Alternating Direction Explicit Research Articles

A fast and efficient numerical algorithm for image segmentation and denoising

<abstract><p>Image segmentation is the process of partitioning an image into homogenous regions, and represents one of the most fundamental and important procedures in image processing. Image denoising is a process to remove unwanted noise from a digital image, enhancing its visual quality. Various algorithms, like non-local means and deep learning-based approaches, have been developed to remove noise while preserving important image details. Currently, the prevalent application of pattern recognition technology is achieved through the implementation of image segmentation algorithms. In this study, we present a new, highly efficient, and fast computational scheme specifically developed for a phase-field mathematical model of image segmentation. The numerical methodology is based on an operator splitting method (OSM). The split operators are solved by using closed-form analytic solutions and a finite difference method (FDM) with an alternating direction explicit (ADE) method. To show the notable efficiency and rapid computational performance of the proposed computational algorithm, we conduct a series of numerical experiments. Through these computational tests, we confirm a significant contribution to the advancement of methodologies employed in the critical domain of image processing.</p></abstract>

An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation

In this article, simultaneous identification of the time-dependent lowest and source terms in a two-dimensional (2D) parabolic equation from knowledge of additional measurements is studied. Existence and uniqueness of the solution is proved by means of the contraction mapping on a small time interval. Since the governing equation is yet ill-posed (very slight errors in the time-average temperature input may cause relatively significant errors in the output potential and source terms), we need to regularize the solution. Therefore, regularization is needed for the retrieval of unknown terms. The 2D problem is discretized using the alternating direction explicit (ADE) method and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine $lsqnonlin$ tool. Finally, we present a numerical example to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the ADE is an efficient and unconditionally stable scheme for reconstructing the potential and source coefficients from minimal data which makes the solution of the inverse problem (IP) unique.

An Explicit Adaptive Finite Difference Method for the Cahn-Hilliard Equation.

In this study, we propose an explicit adaptive finite difference method (FDM) for the Cahn–Hilliard (CH) equation which describes the process of phase separation. The CH equation has been successfully utilized to model and simulate diverse field applications such as complex interfacial fluid flows and materials science. To numerically solve the CH equation fast and efficiently, we use the FDM and time-adaptive narrow-band domain. For the adaptive grid, we define a narrow-band domain including the interfacial transition layer of the phase field based on an undivided finite difference and solve the numerical scheme on the narrow-band domain. The proposed numerical scheme is based on an alternating direction explicit (ADE) method. To make the scheme conservative, we apply a mass correction algorithm after each temporal iteration step. To demonstrate the superior performance of the proposed adaptive FDM for the CH equation, we present two- and three-dimensional numerical experiments and compare them with those of other previous methods.

A fast shape transformation using a phase-field model

We propose a numerical method for a fast shape transformation using a phase-field model. The governing equation is based on the modified Allen–Cahn (AC) equation. We numerically solve the equation by using the operator splitting technique. The alternating direction explicit (ADE) finite difference method is used to reduce the strict temporal step constraint when solving the diffusion term. Therefore, we can use a large temporal step size to simulate a fast shape transformation. The reaction term is solved by the separation of variables, and the fidelity term is solved using the semi-implicit scheme with a frozen coefficient. To demonstrate that the proposed method can simulate the fast shape transformation with simple or complex sources and targets, we perform several numerical experiments in the three-dimensional space. The computational experiments demonstrate that the shape transformation is fast and smooth.

Alternating Direction Explicit Method for a Nonlinear Model in Finance

In this article, at first standard linear Black-Scholes model and then some nonlinear Black-Scholes models will be considered and thereupon alternating direction explicit (ADE) method is applied firstly for solving the standard Black-Scholes model and then for Barles and Soner model which is one of the most complete and comprehensive nonlinear Black-Scholes models. Furthermore, the stability of this method has been considered and its accuracy will be compared with other numerical methods such as finite difference methods. Since in solving nonlinear Black-Scholes models by the ADE methods, we need to solve only some scalar nonlinear equations instead of a full nonlinear system of equations that we should solve in implicit methods, so this method can be a suitable choice for solving such models.

