Search

Parabolic initial–boundary value problems with nonsmooth data show either rapid transitions or reduced smoothness in its solution. For those problems, specific numerical methods are required to avoid spurious oscillations as well as unrealistic smoothing of steep changes in the numerical solution. This article investigates characteristics of the θ -method and introduces a variable- θ method as a synergistic combination of the Crank–Nicolson (CN) method ( θ = 1 ∕ 2 ) and the implicit method ( θ = 1 ). It suppresses spurious oscillations, by evolving the solution implicitly at points where the solution shows a certain portent of oscillations or reduced smoothness, and maintains as a similar accuracy as the CN method with smooth data. An effective strategy is suggested for the detection of points where the solution may introduce spurious oscillations (the wobble set); the resulting variable- θ method is analyzed for its accuracy and stability. Various numerical examples are given to verify its effectiveness.

The main objective of this research is to present a robust numerical framework based upon Isogeometric analysis (IGA) for simulation of thermo-hydro-mechanical (THM) processes in variably saturated soils. The proposed platform employs the Bézier extraction operator to connect IGA to the conventional finite element analysis (FEA), allowing to take advantage of features offered by the two methods. In the first part, the formulation and numerical implementation for fully coupled numerical simulation of THM problems in saturated porous media are presented. The results are compared against analytical solutions and experimental tests available in the literature. In the second part, the proposed method is used to study the temperature effect on the hydro-mechanical response of seabed supporting hydrocarbon pipelines, an aspect that has been overlooked in the majority of previous studies. The results highlight the need for considering nonisothermal behavior in different analysis and design stages of seabed-buried pipelines. In the third part, the proposed IGA-FEA framework is extended to evaluate the nonisothermal elasto-plastic behavior of unsaturated soils. Drucker-Prager yield surface is used as criterion to limit the modified effective stress where the model follows small strain, quasi-static loading conditions. The framework is used to simulate strain localization of unsaturated dense sand subjected to undrained compression loading. In comparison with FEA, the present method smoothly distributes plastic strain over the adjacent elements. The parametric study highlights the importance of considering temperature effects in elasto-plastic analysis of unsaturated soils.

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.

The article is concerned with a mathematical modeling which can improve performances of PDE-based restoration models. Most PDE-based restoration models tend to lose fine structures due to certain degrees of nonphysical dissipation. Sources of such an undesirable dissipation are analyzed for total variation-based restoration models. Based on the analysis, the so-called equalized net diffusion (END) modeling is suggested in order for PDE-based restoration models to significantly reduce nonphysical dissipation. It has been numerically verified that the END-incorporated models can preserve and recover fine structures satisfactorily, outperforming the basic models for both quality and efficiency. Various numerical examples are shown to demonstrate effectiveness of the END modeling.

This article is concerned with effective numerical techniques for partial differential equation (PDE)-based image restoration. Numerical realizations of most PDE-based denoising models show a common drawback: loss of fine structures. In order to overcome the drawback, the article introduces a new time-stepping procedure, called the method of nonflat time evolution (MONTE), in which the timestep size is determined based on local image characteristics such as the curvature or the diffusion magnitude. The MONTE provides PDE-based restoration models with an effective mechanism for the equalization of the net diffusion over a wide range of image frequency components. It can be easily applied to diverse evolutionary PDE-based restoration models and their spatial and temporal discretizations. It has been numerically verified that the MONTE results in a significant reduction in numerical dissipation and preserves fine structures such as edges and textures satisfactorily, while it removes the noise with an improved efficiency. Various numerical results are shown to confirm the claim.

Image denoising processes often lead to significant loss of fine structures such as edges and textures. This paper studies various innovative mathematical and numerical methods applicable for conventional PDE-based denoising models. The method of diffusion modulation is considered to effectively minimize regions of undesired excessive dissipation. Then we introduce a novel numerical technique for residual-driven constraint parameterization, in order for the resulting algorithm to produce clear images whose corresponding residual is as free of image textures as possible. A linearized Crank-Nicolson alternating direction implicit time-stepping procedure is adopted to simulate the resulting model efficiently. Various examples are presented to show efficiency and reliability of the suggested methods in image denoising.

We introduce a novel image zooming algorithm, called the curvature interpolation method (CIM), which is partial-differential-equation (PDE)-based and easy to implement. In order to minimize artifacts arising in image interpolation such as image blur and the checkerboard effect, the CIM first evaluates the curvature of the low-resolution image. After interpolating the curvature to the high-resolution image domain, the CIM constructs the high-resolution image by solving a linearized curvature equation, incorporating the interpolated curvature as an explicit driving force. It has been numerically verified that the new zooming method can produce clear images of sharp edges which are already denoised and superior to those obtained from linear methods and PDE-based methods of no curvature information. Various results are given to prove effectiveness and reliability of the new method.

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%.

This paper introduces edge-forming schemes for image zooming of color images by general magnification factors. In order to remove/reduce artifacts arising in image interpolation, such as image blur and the checkerboard effect, an edge-forming method is suggested to be applied as a postprocess of standard interpolation methods. The method is based on nonconvex nonlinear partial differential equations. The equations are carefully discretized, incorporating numerical schemes of anisotropic diffusion, to be able to form reliable edges satisfactorily. The alternating direction implicit (ADI) method is employed for an efficient simulation of the model. It has been numerically verified that the resulting algorithm can form clear edges in 2 to 3 ADI iterations. Various results are given to show th eeffectiveness and reliability of the algorithm.

