Adaptive Partial Differential Equation Research Articles

Numerical Simulations of a Fluidized Granular Flow Entry Into Water: Insights Into Modeling Tsunami Generation by Pyroclastic Density Currents

AbstractThe tsunami generation potential of pyroclastic density currents (PDCs) entering the sea is poorly understood, due to limited data and observations. Thus far, tsunami generation by PDCs has been modeled in a similar manner to tsunami generation associated with landslides or debris flows, using two‐layer depth‐averaged approaches. Using the adaptive partial differential equation solver Basilisk and benchmarking with published laboratory experiments, this work explores some of the important parameters not yet accounted for in numerical models of PDC‐generated tsunamis. We use assumptions derived from experimental literature to approximate the granular, basal flow component of a PDC as a dense Newtonian fluid flowing down an inclined plane. This modeling provides insight into how the boundary condition of the slope and the viscosity of the dense granular‐fluid influence the characteristics of the waves generated. It is shown that the boundary condition of the slope has a first‐order impact on the interaction dynamics between the fluidized granular flow and water, as well as the energy transfer from the flow to the generated wave. The experimental physics is captured well in the numerical model, which confirms the underlying assumption of Newtonian fluid‐like behavior in the context of wave generation. The results from this study suggest the importance of considering vertical density and velocity stratification in wave generation models. Furthermore, we demonstrate that granular‐fluids denser than water are capable of shearing the water surface and generating significant amplitude waves, despite vigorous overturning.

Analysis and Recognition Method of Internet Image Public Opinion Based on Partial Differential Equation

This article comprehensively and systematically expounds the development trends and basic theory of partial differential methods, analyzes the characteristics of sampling multiscale transformation in detail, and deeply studies the network image denoising and network image restoration methods that perform partial differential diffusion in the pixel domain and the transform domain. An adaptive diffusion method of partial differential equations is proposed. Among them, the key parameters can be adaptively changed according to the curvature and gradient of the local geometric information of the network image, and the diffusion direction and intensity of the diffusion can be controlled. First, using the principle of variation, we derive the Euler equation corresponding to the diffusion method of partial differential equations and analyze its diffusion ability using the local orthogonal coordinate system of the network image. Based on the theoretical analysis of public opinion, this article applies opinion mining technology to the online public opinion early warning system to achieve the purpose of grasping the opinions of netizens in time and guiding the trend of public opinion. Opinion mining is the use of natural language processing technology to automatically extract the emotional tendencies and evaluation objects contained in the subjective text. In the edge area of the network image, the diffusion along the edge direction should have a large diffusion coefficient, and the diffusion along the vertical edge direction should have a small diffusion coefficient; in the flat area of the network image, it diffuses to the surrounding with equal intensity, and the diffusion intensity value is relatively high. Secondly, based on the analysis of the adaptive partial differential equation diffusion method, using the half-point difference format, a numerical method for network image recognition is designed. Both theoretical analysis and experimental results show that the network image recognition model based on adaptive partial differential equation diffusion is more effective than the model based on partial differential equation recognition; at the same time, experiments show that the network image recognition model based on adaptive partial differential equation diffusion is more effective than the network image recognition model based on ordinary diffusion. The network image recognition model based on constant partial differential equation diffusion is more effective in improving the quality of network image recognition.

Improved partial differential equation-based total variation approach to non-subsampled contourlet transform for medical image denoising

This article proposes an improved partial differential equation (PDE)-based total variation (TV) model that enhances grey and coloured brain tumour images obtained by magnetic resonance imaging. A nonsubsampled contourlet transform was applied to images from standard databases that converted into lowpass and highpass (or bandpass) contourlet coefficients. An improved version of the power-law transform method was used on the lowpass contourlet coefficients, and an adaptive threshold method was applied to the highpass (or bandpass) contourlet coefficients. The inverse contourlet transform was performed on all the enhanced contourlet coefficients to generate a complete brain tumour image. Finally, the PDE-based TV model was applied to this image to produce the denoised image. The performance of the suggested method was calculated in terms of the peak signal-to-noise ratio, mean square error, and structural similarity index. This method achieved the best peak signal-to-noise ratio, mean square error, and structural similarity index of 77.9846 dB, 0.00012612, and 97.895%, respectively, compared to the conventional PDE+modified transform-based gamma correction, adaptive PDE+generalized cross-validation, parallel magnetic resonance imaging, and Berkeley wavelet transform+support vector machine methods.