Numerical Treatment of a Two-Dimensional Vertically-Averaged Groundwater Flow Model Using Alternating Direction Explicit Methods

Leachate from a landfill can flow down and contaminate groundwater. Mathematical models are often used to describe groundwater flow, which can help designers to identify an appropriate location for a landfill area. We focused on simulation of hydraulic head of groundwater under landfill construction in a rural area. We used a partial differential equation model called two-dimensional vertically-averaged groundwater flow model to explain groundwater volume, in particular, to explain the hydraulic head of the groundwater which strongly affected the groundwater quality. We applied two alternating direction explicit methods to the groundwater flow model under general boundary conditions and rates of change of boundary conditions to obtain an approximate hydraulic head of groundwater. An alternating direction explicit method for all types of boundaries is used to find approximate solutions. The traditional alternative direction explicit method was modified to form our modified alternative direction explicit method. The results of all method are closed. The traditional alternative direction explicit method and the modified alternative direction explicit method led to more accurate solutions than forward time centered space in heterogeneous aquifer, two methods ensured that a stable solution of all interior points can be obtained by using a fine grid spacing.

An alternating direction explicit method for time evolution equations with applications to fractional differential equations

We derive and analyze the alternating direction explicit (ADE) method for time evolution equations with the time-dependent Dirichlet boundary condition and with the zero Neumann boundary condition. The original ADE method is an additive operator splitting (AOS) method, which has been developed for treating a wide range of linear and nonlinear time evolution equations with the zero Dirichlet boundary condition. For linear equations, it has been shown to achieve the second order accuracy in time yet is unconditionally stable for an arbitrary time step size. For the boundary conditions considered in this work, we carefully construct the updating formula at grid points near the boundary of the computational domain and show that these formulas maintain the desired accuracy and the property of unconditional stability. We also construct numerical methods based on the ADE scheme for two classes of fractional differential equations. We will give numerical examples to demonstrate the simplicity and the computational efficiency of the method.

High‐order ADE scheme for solving the fluid diffusion equation in non‐uniform grids and its application in coupled hydro‐mechanical simulation

SummaryTo improve the stability and efficiency of explicit technique, one proposed method is to use an unconditionally stable alternating direction explicit (ADE) scheme. However, the standard ADE scheme is only moderately accurate and restricted to uniform grids. This paper derives a novel high‐order ADE scheme capable of solving the fluid diffusion equation in non‐uniform grids. The new scheme is derived by performing a fourth‐order finite difference approximation to the spatial derivatives of the diffusion equation in non‐uniform grid. The implicit Crank‐Nicolson technique is then applied to the resulting approximation, and the subsequent equation is split into two alternating direction sweeps, giving rise to a new high‐order ADE scheme. Because the new scheme can be potentially applied in coupled hydro‐mechanical (H‐M) simulation, the pore pressure solutions from the new scheme are then sequentially coupled with an existing geomechanical simulator in the computer program Fast Lagrangian Analysis of Continua. This coupling procedure is called the sequentially explicit coupling technique based on the fourth‐order ADE scheme (SEA‐4). Verifications of well‐known consolidation problems showed that the new ADE scheme and SEA‐4 can reduce computer runtime by 46% to 75% to that of Fast Lagrangian Analysis of Continua's basic scheme. At the same time, the techniques still maintained average percentage error of 1.6% to 3.5% for pore pressure and 0.2% to 1.5% for displacement solutions and were still accurate under typical grid non‐uniformities. This result suggests that the new high‐order ADE scheme can provide an efficient explicit technique for solving the flow equation of a coupled H‐M problem, which will be beneficial for large‐scale and long‐term H‐M problems in geoengineering.

