognition is time consuming, so performing point and range queries in high dimension is reconsidered for object recognition. This paper proposes optimized dimensionality reduction using structured sparse principle component analysis. The proposed method retains high dimensional feature structures, removes redundant features that do not contribute to similarity, and classifies the query image in a large database. The qualitative and quantitative experimental results, including a comparison with the current state-of-the-art visual object recognition algorithms, verify that the proposed recognition algorithm performs favorably in reducing the query image dimension and number of training images.

A Stable and Convergent Hodge Decomposition Method for Fluid–Solid Interaction

Fluid–solid interaction has been a challenging subject due to their strong nonlinearity and multidisciplinary nature. Many of the numerical methods for solving FSI problems have struggled with non-convergence and numerical instability. In spite of comprehensive studies, it has still been a challenge to develop a method that guarantees both convergence and stability. Our discussion in this work is restricted to the interaction of viscous incompressible fluid flow and a rigid body. We take the monolithic approach by Gibou and Min (J Comput Phys 231:3245–3263, 2012) that results in an augmented Hodge projection. The projection updates not only the fluid vector field but also the solid velocities. We derive the equivalence between the augmented Hodge projection and the Poisson equation with non-local Robin boundary condition. We prove the existence, uniqueness, and regularity for the weak solution of the Poisson equation, through which the Hodge projection is shown to be unique and orthogonal. We also show the stability of the projection in the sense that the projection does not increase the total kinetic energy of the fluid or the solid. Finally, we discuss a numerical method as a discrete analogue to the Hodge projection, then we show that the unique decomposition and orthogonality also hold in the discrete setting. As one of our main results, we prove that the numerical solution is convergent with at least first-order accuracy. We carry out numerical experiments in two and three dimensions, which validate our analysis and arguments.

Comparison of eigenvalue ratios in artificial boundary perturbation and Jacobi preconditioning for solving Poisson equation

Abstract The Shortley–Weller method is a standard finite difference method for solving the Poisson equation with Dirichlet boundary condition. Unless the domain is rectangular, the method meets an inevitable problem that some of the neighboring nodes may be outside the domain. In this case, an usual treatment is to extrapolate the function values at outside nodes by quadratic polynomial. The extrapolation may become unstable in the sense that some of the extrapolation coefficients increase rapidly when the grid nodes are getting closer to the boundary. A practical remedy, which we call artificial perturbation, is to treat grid nodes very near the boundary as boundary points. The aim of this paper is to reveal the adverse effects of the artificial perturbation on solving the linear system and the convergence of the solution. We show that the matrix is nearly symmetric so that the ratio of its minimum and maximum eigenvalues is an important factor in solving the linear system. Our analysis shows that the artificial perturbation results in a small enhancement of the eigenvalue ratio from O ( 1 / ( h ⋅ h m i n ) to O ( h − 3 ) and triggers an oscillatory order of convergence. Instead, we suggest using Jacobi or ILU-type preconditioner on the matrix without applying the artificial perturbation. According to our analysis, the preconditioning not only reduces the eigenvalue ratio from O ( 1 / ( h ⋅ h m i n ) to O ( h − 2 ) , but also keeps the sharp second order convergence.

On Solving the Singular System Arisen from Poisson Equation with Neumann Boundary Condition

We consider solving the singular linear system arisen from the Poisson equation with the Neumann boundary condition. To handle the singularity, there are two usual approaches: one is to fix a Dirichlet boundary condition at one point, and the other seeks a unique solution in the orthogonal complement of the kernel. One may incorrectly presume that the two solutions are the similar to each other. In this work, however, we show that their solutions differ by a function that has a pole at the Dirichlet boundary condition. The pole of the function is comparable to that of the fundamental solution of the Laplace operator. Inevitably one of them should contain the pole, and accordingly has inferior accuracy than the other. According to our novel analysis in this work, it is the fixing method that contains the pole. The projection method is thus preferred to the fixing method, but it also contains cons: in finding a unique solution by conjugate gradient method, it requires extra steps per each iteration. In this work, we introduce an improved method that contains the accuracy of the projection method without the extra steps. We carry out numerical experiments that validate our analysis and arguments.

User-drawn sketch-based 3D object retrieval using distributed compressive sensing

User-drawn sketch-based 3D object retrieval system is popularly used for the basis of Computer Aided Design, scientific simulation and visualization, and serious game. The sketch-based 3D object retrieval is dependent on the descriptors and similarity measure to find and order the 3D objects what we want to see. In this paper we present a hybrid shape descriptor that integrates the local and global features to effectively retrieve the 3D objects from 2D sketch query image. Integrated shape descriptor with the high-dimensional sparse feature vector is then optimized using distributed compressive sensing to improve the performance of the view-based 3D object retrieval. The experimental results demonstrate the effectiveness and advantages of our proposed framework.

Simultaneous detection of pedestrians, pose, and the camera viewpoint from 3D models

This paper describes a pedestrian detection trained from the projected suggestive contours of the 3D models and an estimation of its 3D pose instead of using multiple 2D training images. The first part explains the 3D mesh model training for pedestrian detection; the suggestive contours projected to various viewpoints enables to avoid hand-crafted training of 2D images. The second part depicts extracting the features and measuring the similarity in the space of diffusion tensor fields. By measuring the similarity and ordering the trained 3D models, the 3D camera viewpoint and pose of the detected pedestrians can also be estimated. Experiments show the effectiveness of our method.