Adaptive PDE Research Articles

Adaptive event-triggered PDE control for load-moving cable systems

Motivated by lateral vibration suppression of a mining cable elevator, which is a load-moving cable system, where the load moves along a viscoelastic guideway whose stiffness and damping coefficients are unknown, we present event-triggered adaptive output-feedback boundary control design of a hyperbolic PDE–ODE coupled system using the measurements at the PDE actuated boundary and the ODE, where the PDE subsystem is a class of 2 × 2 coupled hyperbolic PDEs with spatially-varying coefficients and on a time-varying domain, and a high uncertainty exist in the system matrix of the ODE subsystem at the uncontrolled boundary of the PDE. A continuous-in-time observer-based adaptive backstepping control law is designed where the control gains can be self-tuned to adjust the system matrix of the ODE into a given target system matrix, based on which an observer-based dynamic event-triggering mechanism is built and the existence of a minimal dwell-time is proved. The asymptotic stability of the overall adaptive event-based output-feedback closed-loop system is proved via Lyapunov analysis. In numerical simulation, the performance of the proposed controller is verified in lateral vibration suppression of a mining cable elevator.

Abnormal spatio‐temporal source estimation for a linear unstable parabolic distributed parameter system: An adaptive PDE observer perspective

Detection and estimation of abnormalities for distributed parameter system (DPS) have wide applications in industry, e.g., battery thermal fault diagnosis, quality monitoring of hot-rolled strip laminar cooling process. In this paper, the abnormal spatio-temporal (S-T) source detection and estimation problem for a linear unstable DPS is first studied. The proposed methodology consists of two steps: first, an abnormality detection filter (ADF) which generates a residual signal for abnormality detection in the time domain is constructed using pointwise measurement; Then, an adaptive Luenberger-type PDE observer including an adaptive estimation algorithm is designed and triggered only when an alarm raises from the ADF. Theoretic analysis based on the spatial domain decomposition approach is presented to show the convergence of the estimation errors. Finally, an illustrative example is presented to show the performance of the proposed method.

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.

Multiple Facial Image Editing Using Edge-Aware PDE Learning

This paper introduces a novel facial editing tool, called edge-aware mask, to achieve multiple photo-realistic rendering effects in a unified framework. The edge-aware masks facilitate three basic operations for adaptive facial editing, including region selection, edit setting and region blending. Inspired by the state-of-the-art edit propagation and partial differential equation PDE learning method, we propose an adaptive PDE model with facial priors for masks generation through edge-aware diffusion. The edge-aware masks can automatically fit the complex region boundary with great accuracy and produce smooth transition between different regions, which significantly improves the visual consistence of face editing and reduce the human intervention. Then, a unified and flexible facial editing framework is constructed, which consists of layer decomposition, edge-aware masks generation, and layer/mask composition. The combinations of multiple facial layers and edge-aware masks can achieve various facial effects simultaneously, including face enhancement, relighting, makeup and face blending etc. Qualitative and quantitative evaluations were performed using different datasets for different facial editing tasks. Experiments demonstrate the effectiveness and flexibility of our methods, and the comparisons with the previous methods indicate that improved results are obtained using the combination of multiple edge-aware masks.

SAR Image Despeckling Based on Adaptive PDE Filter and Histogram

PDE is suitable for dealing with degraded SAR image with speckle noise because it can enhance local structure and preserve background and shadow. However, PDE does not take into account the non stationary property and inherent imprecision of SAR image. In this paper, we propose an improved algorithm which combines PDE with histogram of image local coefficient of variation to make different strategies on different regions. The proposed technique compares w.r.t. enhancing Gamma Map filter with better results both in evaluation criteria and visual effect.

A note on reference solution based $hp$-adaptive PDE solvers

A note on reference solution based $hp$-adaptive PDE solvers

Solving Burgersʼ equation using optimal rational approximations

We solve viscous Burgersʼ equation using a fast and accurate algorithm—referred to here as the reduction algorithm—for computing near optimal rational approximations.Given a proper rational function with n poles, the reduction algorithm computes (for a desired L∞-approximation error) a rational approximation of the same form, but with a (near) optimally small number m≪n of poles. Although it is well known that (nonlinear) optimal rational approximations are much more efficient than linear representations of functions via a fixed basis (e.g. wavelets), their use in numerical computations has been limited by a lack of efficient, robust, and accurate algorithms. The reduction algorithm presented here computes reliably (near) optimal rational approximations with high accuracy (e.g., ≈10−14) and a complexity that is essentially linear in the number n of original poles. A key tool is a recently developed algorithm for computing small con-eigenvalues of Cauchy matrices with high relative accuracy, an impossible task for standard algorithms without extended precision.Using the reduction algorithm, we develop a numerical calculus for rational representations of functions. Indeed, while operations such as multiplication and convolution increase the number of poles in the representation, we use the reduction algorithm to maintain an optimally small number of poles.To demonstrate the efficiency, robustness, and accuracy of our approach, we solve Burgersʼ equation with small viscosity ν. It is well known that its solutions exhibit moving transition regions of width O(ν), so that this equation provides a stringent test for adaptive PDE solvers. We show that optimal rational approximations capture the solutions with high accuracy using a small number of poles. In particular, we solve the equation with local accuracy ϵ=10−9 for viscosity as small as ν=10−5.