Axisymmetric alternating direction explicit scheme for efficient coupled simulation of hydro-mechanical interaction in geotechnical engineering—Application to circular footing and deep tunnel in saturated ground

Explicit solution techniques have been widely used in geotechnical engineering for simulating the coupled hydro-mechanical (H-M) interaction of fluid flow and deformation induced by structures built above and under saturated ground, i.e. circular footing and deep tunnel. However, the technique is only conditionally stable and requires small time steps, portending its inefficiency for simulating large-scale H-M problems. To improve its efficiency, the unconditionally stable alternating direction explicit (ADE) scheme could be used to solve the flow problem. The standard ADE scheme, however, is only moderately accurate and is restricted to uniform grids and plane strain flow conditions. This paper aims to remove these drawbacks by developing a novel high-order ADE scheme capable of solving flow problems in non-uniform grids and under axisymmetric conditions. The new scheme is derived by performing a fourth-order finite difference (FD) approximation to the spatial derivatives of the axisymmetric fluid–diffusion equation in a non-uniform grid configuration. The implicit Crank-Nicolson technique is then applied to the resulting approximation, and the subsequent equation is split into two alternating direction sweeps, giving rise to a new axisymmetric ADE scheme. The pore pressure solutions from the new scheme are then sequentially coupled with an existing geomechanical simulator in the computer code fast Lagrangian analysis of continua (FLAC). This coupling procedure is called the sequentially-explicit coupling technique based on the fourth-order axisymmetric ADE scheme or SEA-4-AXI. Application of SEA-4-AXI for solving axisymmetric consolidation of a circular footing and of advancing tunnel in deep saturated ground shows that SEA-4-AXI reduces computer runtime up to 42%–50% that of FLAC's basic scheme without numerical instability. In addition, it produces high numerical accuracy of the H-M solutions with average percentage difference of only 0.5%–1.8%.

A PDE Software Framework in C++11 for a Class of Path-Dependent Options

In this article we apply the Alternating Direction Explicit (ADE) finite difference method to a class of partial differential equations (PDEs) occurring in computational finance. We take the results of Wilmott, Lewis, and Duffy (2014) and we design a softward framework based on it using system decomposition methods in combination with the multiparadigm language features in C++1.

A fast Huygens sweeping method for capturing paraxial multi-color optical self-focusing in nematic liquid crystals

We propose a numerically efficient algorithm for simulating the multi-color optical self-focusing phenomena in nematic liquid crystals. The propagation of the nematicon is modeled by a parabolic wave equation coupled with a nonlinear elliptic partial differential equation governing the angle between the crystal and the direction of propagation. Numerically, the paraxial parabolic wave equation is solved by a fast Huygens sweeping method, while the nonlinear elliptic PDE is handled by the alternating direction explicit (ADE) method. The overall algorithm is shown to be numerically efficient for computing high frequency beam propagations.

An efficient parallel numerical modeling of bioheat transfer in realistic tissue structure

A fast and accurate numerical method of solving bioheat transfer problems is critical for many biomedical applications, such as efficient implementation of thermal therapy planning in a clinical setup, and evaluation of the thermal effects induced by specific energy absorption rate (SAR) from external electromagnetic field. This paper developed a parallel finite-difference method based on alternating direction explicit (ADE) scheme to solve three dimensional transient bioheat equation for the realistic tissue structure. An optimized processing pipeline for accurate presentation of the irregular boundary condition was established to considerably reduce the staircase errors induced by cubic voxel. The voxels order aligned along diagonal direction was designed to allow fast parallel strategy for the lower and upper matrix solution involved in ADE scheme for irregular tissue. The test cases including a real head model have indicated that the developed method can support large time step and cubic voxel approximation of irregular interface and boundary, and also produce significant parallel efficiency.

Non symmetric alternating direction explicit method applied to the calculation of ADS transients

ADS systems – Accelerator Driven Systems, along with Generation IV reactors, have been receiving great attention, especially due to their capability of transmutation and eventually incineration of nuclear waste. To study ADS safety aspects, kinetic models are used to analyze transients. Currently, there are quite a few studies involving transient in ADS systems. Furthermore, most existing studies make use of the Point Kinetics equations. In this context, this work presents a proposal to implement a space dependent kinetics formalism, the Non-Symmetric Alternating Direction Explicit Method, to analyze ADS systems under transient conditions. The computer code developed in this study has satisfactorily reproduced results in the literature of the transients tested. The results are in accordance to what should be expected from the physical point of view.