An adaptive VPDE image denoising model based on structure tensor

Image denoising is one of the challenging topics in image processing. Since the formation mechanism of the degraded image is unknown, it is difficult to achieve effective image denoising. Most existing denoising methods are easy to blur the edges of the image so that it is difficult to maintain more image detail information. In recent years, partial differential equation (PDE) has been applied to image denoising, which has shown good application prospects. However, the image denoising methods based on PDE are easy to appear staircase effect and edge blur, and can not achieve satisfactory denoising effect. To avoid some problems, an adaptive variational PDE (VPDE) image denoising model based on the structure tensor is proposed and analyzed. In the proposed model, since the structure tensor can provide more local structural information and direction information of the gradient, the eigenvalues of the structure tensor of an image are used to construct the norm parameter. Through theoretical analysis and experiments, we can confirm that the proposed norm parameter has the following adaptive characteristics: (1) In the flat region of the image, the proposed norm parameter tends to 2, so which has the characteristic of isotropic diffusion. (2) At the edge region of the image, the norm parameter tends to 1, and which spreads only along the tangential direction of the image edge. The experimental results from both subjective and objective aspects show that using these adaptive characteristics, the proposed image denoising model can effectively avoid staircase effect, preserve the image details, and obtain satisfactory denoising effect.

Intravascular optical coherence tomography image segmentation based on Gaussian mixture model and adaptive fourth-order PDE

The accuracy of the fibrotic plaque segmentation is vital in identifying the coronary artery stenosis. In this paper, we address an automated approach (APDE-GMM) for separating the fibrotic plaque area of intravascular optical coherence tomography (IV-OCT) images. Under this approach, an objective function consisting of a new energy functional with Rayleigh distribution and the negative log-likelihood function of Gaussian mixture model (GMM) is developed. Also, the study presents an adaptive diffusivity function where the gradient threshold can be associated to suppress the effect of speckle noise. The parameter estimation is carried out by the expectation–maximization technology. In addition, this paper derives a fourth-order partial differential equation (PDE) via Euler–Lagrange equation to obtain the optimal solutions. It has been compared to other segmentation approaches on synthetic and clinical IV-OCT images. The results demonstrate that APDE-GMM segmentates more accurately.

A regional adaptive variational PDE model for computed tomography image reconstruction

Improving CT images by increasing the number of scans, hence increasing the ionizing radiation dose, can increase the probability of inducing cancer in the patient. Using fewer images but improving them by accurate reconstruction is better solution.In this paper, an adaptive variational Partial Differential Equation (PDE) model is proposed for image reconstruction. L2 energy of the image gradient and the Total Variation (TV) are combined to form a new functional, which is introduced to an optimization problem. The dynamic behaviors of the model are formed by a threshold function, and then the L2 term is applied in the lower-density region to increase reconstruction speed, and the TV term is applied in the higher-density region to preserve the most important image features. The threshold function is asymptotically controlled by an evolutionary PDE and is more suitable for complex images. The efficiency and accuracy of the proposed model are demonstrated in numerical experiments.