The paper is concerned with PDE-based image restoration. A new model is introduced by hybridizing a nonconvex variant of the total variation minimization (TVM) and the motion by mean curvature (MMC) in order to deal with the mixture of the impulse and Gaussian noises reliably. We suggest the essentially nondissipative (ENoD) difference schemes for the MMC component to eliminate the impulse noise with a minimum (ideally no) introduction of dissipation. The MMC-TVM hybrid model and the ENoD schemes are applied for both gray-scale and color images. For color image denoising, we consider the chromaticity-brightness decomposition with the chromaticity formulated in the angle domain. An incomplete Crank-Nicolson alternating direction implicit time-stepping procedure is adopted to solve those differential equations efficiently. Numerical experiments have shown that the new hybrid model and the numerical schemes can remove the mixture of the impulse and Gaussian noises, efficiently and reliably, preserving edges quite satisfactorily.

It is often noted that the Helmholtz equation is extremely difficult to solve, in particular, for high-frequency solutions for heterogeneous media. Since stability for second-order discretization methods requires one to choose at least 10–12 grid points per wavelength, the discrete problem on the possible coarsest mesh is huge. In a realistic simulation, one is required to choose 20–30 points per wavelength to achieve a reasonable accuracy; this problem is hard to solve. This article is concerned with the high-frequency asymptotic decomposition of the wavefield for an efficient and accurate simulation for the high-frequency numerical solution of the Helmholtz equation. It has been numerically verified that the new method is accurate enough even when one chooses 4–5 grid points per wavelength.

In this dissertation, a new numerical method for solving fractional calculus problems is presented. The method is based upon the fractional Taylor vector approximations. The operational matrix of the fractional integration for the fractional Taylor vector is introduced. This matrix is then utilized to reduce the solution of the fractional calculus problems to the solution of a system of algebraic equations. This method is used to solve fractional differential equations, Bagley-Torvik equations, fractional integro-differential equations, and fractional duffing problems. Illustrative examples are included to demonstrate the validity and applicability of this technique.

This article is concerned with the fourth-order compact scheme for an accurate simulation of acoustic waves in the frequency domain. The compact schemes have been known for a long time; however, they have exhibited difficulties, in particular, in dealing with general boundary conditions. This article introduces a new formulation for the compact scheme for the Helmholtz equation and an effective strategy of incorporating absorbing boundary conditions. It has been numerically verified that the resulting compact scheme is fourth order in general heterogeneous media and improves the accuracy of the numerical solution dramatically, by more than two digits, over the standard second-order scheme.

A frequency-stepping algorithm for solving multifrequency (acoustic) wave propagation is considered. A two-grid method is employed for the problems of single frequency. For high frequency applications, the coarse grid problem is still huge, since one has to choose at least six to eight grid points per wavelength for a stability reason. The coarse grid problem is solved by a nonoverlapping domain decomposition (DD) method. The solution of the former frequency problem is used as the initial guess for the solution of the next larger frequency problem. Such an algorithm turns out to be efficient for multifrequency, as well as single-frequency problems, as shown in numerical results. Also, it is easily parallelizable with a high efficiency.

This article is concerned with new strategies with which explicit time-stepping procedures of PDE-based restoration models converge with a similar efficiency to implicit algorithms. Conventional explicit algorithms often require hundreds of iterations to converge. In order to overcome the difficulty and to further improve image quality, the article introduces new spatially variable constraint term and timestep size, as a method of nonflat time evolution (MONTE). It has been verified that the explicit time-stepping scheme incorporating MONTE converges in only 4-15 iterations for all restoration examples we have tested. It has proved more effective than the additive operator splitting (AOS) method in both computation time and image quality (measured in PSNR), for most cases. Since the explicit MONTE procedure is efficient in computer memory, requiring only twice the image size, it can be applied particularly for huge data sets with a great efficiency in computer memory as well.

In tomographic image processing of seismic data, the first-arrival traveltime (FATT) is often different from those of more energetic wavefronts in realistic media. Since the traveltime of most-energetic wavefront (METT) dominates the data, computing the METT is recognized as an essential element in modern seismic imaging techniques. Solving the full wave equation is extremely expensive to be impractical even for large-size computers to carry out; the solution of the eikonal equation for which the corresponding amplitude is continuous is conjectured to be the METT.

When the solution and problem coefficients are highly oscillatory, the computed solution may not show characteristics of the original physical problem unless the numerical mesh is sufficiently fine. In the case, the coarse grid problem of a multigrid (MG) algorithm must be still huge and poorly-conditioned, and therefore, it is hard to solve by either a direct method or an iterative scheme. This article suggests a MG algorithm for such problems in which the coarse grid problem is slightly modified by an artificial damping (compressibility) term. It has been numerically observed that the artificial damping, even if slight, makes the coarse grid problem much easier to solve, without deteriorating the overall convergence rate of the MG method. For most problems, 2–6 times speed up have been observed.

An artificial damping algorithm for solving the Helmholtz problem is considered. When the imaginary part of the wave number is small, the problem is known to be difficult to solve. In this paper, we propose an efficient artificial damping algorithm which can be viewed as a rational iteration. Each damped problem is solved incompletely by a nonoverlapping domain decomposition method. Convergence arguments are presented. Numerical results run on an nCUBE2 are presented to demonstrate the effectiveness of the method.