Approximation of stochastic advection diffusion equations with Stochastic Alternating Direction Explicit methods

The numerical solutions of stochastic partial differential equations of Itô type with time white noise process, using stable stochastic explicit finite difference methods are considered in the paper. Basically, Stochastic Alternating Direction Explicit (SADE) finite difference schemes for solving stochastic time dependent advection-diffusion and diffusion equations are represented and the main properties of these stochastic numerical methods, e.g. stability, consistency and convergence are analyzed. In particular, it is proved that when stable alternating direction explicit schemes for solving linear parabolic PDEs are developed to the stochastic case, they retain their unconditional stability properties applying to stochastic advection-diffusion and diffusion SPDEs. Numerically, unconditional stable SADE techniques are significant for approximating the solutions of the proposed SPDEs because they do not impose any restrictions for refining the computational domains. The performance of the proposed methods is tested for stochastic diffusion and advection-diffusion problems, and the accuracy and efficiency of the numerical methods are demonstrated.

Parallelization of 2-D IADE-DY Scheme on Geranium Cadcam Cluster for Heat Equation

A parallel implementation of the Iterative Alternating Direction Explicit method by D’Yakonov (IADE-DY) for solving 2-D heat equation on a distributed system of Geranium Cadcam cluster (GCC) using the Message Passing Interface (MPI) is presented. The implementation of the scheduling of n tri-diagonal system of equations with the above method was used to show improvement on speedup, effectiveness, and efficiency. The Master/Worker paradigm and Single Program Multiple Data (SPMD) model is employed to manage the whole computation based on the use of domain decomposition. The completion of the execution can need task recovery and favorable configuration. The above mentioned details consist of a main report about the numerical validation of the parallelization through simulation to demonstrate the proposed method effectiveness on the cluster system. It was found that the rate of convergence decreases as the number of processors increases. The result of this paper suggests that the 2-D IADE-DY scheme is a good approach to solving problems, particularly when it is simulation with more processors.

Alternating-direction explicit scheme for the 2D finite-difference time-domain method for TEz waves

In this article, a new explicit finite-difference time-domain (FDTD) method is proposed to eliminate the Courant–Friedrich–Levy (CFL) condition restraint. This new algorithm is based on an alternating-direction explicit (ADE) method and Crank–Nicolson (CN) scheme. Called CN-ADE-FDTD method, and we present two versions of the proposed method. Numerical stability analysis of the new algorithm was also presented. Furthermore, the results by the CN-ADE-FDTD method are compared with the results by the conventional FDTD method. As a result, it is confirmed that the proposed method is unconditionally stable and superior to the conventional one. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:2689–2694, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26346

Parallel Implementation of 2-D Telegraphic Equation on MPI/PVM Cluster

In this paper, a parallel implementation of the Iterative Alternating Direction Explicit method by D’Yakonov (IADE-DY) to solve 2-D telegraphic problem on a distributed system using Message Passing Interface (MPI) and Parallel Virtue Machine (PVM) are presented. The parallelization of the program is implemented by a domain decomposition strategy. A Single Program Multiple Data (SPMD) model is employed for the implementation. The implementation is discussed in relation to means of the parallel performance strategies and analysis. The model enhances overlap communication and computation to avoid unnecessary synchronization, hence, the method yields significant speedup. The level of speedup observed from tables as the mesh increases are in the range of 5–10%. Improvement has been achieved by numbers of tables and figures in our experiment. We present some analyses that are helpful for speedup and efficiency. It is concluded that the efficiency is strongly dependent on the grid size, block numbers and the number of processors for both MPI and PVM. Different strategies to improve the computational efficiency are proposed.

