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After a theory of morphogenesis in chemical cells was introduced in the 1950s, much attention had been devoted to the numerical solution of reaction-diffusion (RD) partial differential equations (PDEs). The Crank–Nicolson (CN) method has been a common second-order time-stepping procedure. However, the CN method may introduce spurious oscillations for nonsmooth data unless the time step size is sufficiently small. This article studies a nonoscillatory second-order time-stepping procedure for RD equations, called a variable-θmethod, as a perturbation of the CN method. In each time level, the new method detects points of potential oscillations to implicitly resolve the solution there. The proposed time-stepping procedure is nonoscillatory and of a second-order temporal accuracy. Various examples are given to show effectiveness of the method. The article also performs a sensitivity analysis for the numerical solution of biological pattern forming models to conclude that the numerical solution is much more sensitive to the spatial mesh resolution than the temporal one. As the spatial resolution becomes higher for an improved accuracy, the CN method may produce spurious oscillations, while the proposed method results in stable solutions.

Abstract Elliptic obstacle problems are formulated to find either superharmonic solutions or minimal surfaces that lie on or over the obstacles, by incorporating inequality constraints. In order to solve such problems effectively using finite difference (FD) methods, the article investigates simple iterative algorithms based on the successive over-relaxation (SOR) method. It introduces subgrid FD methods to reduce the accuracy deterioration occurring near the free boundary when the mesh grid does not match with the free boundary. For nonlinear obstacle problems, a method of gradient-weighting is introduced to solve the problem more conveniently and efficiently. The iterative algorithm is analyzed for convergence for both linear and nonlinear obstacle problems. An effective strategy is also suggested to find the optimal relaxation parameter. It has been numerically verified that the resulting obstacle SOR iteration with the optimal parameter converges about one order faster than state-of-the-art methods and the subgrid FD methods reduce numerical errors by one order of magnitude, for most cases. Various numerical examples are given to verify the claim.

Digital elevation model (DEM) is a three-dimensional (3D) representation of a terrain's surface - for a planet (including Earth), moon, or asteroid - created from point cloud data which measure terrain elevation. Its modeling requires surface reconstruction for the scattered data, which is an ill-posed problem and most computational algorithms become overly expensive as the number of sample points increases. This article studies an effective partial differential equation (PDE)-based algorithm, called the curvature interpolation method (CIM). The new method iteratively utilizes curvature information, estimated from an intermediate surface, to construct a reliable image surface that contains all of the data points. The CIM is applied for DEM for point cloud data acquired by light detection and ranging (LiDAR) technology. It converges to a piecewise smooth image, requiring O(N) operations independently of the number of sample points, where $N$ is the number of grid points.

This article concerns mathematical image denoising methods incorporating fourth-order partial differential equations (PDEs). We introduce and analyze piecewise planarity conditions (PPCs) with which unconstrained fourth-order variational models in continuum converge to a piecewise planar image. It has been observed that fourth-order variational models holding PPCs can restore better images than models without PPCs and second-order models. Numerical schemes are presented in detail and various examples in image denoising are provided to verify the claim.

Mathematical restoration models, in particular, total variation-based models can easily lose fine structures during image denoising. In order to overcome the drawback, this article introduces two strategies: the non-convex (NC) diffusion and the texture-free residual (TFR) parameterization. A non-standard numerical procedure is suggested and its stability is analyzed to effectively solve the NC diffusion model which is mathematically unstable. It has been numerically verified that the resulting algorithm incorporating the NC diffusion and TFR parameterization is able to not only reduce the noise satisfactorily but also enhance edges effectively, at the same time.

This article presents results of a parameter study for a new denoising model, using parallel computing and advanced dynamic load balancing techniques for performance improvement of implementations. A denoising model is suggested hybridizing total variation and Laplacian mean-curvature; the fourth-order model and its numerical procedure introduce a number of parameters. As a preliminary step in the model development, a parameter study has been undertaken in order to discover solitary and interactive effects of the parameters on model accuracy. Such a parameter study is necessarily time-consuming due to the huge number of combinations of the parameter values to be tested. In addition, the study has to be performed on various images, thereby increasing the overall investigation time. The performance of this first parallel implementation of a new hybrid model for image denoising is evaluated when the application is running on heterogeneous environments. The hybrid model is simulated on a general-purpose Linux cluster for which the parallel efficiency exceeds 96%. [ABSTRACT FROM AUTHOR]

The powder injection molding (PIM) process is complicated since it is a hybrid technology embracing plastic injection molding, powder metallurgy (PM}, and ceramic processing, In developing this technology, engineers have relied heavily on trial-and-error experiments. As a fundamental understanding of the inherent physical phenomena emerges, process modeling has progressed by utilizing new and customized simulation tools. These computational tools allow for parallel design of the component and process via analysis, prediction, and optimization modules. This contribution reviews computer simulation efforts and the future of this methodology. [ABSTRACT FROM AUTHOR]