Denoising and segmentation of MR images using fourth order non‐linear adaptive PDE and new convergent clustering

AbstractAt present, digital image processing plays a vital role in medical imaging areas and specifically in magnetic resonance imaging (MRI) of brain images such as axial and coronal sections. This article mainly focused on the MRI brain images. The existing methods such as total variation (MC), parallel MRI, modified pyramidal dual‐tree direction filter, adaptive dictionary selection algorithm, classifier methods, and fuzzy clustering techniques are poor in image eminence and precision. Thus, this article presents a novel approach consisting of denoising followed by segmentation. The objective of these proposed methods was visual eminence improvement of medical images to examine tumor extent using an adaptive partial differential equation (APDE)‐based analysis with soft threshold function in denoising. The fourth order, nonlinear APDE was used to denoise the image depending on gradient and Laplacian operators associated with the new adaptive Haar‐type wavelet transform. A second approach was the new convergent K‐means clustering for segmentation. The convergent K‐means procedure diminishes the summation of the squared deviations of structures in a cluster from the center. The significance of these proposed methods was to compute their performances in terms of mean squared error, peak signal‐to‐noise ratio, structure similarity, segmentation accuracy, false hit, missed‐term, and elapsed time. The results were analyzed with the MATLAB software.

PDE Boundary Control of Multi-Input LTI Systems With Distinct and Uncertain Input Delays

Taking advantage of the concept of a transport partial differential equation (PDE) representation of the delayed input, we propose a predictor-feedback control for the stabilization of multi-input linear time-invariant (LTI) systems with distinct input delays. Different from recently published results about predictor feedbacks for multi-input time delay systems that require a perfect knowledge of the length of delays, the LTI plants taken into account in this paper have the uncertain and distinct time delay in each individual input channel. To deal with the problem of the unknown multi-input delays and the consequently unmeasurable distributed input, an adaptive PDE boundary control scheme including an estimation of the unmeasured transport PDE state and an online identification of the unknown delay is developed. The conventional backstepping transformation for the known delay case is also replaced by a unity-rescaled backstepping transformation for the case of delay adaptation. With Lyapunov-based analysis, we show local stability of the delay-adaptive prediction-based closed-loop systems.

Multiscale Tikhonov-Total Variation Image Restoration Using Spatially Varying Edge Coherence Exponent.

Edge preserving regularization using partial differential equation (PDE)-based methods although extensively studied and widely used for image restoration, still have limitations in adapting to local structures. We propose a spatially adaptive multiscale variable exponent-based anisotropic variational PDE method that overcomes current shortcomings, such as over smoothing and staircasing artifacts, while still retaining and enhancing edge structures across scale. Our innovative model automatically balances between Tikhonov and total variation (TV) regularization effects using scene content information by incorporating a spatially varying edge coherence exponent map constructed using the eigenvalues of the filtered structure tensor. The multiscale exponent model we develop leads to a novel restoration method that preserves edges better and provides selective denoising without generating artifacts for both additive and multiplicative noise models. Mathematical analysis of our proposed method in variable exponent space establishes the existence of a minimizer and its properties. The discretization method we use satisfies the maximum-minimum principle which guarantees that artificial edge regions are not created. Extensive experimental results using synthetic, and natural images indicate that the proposed multiscale Tikhonov-TV (MTTV) and dynamical MTTV methods perform better than many contemporary denoising algorithms in terms of several metrics, including signal-to-noise ratio improvement and structure preservation. Promising extensions to handle multiplicative noise models and multichannel imagery are also discussed.

A novel adaptive partial differential equation model for image segmentation

Many partial differential equation (PDE) models have been proposed for image segmentation. However, most of them cannot handle the complicated images such as inhomogeneous images. To overcome the limitation of traditional PDE-based models, a novel PDF model is proposed in this paper and then successfully applied to inhomogeneous image segmentation task. The new PDE model contains the global statistical information and local statistical information, which are balanced using an adaptive weight parameter. Some inhomogeneous images are selected for evaluations. Experimental results show that the proposed method is more effective to segment images with in-homogeneity or low contrast, without choosing the weight parameter between global and local information.

