scholarly journals Preservation of Fine Structures in PDE-Based Image Denoising

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Hakran Kim ◽  
Velinda R. Calvert ◽  
Seongjai Kim
Keyword(s):  
Numerical Methods ◽  
Image Denoising ◽  
Numerical Technique ◽  
Significant Loss ◽  
Implicit Time ◽  
Alternating Direction Implicit ◽  
Time Stepping ◽  
Alternating Direction ◽  
Fine Structures ◽  
Efficiency And Reliability

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.

scholarly journals Comparison of numerical methods to price zero coupon bonds in a two-factor CIR model

2021 ◽  
Vol 20 (1) ◽  
pp. 109-147
Author(s):  
S. Emslie ◽  
S. Mataramvura
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Factor Model ◽  
The Other ◽  
Finite Difference Schemes ◽  
Bond Price ◽  
Cir Model ◽  
Alternating Direction Implicit ◽  
Alternating Direction ◽  
Runge Kutta

In this paper we price a zero coupon bond under a Cox–Ingersoll–Ross (CIR) two-factor model using various numerical schemes. To the best of our knowledge, a closed-form or explicit price functional is not trivial and has been less studied. The use and comparison of several numerical methods to determine the bond price is one contribution of this paper. Ordinary differential equations (ODEs) , finite difference schemes and simulation are the three classes of numerical methods considered. These are compared on the basis of computational efficiency and accuracy, with the second aim of this paper being to identify the most efficient numerical method. The numerical ODE methods used to solve the system of ODEs arising as a result of the affine structure of the CIR model are more accurate and efficient than the other classes of methods considered, with the Runge–Kutta ODE method being the most efficient. The Alternating Direction Implicit (ADI) method is the most efficient of the finite difference scheme methods considered, while the simulation methods are shown to be inefficient. Our choice of considering these methods instead of the other known and apparently new numerical methods (eg Fast Fourier Transform (FFT) method, Cosine (COS) method, etc.) is motivated by their popularity in handling interest rate instruments. Keywords: Cox–Ingersoll–Ross model; numerical methods; Runge–Kutta method; zero-coupon bonds; Alternating Direction Implicit method

ADI Galerkin FEMs for the 2D nonlinear time-space fractional diffusion-wave equation

2017 ◽  
Vol 08 (03) ◽  
pp. 1750025 ◽  
Author(s):  
Meng Li ◽  
Chengming Huang
Keyword(s):  
Wave Equations ◽  
Numerical Technique ◽  
Fractional Diffusion ◽  
Spatial Direction ◽  
Diffusion Wave ◽  
Alternating Direction Implicit ◽  
Galerkin Finite Element ◽  
Time Space ◽  
Alternating Direction ◽  
Temporal Derivative

In this paper, we study a new numerical technique for a class of 2D nonlinear fractional diffusion-wave equations with the Caputo-type temporal derivative and Riesz-type spatial derivative. Galerkin finite element scheme is used for the discretization in the spatial direction, and the temporal component is discretized by a new alternating direction implicit (ADI) method. Next, we strictly prove that the numerical method is stable and convergent. Finally, to confirm our theoretical analysis, some numerical examples in 2D space are presented.

scholarly journals Alternating Direction Implicit (ADI) Methods for Solving Two-Dimensional Parabolic Interface Problems with Variable Coefficients

Computation ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 79
Author(s):  
Chuan Li ◽  
Guangqing Long ◽  
Yiquan Li ◽  
Shan Zhao
Keyword(s):  
Convergence Rates ◽  
Time Integration ◽  
Variable Coefficients ◽  
Interface Problems ◽  
Implicit Time ◽  
Two Dimensional ◽  
Alternating Direction Implicit ◽  
Fully Implicit ◽  
Alternating Direction ◽  
Adi Methods

The matched interface and boundary method (MIB) and ghost fluid method (GFM) are two well-known methods for solving elliptic interface problems. Moreover, they can be coupled with efficient time advancing methods, such as the alternating direction implicit (ADI) methods, for solving time-dependent partial differential equations (PDEs) with interfaces. However, to our best knowledge, all existing interface ADI methods for solving parabolic interface problems concern only constant coefficient PDEs, and no efficient and accurate ADI method has been developed for variable coefficient PDEs. In this work, we propose to incorporate the MIB and GFM in the framework of the ADI methods for generalized methods to solve two-dimensional parabolic interface problems with variable coefficients. Various numerical tests are conducted to investigate the accuracy, efficiency, and stability of the proposed methods. Both the semi-implicit MIB-ADI and fully-implicit GFM-ADI methods can recover the accuracy reduction near interfaces while maintaining the ADI efficiency. In summary, the GFM-ADI is found to be more stable as a fully-implicit time integration method, while the MIB-ADI is found to be more accurate with higher spatial and temporal convergence rates.

scholarly journals Numerical methods for time-fractional convection-diffusion problems with high-order accuracy

2021 ◽  
Vol 19 (1) ◽  
pp. 782-802
Author(s):  
Gang Dong ◽  
Zhichang Guo ◽  
Wenjuan Yao
Keyword(s):  
High Order Accuracy ◽  
Order Accuracy ◽  
Alternating Direction Implicit ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Time Stepping ◽  
Alternating Direction ◽  
Special Cases ◽  
Compact Adi Scheme ◽  
Convergence Of The Scheme

Abstract In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α \alpha ( 1 < α < 2 1\lt \alpha \lt 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 {H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}) , where τ \tau is the temporal grid size and h 1 {h}_{1} , h 2 {h}_{2} are spatial grid sizes in the x x and y y directions, respectively. It is proved that the method can even attain ( 1 + α ) \left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.

Implicit Multigrid Computations of Buoyant Drops Through Sinusoidal Constrictions

2004 ◽  
Vol 71 (6) ◽  
pp. 857-865 ◽  
Author(s):  
Metin Muradoglu ◽  
Seckin Gokaltun
Keyword(s):  
Multigrid Method ◽  
Front Tracking ◽  
Computational Domain ◽  
Governing Equations ◽  
Alternating Direction Implicit ◽  
Front Tracking Method ◽  
Time Stepping ◽  
Alternating Direction ◽  
Dual Time Stepping ◽  
Constricted Channel

Two-dimensional computations of dispersed multiphase flows involving complex geometries are presented. The numerical algorithm is based on the front-tracking method in which one set of governing equations is written for the whole computational domain and different phases are treated as a single fluid with variable material properties. The front-tracking methodology is combined with a newly developed finite volume solver based on dual time-stepping, diagonalized alternating direction implicit multigrid method. The method is first validated for a freely rising drop in a straight channel, and it is then used to compute a freely rising drop in various constricted channels. Interaction of two buoyancy-driven drops in a continuously constricted channel is also presented.

scholarly journals Metode Numerik Untuk Menentukan Harga Opsi Dengan Model Volatilitas Leland

2020 ◽  
Vol 2 (2) ◽  
pp. 41-51
Author(s):  
Arsyad L
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Transaction Cost ◽  
Option Price ◽  
Implicit Time ◽  
Implicit Methods ◽  
Time Stepping ◽  
Volatility Model ◽  
Upwind Finite Difference

Option price under transaction cost with leland volatility model is the solution of a non linear diferential equations. To solve this equation used numerical methods based on an upwind finite difference for spatial discretization as well as the use of explicit and implicit methods for discretizing time-stepping. upwind finite difference method with explicit time-stepping scheme proved to be unstable so as not konvegen. While the use of implicit time-stepping scheme is proved monotonous, consistent and stable so that converge to the viscosity solution.

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