Mathematical Methods for Images and Surfaces 2011

Mathematical Methods for Images and Surfaces 2011

Transition of Liesegang Precipitation Systems: Simulations with an Adaptive Grid PDE Method

AbstractThe dynamics of the Liesegang type pattern formation is investigated in a centrally symmetric two-dimensional setup. According to the observations in real experiments, the qualitative change of the dynamics is exhibited for slightly different initial conditions. Two kinds of chemical mechanisms are studied; in both cases the pattern formation is described using a phase separation model including the Cahn-Hilliard equations. For the numerical simulations we make use of an adaptive grid PDE method, which successfully deals with the computationally critical cases such as steep gradients in the concentration distribution and investigation of long time behavior. The numerical simulations show a good agreement with the real experiments.

A free-space adaptive FMM-Based PDE solver in three dimensions

We present a kernel-independent, adaptive fast multipole method (FMM) of arbitrary order accuracy for solving elliptic PDEs in three dimensions with radiation boundary conditions. The algorithm requires only a Green’s function evaluation routine for the governing equation and a representation of the source distribution (the right-hand side) that can be evaluated at arbitrary points. The performance of the FMM is accelerated in two ways. First, we construct a piecewise polynomial approximation of the right-hand side and compute far-field expansions in the FMM from the coefficients of this approximation. Second, we precompute tables of quadratures to handle the near-field interactions on adaptive octree data structures, keeping the total storage requirements in check through the exploitation of symmetries. We present numerical examples for the Laplace, modified Helmholtz and Stokes equations.

An Adaptive PDE Image Processing Method Based on <I>L</I><SUB>p</SUB> Norm

摘要: 提出了一种新的基于Lp范数的自适应偏微分方程图像处理模型, 改进了Tony Chan的TV变分模型和张红英的p-Laplace模型. TV模型对图像采用全局约束, 而新的扩散方程在图像不同的位置上采用不同的约束, 具有局部自适应的特性, 在扩散的同时更好地保持了图像的边缘信息, 进而将其应用到图像恢复(去噪, 去除模糊)中去. 实验结果表明新模型的综合性能优于Tony Chan和张等现有的模型. 关键词: 偏微分方程 / Lp范数 / 图像去噪 / 图像恢复

Verifying approximate solutions to differential equations

It is now standard practice in computational science for large-scale simulations to be implemented and investigated in a problem solving environment (PSE) such as MATLAB or MAPLE. In such an environment, a scientist or engineer will formulate a mathematical model, approximate its solution using an appropriate numerical method, visualize the approximate solution and verify (or validate) the quality of the approximate solution. Traditionally, we have been most concerned with the development of effective numerical software for generating the approximate solution and several efficient and reliable numerical libraries are now available for use within the most widely used PSEs. On the other hand, the visualization and verification tasks have received little attention, even though each often requires as much computational effort as is involved in generating the approximate solution. In this paper, we will investigate the effectiveness of a suite of tools that we have recently introduced in the MATLAB PSE to verify approximate solutions of ordinary differential equations. We will use the notion of ‘effectivity index’, widely used by researchers in the adaptive mesh PDE community, to quantify the credibility of our verification tools. Numerical examples will be presented to illustrate the effectiveness of these tools when applied to a standard numerical method on two model test problems.

Nonlinear Methods of Approximation

Abstract. Our main interest in this paper is nonlinear approximation. The basic idea behind nonlinear approximation is that the elements used in the approximation do not come from a fixed linear space but are allowed to depend on the function being approximated. While the scope of this paper is mostly theoretical, we should note that this form of approximation appears in many numerical applications such as adaptive PDE solvers, compression of images and signals, statistical classification, and so on. The standard problem in this regard is the problem of m -term approximation where one fixes a basis and looks to approximate a target function by a linear combination of m terms of the basis. When the basis is a wavelet basis or a basis of other waveforms, then this type of approximation is the starting point for compression algorithms. We are interested in the quantitative aspects of this type of approximation. Namely, we want to understand the properties (usually smoothness) of the function which govern its rate of approximation in some given norm (or metric). We are also interested in stable algorithms for finding good or near best approximations using m terms. Some of our earlier work has introduced and analyzed such algorithms. More recently, there has emerged another more complicated form of nonlinear approximation which we call highly nonlinear approximation. It takes many forms but has the basic ingredient that a basis is replaced by a larger system of functions that is usually redundant. Some types of approximation that fall into this general category are mathematical frames, adaptive pursuit (or greedy algorithms), and adaptive basis selection. Redundancy on the one hand offers much promise for greater efficiency in terms of approximation rate, but on the other hand gives rise to highly nontrivial theoretical and practical problems. With this motivation, our recent work and the current activity focuses on nonlinear approximation both in the classical form of m -term approximation (where several important problems remain unsolved) and in the form of highly nonlinear approximation where a theory is only now emerging.

