Multiscale Discretization Research Articles

The Use of the Multi-Scale Discrete Wavelet Transform and Deep Neural Networks on ECGs for the Diagnosis of 8 Cardio-Vascular Diseases

Cardiovascular diseases (CVD) continues to be the leading cause of death worldwide, with over 17 million deaths each year. In 2015, approximately 422 million people suffered from cardiovascular disease (CVD). Reading and analyzing electrocardiograms (ECGs) can be time consuming, and the development of decision support tools based on automated systems can facilitate and speed up the diagnosis of ECGs. In this paper, we propose a 12 leads ECG signals classification using Multi-level Discrete Wavelet Transform and ResNet34 Deep Learning algorithm which classifies 8 types of cardiovascular diseases: Atrial fibrillation (AF), 1st degree atrioventricular block (AV), Left bundle branch block (LBBB), Right bundle branch block (RBBB), Premature ventricular contraction (PVC), Premature atrial contraction (PAC), ST segment depression (STD), and ST segment elevation (STE). The ECGs are preprocessed, and different features are extracted using Multi-level Discrete Wavelet Transform. The model is trained on a database of more than 6000 electrocardiograms which includes 9 types of 12-lead ECGs: a normal type and the 8 abnormal ones which correspond to the diseases mentioned above.

Numerical homogenization of fractal interface problems

We consider the numerical homogenization of a class of fractal elliptic interface problems inspired by related mechanical contact problems from the geosciences. A particular feature is that the solution space depends on the actual fractal geometry. Our main results concern the construction of projection operators with suitable stability and approximation properties. The existence of such projections then allows for the application of existing concepts from localized orthogonal decomposition (LOD) and successive subspace correction to construct first multiscale discretizations and iterative algebraic solvers with scale-independent convergence behavior for this class of problems.

Development of a multiscale discretization method for the geographical detector model

ABSTRACT The geographical detector model (GDM) is based on the spatial variance analysis of geographical strata of variables to assess the association between the independent variables () and dependent variables (). The independent variables of the GDM must be discretized into classes. However, current discretization methods employ univariate analysis, which may lead to inaccurate results. The aim of this study was to develop a novel bivariate optimal discretization approach, known as the multiscale discretization (MSD) method. The objective of the MSD method is to determine an appropriate set of thresholds for , thereby minimizing the variance of within the spatial partitions determined by the discrete . We successfully applied the MSD method to assess the relationship between the precipitation and enhanced vegetation index on the African continent, as well as the habitat range of pandas in Ya’an County, Sichuan Province, China. The results demonstrate that the MSD is a feasible, robust, and rapid method for converting continuous data into discrete data, with globally optimal discretization results. Furthermore, the MSD method can evaluate the degree of association between and more accurately, and can optimize the results of the GDM.

Correction to: Development and validation of a new friction model for cutting processes

The authors wish to thank the German Research Foundation (DFG) for funding the research within the project Modelling of broaching processes by multi-scale discretization (KL500/159-1).

Multiscale Discrete Framelet Transform for Graph-Structured Signals

Graph-structured signal enables rich description of data defined in the domain with irregular structure, which has seen its rapid growth in many applications including social, energy, transportation, sensor, neuronal networks, and many others. This paper aims at generalizing discrete framelet transform defined for regular grids in Euclidean space to finite undirected weighted graphs. By leveraging the intuition from classic framelet transform for signals on regular grids, we proposed an approach for constructing multiscale undecimal framelet transform for signals defined on finite graphs with a perfect reconstruction property. The proposed method is based on the definition of basic blocks involved in framelet transform, including graph shift operator, convolution, and band-limited down/up-sampling. These blocks enable a painless construction of a class of multilevel undecimal framelet transforms in vertex domain by directly calling wavelet filter banks of existing wavelet biframes and tight frames. The proposed discrete framelet transform on graphs keeps most desired properties of its counterpart on regular grids, and one can see its usage in applications.

Validation study of computational fluid dynamics models of hemodynamics in the human aorta

AbstractWe report on conducting a rigorous validation study where we attempt to compare simulated hemodynamics in a mock model of the human aorta with experimental bi‐plane particle image velocimetry (PIV) data. Within the cardiac modeling framework CARP (Cardiac Arrhythmia Research Package) [12] we implemented a finite element CFD solver based on the residual‐based variational multiscale (RBVMS) discretization of the incompressible Navier‐Stokes equations. CFD simulation results compare favorably with experimental PIV data suggesting that our CFD analysis tools are representative of blood flow in the human aorta.

An Overlapping Domain Decomposition Method Based on Calderón Preconditioned CFIE

In this letter, an overlapping domain decomposition method (ODDM) based on the Calderon preconditioned combined field integral equation (CP-CFIE) is proposed. In contrast to previous ODDM using the combined field integral equation (CFIE), the proposed method possesses excellent robustness in the case of multiscale discretization. Several numerical examples are provided to demonstrate the correctness and performance of the proposed method.

Dual-Horizon Peridynamics Analysis of Debonding Failure in FRP-to-Concrete Bonded Joints

A dual-horizon peridynamics (DH-PD) model is proposed for the simulation of debonding process in fiber-reinforced polymer (FRP)-to-concrete bonded joints. In the novel procedure of implementing DH-PD in the framework of finite element method (FEM), truss elements are employed to represent the bonds and dual-bonds. The quadtree approach is utilized to generate the multi-scale discretization and a volume correction scheme based on the background grid is proposed for the non-uniform grid. A benchmark numerical example is performed to test the accuracy and efficiency of the developed model in analysis of the bond behavior. The predicted results are consistent with the experimental findings, the FEM results and the analytical solutions. Additionally, these results demonstrate that the bond strength and the debonding ductility are visibly affected by concrete strength and the thickness of FRP plate, while the adhesive thickness has no significant impact on the debonding behavior.