Total Variation Differential Equation with Wavelet Transform for Image Restoration

To overcome the staircasing effects and simultaneously avoid edge blurring, We present an adaptive partial differential equation combining the total variation with wavelet transform for image restoration. A noise removal algorithm based on variation method and partial difference equations is proposed in this paper. The combining algorithm takes the advantage of both filters since it is able to preserve edges while avoiding the staircase effect in smooth regions. The TV method provides fast adaptive wavelet-based solvers for the TV model. Our approach not only employs a wavelet collocation method applied to the TV model using two-dimensional anisotropic tensor product of wavelets, but also proposes the differential equation for image restoration. Most conventional image processors consider little the influence of human vision psychology. The algorithm inherently not only combines the restoration property of wavelet compression algorithms with that of the TV model, but also gives a relative new TV functional which considers the influence of human vision psychology. We present a detailed description of our method which indicates that a combination of wavelet based restoration techniques with the TV model produces superior results. Experimental results illustrate the effectiveness of the model image restoration. DOI : http://dx.doi.org/10.11591/telkomnika.v12i6.5495 Full Text: PDF

Alternating Schwarz methods for partial differential equation-based mesh generation

To solve boundary value problems with moving fronts or sharp variations, moving mesh methods can be used to achieve reasonable solution resolution with a fixed, moderate number of mesh points. Such meshes are obtained by solving a nonlinear elliptic differential equation in the steady case, and a nonlinear parabolic equation in the time-dependent case. To reduce the potential overhead of adaptive partial differential equation-(PDE) based mesh generation, we consider solving the mesh PDE by various alternating Schwarz domain decomposition methods. Convergence results are established for alternating iterations with classical and optimal transmission conditions on an arbitrary number of subdomains. An analysis of a colouring algorithm is given which allows the subdomains to be grouped for parallel computation. A first result is provided for the generation of time-dependent meshes by an alternating Schwarz algorithm on an arbitrary number of subdomains. The paper concludes with numerical experiments illustrating the relative contraction rates of the iterations discussed.

Adaptive Partial Differential Equation Observer for Battery State-of-Charge/State-of-Health Estimation Via an Electrochemical Model

This paper develops an adaptive partial differential equation (PDE) observer for battery state-of-charge (SOC) and state-of-health (SOH) estimation. Real-time state and parameter information enables operation near physical limits without compromising durability, thereby unlocking the full potential of battery energy storage. SOC/SOH estimation is technically challenging because battery dynamics are governed by electrochemical principles, mathematically modeled by PDEs. We cast this problem as a simultaneous state (SOC) and parameter (SOH) estimation design for a linear PDE with a nonlinear output mapping. Several new theoretical ideas are developed, integrated together, and tested. These include a backstepping PDE state estimator, a Padé-based parameter identifier, nonlinear parameter sensitivity analysis, and adaptive inversion of nonlinear output functions. The key novelty of this design is a combined SOC/SOH battery estimation algorithm that identifies physical system variables, from measurements of voltage and current only.

A local structural adaptive partial differential equation for image denoising

In this paper, we propose a local and contextual controlled (LCC) fourth order partial differential equation (PDE) method for noise removal. First, a region based fourth order PDE is proposed, which conductance coefficients are chosen adaptively in terms of the domain type. Thus, the proposed method can preserve the advantages of forth order PDE and avoid leaving isolated black and white speckles. Furthermore, the region based fourth order PDE and two discontinuity measures are incorporated into a LCC fourth order PDE, where the joint use of the two discontinuity measures leads to a complementary effect for edge preservation. The proposed LCC fourth order PDE is tested on two typical images, including texture image and natural scene image, which show the superiority of the proposed method.

On a System of Adaptive Coupled PDEs for Image Restoration

In this paper, we consider a coupled system of partial differential equations (PDEs) based model for image restoration. Both the image and the edge variables are incorporated by coupling them into two different PDEs. It is shown that the initial-boundary value problem has global in time dissipative solutions (in a sense going back to P.-L. Lions), and several properties of these solutions are established. Some numerical examples are given to highlight the denoising nature of the proposed model along with some comparison results.