A New Paradigm for Parallel Adaptive Meshing Algorithms

We present a new approach to the use of parallel computers with adaptive finite element methods. This approach addresses the load balancing problem in a new way, requiring far less communication than current approaches. It also allows existing sequential adaptive PDE codes such as PLTMG and MC to run in a parallel environment without a large investment in recoding. In this new approach, the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor, using a sequential adaptive solver, without requiring any modifications to the sequential solver. The small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh, which are then used in a weighted recursive spectral bisection of the initial mesh. The bulk of the calculation then takes place independently on each processor, with no communication, using possibly the same sequential adaptive solver. Each processor adapts its region of the mesh independent...

A New Paradigm for Parallel Adaptive Meshing Algorithms

We present a new approach to the use of parallel computers with adaptive finite element methods. This approach addresses the load balancing problem in a new way, requiring far less communication than current approaches. It also allows existing sequential adaptive PDE codes such as PLTMG and MC to run in a parallel environment without a large investment in recoding. In this new approach, the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor, using a sequential adaptive solver, without requiring any modifications to the sequential solver. The small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh, which are then used in a weighted recursive spectral bisection of the initial mesh. The bulk of the calculation then takes place independently on each processor, with no communication, using possibly the same sequential adaptive solver. Each processor adapts its region of the mesh independently, and a nearly load-balanced mesh distribution is usually obtained as a result of the initial weighted spectral bisection. Only the initial fan-out of the mesh decomposition to the processors requires communication. Two additional steps requiring boundary exchange communication may be employed after the individual processors reach an adapted solution, namely, the construction of a global conforming mesh from the independent subproblems, followed by a final smoothing phase using the subdomain solutions as an initial guess. We present a series of convincing numerical experiments which illustrate the effectiveness of this approach. The justification of the initial refinement prediction step, as well as the justification of skipping the two communication-intensive steps, is supported by some recent [J. Xu and A. Zhou, Math. Comp., to appear] and not so recent [J. A. Nitsche and A. H. Schatz, Math. Comp., 28 (1974), pp. 937--958; A. H. Schatz and L. B. Wahlbin, Math. Comp., 31 (1977), pp. 414--442; A. H. Schatz and L. B. Wahlbin, Math. Comp., 64 (1995), pp. 907--928] results on local a priori and a posteriori error estimation.

Parallel multigrid in an adaptive PDE solver based on hashing and space-filling curves

Partial differential equations can be solved efficiently by adaptive multigrid methods on a parallel computer. We report on the concept of hash-table storage techniques to set up such a program. The code requires substantial less amount of memory than implementations based on tree type data structures and is easier to program in the sequential case. The parallelization takes place by a space-filling curve domain decomposition intimately connected to the hash table. The new data structure simplifies the parallelization of the code substantially and introduces a cheap way to solve the load balancing and mapping problem. We report on the main features of the method and give the results of numerical experiments with the new parallel solver on a cluster of 64 Pentium II/400MHz connected by a Myrinet in a fat tree topology.

Recursive mesh refinement on hypercubes

Adaptive methods for PDEs can be viewed as a graph problem. An efficient parallel implementation of adaptive PDE methods then requires distributing the nodes of the associated graph uniformly across the processors so that the resulting cost of communication between processors is low.

Concepts of an adaptive hierarchical finite element code

This paper presents the mathematical concepts underlying the new adaptive finite element code KASKADE, which, in its present form, applies to linear scalar second-order 2D elliptic problems on general domains. The starting point for this new development is the recent work on hierarchical finite element bases by H. Yserentant ( Numer. Math. 49, 379–412 (1986)). It is shown that this approach permits a flexible balance among iterative solver, local error estimator, and local mesh refinement device—the main components of an adaptive PDE code. Without use of standard multigrid techniques, the same kind of computational complexity is achieved—independent of any uniformity restrictions on the applied meshes. In addition, the method is extremely simple and all computations are purely local, making the method particularly attractive in view of parallel computing. The algorithmic approach is illustrated by a well-known critical test problem.