Unconditionally Stable and Second-Order Accurate Explicit Finite Difference Schemes Using Domain Transformation: Part I One-Factor Equity Problems

We introduce a class of stable and second-order accurate finite difference schemes that resolve a number of problems when approximating the solution of option pricing models in computational finance using the finite difference method (FDM). In particular, we show how to avoid having to apply truncation methods to the domain of integration as well as the resulting ad-hoc experimentation in choosing suitable numerical boundary conditions that we must apply on the boundary of the truncated domain. Instead, we transform the PDE option model (which is originally defiined on a semi-infinite interval) to a PDE that is defined on a bounded interval. We then apply the elegant Fichera theory to help us determine which boundary conditions to apply to the transformed PDE. We show that the new system is well-posed by proving energy inequalities in the space of square-integrable functions. Having done that, we adapt the Alternating Direction Explicit (ADE) (a method that dates from the first half of the last century) method to compute an approximation to the solution of the transformed PDE. We prove that the explicit scheme is unconditionally stable and second-order accurate. The scheme is very easy to model, to program and to parallelise and the generalization to n-factor problems is easily motivated. We examine equity problems and we compare the ADE approach (in terms of accuracy and speedup) with more traditional finite difference schemes and with the Monte Carlo method. Finally, we stress-test the scheme by varying critical parameters over a spectrum of values and the results are noted. The methods in this article - in particular the combination of pure and applied techniques - are relatively new in the computational finance literature in our opinion, in particular the realization that a PDE on a semi-infinite interval can be transformed to one on a bounded interval and that the new PDE can be approximated by schemes other than Crank-Nicolson and its workarounds. Finally, we shall report on n-factor models in future articles.

A new unconditionally stable explicit scheme for the convection–diffusion equation with Robin boundary conditions

An alternating direction explicit (ADE) scheme to solve the unsteady convection–diffusion equation with Robin boundary conditions is presented and discussed in this paper. It was derived based on the local series expansion method and proved unconditionally stable by von Neumann stability analysis. Thereafter, the ADE scheme is compared with the conventional schemes, and a comparison between the amplification factor of all schemes and the exact one shows that the proposed scheme can simulate well both convection- and diffusion-dominated problems. Finally, the proposed method was validated by a numerical experiment which indicates that, for large cell Reynolds numbers, the proposed scheme, which has unconditional stability, is more accurate than implicit schemes and most explicit schemes. It is also shown that the proposed scheme is simple to implement, economical to use, effective for dealing with Robin boundary conditions and easy to apply to multidimensional problems.

Numerical methods for conjunctive two-dimensional surface and three-dimensional sub-surface flows

SUMMARY Sophisticated catchment runoff problems necessitate conjunctive modeling of overland flow and subsurface flow. In this paper, finite difference numerical methods are studied for simulation of catchment runoff of two-dimensional surface flow interacting with three-dimensional unsaturated and saturated sub-surface flows. The equations representing the flows are mathematically classified as a type of heat diffusion equation. Therefore, two- and three-dimensional numerical methods for heat diffusion equations were investigated for applications to the surface and sub-surface flow sub-models in terms of accuracy, stability, and calculation time. The methods are the purely explicit method, Saul’yev’s methods, the alternating direction explicit (ADE) methods, and the alternating direction implicit (ADI) methods. The methods are first examined on surface and sub-surface flows separately; subsequently, 12 selected combinations of methods were investigated for modeling the conjunctive flows. Saul’yev’s downstream (S-d) method was found to be the preferred method for two-dimensional surface flow modeling, whereas the ADE method of Barakat and Clark is a less accurate, stable alternative. For the three-dimensional sub-surface flow model, the ADE method of Larkin (ADE-L) and Brian’s ADI method are unconditionally stable and more accurate than the other methods. The calculations of the conjunctive models utilizing the S-d surface flow sub-model give excellent results and confirm the expectation that the errors of the surface and sub-surface sub-models interact. The surface sub-model dominates the accuracy and stability of the conjunctive model, whereas the sub-surface sub-model dominates the calculation time, suggesting the desirability of using a smaller time increment for the surface sub-model. Copyright © 2000 John Wiley & Sons, Ltd.