Selective Extraction of Entangled Textures via Adaptive PDE Transform.

Texture and feature extraction is an important research area with a wide range of applications in science and technology. Selective extraction of entangled textures is a challenging task due to spatial entanglement, orientation mixing, and high-frequency overlapping. The partial differential equation (PDE) transform is an efficient method for functional mode decomposition. The present work introduces adaptive PDE transform algorithm to appropriately threshold the statistical variance of the local variation of functional modes. The proposed adaptive PDE transform is applied to the selective extraction of entangled textures. Successful separations of human face, clothes, background, natural landscape, text, forest, camouflaged sniper and neuron skeletons have validated the proposed method.

ALPS: A framework for parallel adaptive PDE solution

Adaptive mesh refinement and coarsening (AMR) is essential for the numerical solution of partial differential equations (PDEs) that exhibit behavior over a wide range of length and time scales. Because of the complex dynamic data structures and communication patterns and frequent data exchange and redistribution, scaling dynamic AMR to tens of thousands of processors has long been considered a challenge. We are developing ALPS, a library for dynamic mesh adaptation of PDEs that is designed to scale to hundreds of thousands of compute cores. Our approach uses parallel forest-of-octree-based hexahedral finite element meshes and dynamic load balancing based on space-filling curves. ALPS supports arbitrary-order accurate continuous and discontinuous finite element/spectral element discretizations on general geometries. We present scalability and performance results for two applications from geophysics: seismic wave propagation and mantle convection.

Code-generator-based software package for defining and solving one-dimensional, dynamic, catalytic reactor models

A code-generator-based software is presented which automatically generates FORTRAN subroutines describing a tubular catalytic reactor from simply structured xml-input files specifying the reacting species and reaction network. These routines are then linked to the adaptive partial differential equation (PDE) solver BACOL, which has been extended to allow direct execution from within MATLAB. A unique feature of the presented subroutine-generator is the option for an automated simplification of heterogeneous reaction networks into global Langmuir–Hinshelwood-type reaction rates using a quasi-equilibrium approximation. Using a water–gas shift reaction mechanism published by Mhadewshwar et al., the high-computational performance of the presented software is demonstrated and the impact of the automated approximations upon the solution as well as the simulation’s computational performance are investigated.

On Resistive MHD Models with Adaptive Moving Meshes

In this paper we describe an adaptive moving mesh technique and its application to convection-diffusion models from magnetohydrodynamics (MHD). The method is based on a coordinate transformation between physical and computational coordinates. The transformation can be viewed as a solution of adaptive mesh partial differential equations (PDEs) which are derived from the minimization of a mesh-energy integral. For an efficient implementation we have used an approach in which the numerical solution of the physical PDE model and the adaptive PDEs are decoupled. Further, to avoid solving large nonlinear systems, an implicit-explicit method is applied for the time integration in combination with the iterative method Bi-CGSTAB. The adaptive mesh can be viewed as a 2D variant of the equidistribution principle, and it has the ability to track individual features of the physical solutions in the developing plasma flows. The results of a series of numerical experiments are presented which cover several aspects typifying resistive magnetofluid-dynamics.

Adaptive moving mesh computations for reaction–diffusion systems

In this paper we describe an adaptive moving mesh technique and its application to reaction–diffusion models from chemistry. The method is based on a coordinate transformation between physical and computational coordinates. The transformation can be viewed as a solution of adaptive mesh partial differential equations (PDEs) which is derived from the minimization of a mesh-energy integral. For an efficient implementation we have used an approach in which the numerical solution of the physical PDEs and the adaptive PDEs are decoupled. Further, to avoid solving large nonlinear systems, a second-order implicit–explicit time-integration method in combination with the iterative method Bi-CGSTAB is applied in the method-of-lines procedure. Numerical examples are given in one and two space dimensions.