Basic and extendable framework for effective charge transport in electrochemical systems

We consider basic and easily extendible transport formulations for lithium batteries consisting of an anode (Li-foil), a separator (polymer electrolyte), and a composite cathode (composed of electrolyte and intercalation particles). Our mathematical investigations show the following novel features: (i) complete and very basic description of mixed transport processes relying on a neutral, binary symmetric electrolyte resulting in a non-standard Poisson equation for the electric potential together with interstitial diffusion approximated by classical diffusion; (ii) upscaled and basic composite cathode equations allowing to take geometric and material features of electrodes into account; (iii) the derived effective macroscopic model can be numerically solved with well-known numerical strategies for homogeneous domains and hence does not require to solve a high-dimensional numerical problem or to depend on computationally involved multiscale discretization strategies where highly heterogeneous and realistic, nonlinear, and reactive boundary conditions are still unexplored. We believe that the here proposed basic and easily extendible formulations will serve as a basic and simple setup towards a systematic theoretical and experimental understanding of complex electrochemical systems and their optimization, e.g. Li-batteries.

Multiscale finite element discretizations based on local defect correction for the biharmonic eigenvalue problem of plate buckling

In this paper, we study the multiscale finite element discretizations about the biharmonic eigenvalue problem of plate buckling. On the basis of the work of Dai and Zhou (SIAM J. Numer. Anal. 46[1] [2008] 295‐324), we establish a three‐scale scheme, a multiscale discretization scheme, and the associated parallel version based on local defect correction. We first prove a local priori error estimate of finite element approximations, then give the error estimates of multiscale discretization schemes. Theoretical analysis and numerical experiments indicate that our schemes are suitable and efficient for eigenfunctions with local low smoothness.

Multiscale coarse spaces for overlapping Schwarz methods based on the ACMS space in 2D

Two-level overlapping Schwarz domain decomposition methods for second-order elliptic problems in two dimensions are proposed using coarse spaces constructed from the Approximate Component Mode Synthesis (ACMS) multiscale discretization approach. These coarse spaces are based on eigenvalue problems using Schur complements on subdomain edges. It is then shown that the convergence of the resulting preconditioned Krylov method can be controlled by a user-specified tolerance and thus can be made independent of heterogeneities in the coefficient of the partial differential equation. The relations of this new approach to other known adaptive coarse space approaches for overlapping Schwarz methods are also discussed. Compared to one of the competing adaptive approaches, the new coarse space can be significantly smaller. Compared to other competing approaches, the eigenvalue problems are significantly cheaper to solve, i.e., the dimension of the eigenvalue problems is minimal among the competing adaptive approaches under consideration. Our local eigenvalue problems can be solved using one iteration of LobPCG for essentially the same cost as a Cholesky-decomposition of a Schur complement on a subdomain edge.

Heterogeneous multiscale method for optimal control problem governed by parabolic equation with highly oscillatory coefficients

In this paper, we investigate the heterogeneous multiscale method (HMM) for the optimal control problem governed by the parabolic equation with highly oscillatory coefficients. The state variable and adjoint state variable are approximated by the multiscale discretization scheme that relies on coupled macro and micro finite elements, while the control variable is discretized by the piecewise constants. By applying the well-known Lions’ Lemma to the discretized optimal control problem, we obtain the necessary and sufficient optimality conditions. A priori error estimates are derived for the state, co-state and the control with uniform bounded constants. Finally, numerical results are presented to illustrate our theoretical findings.

The effect of temperature on the elastic precursor decay in shock loaded FCC aluminium and BCC iron

This article offers a comprehensive experimental and theoretical study of the causes of thermal hardening in FCC Al and BCC Fe at high strain rates, with the aim to shed light on important mechanisms governing deformation and failures in materials subjected to shocks and impacts at very high strain rates. Experimental evidence regarding the temperature dependence of the dynamic yield point of FCC Al and BCC Fe shock loaded at 107 s−1 is provided. The dynamic yield point of Al increases with temperature in the range 125K–795K; for the same loading and temperate range, the dynamic yield point of BCC Fe remains largely insensitive. A Multiscale Discrete Dislocation Plasticity (MDDP) model of both Fe and Al is developed, leading to good agreement with experiments. The importance of the Peierls barrier in Fe is highlighted, showing it is largely responsible for the temperature insensitivity in BCC metals. The relevance of the mobility of edge components in determining the plastic response of both FCC Al and BCC Fe at different temperatures is discussed, which leads to developing a mechanistic explanation of the underlying mechanisms leading to the experimental behaviour using Dynamic Discrete Dislocation Plasticity (D3P). It is shown that the main contributing factor to temperature evolution of the dynamic yield point is not the mobility of dislocations, but the temperature variation of the shear modulus, the decrease of which is correlated to the experimental behaviour observed for both FCC Al and BCC Fe.

Directional Frames for Image Recovery: Multi-scale Discrete Gabor Frames

Sparsity-driven image recovery methods assume that images of interest can be sparsely approximated under some suitable system. As discontinuities of 2D images often show geometrical regularities along image edges with different orientations, an effective sparsifying system should have high orientation selectivity. There have been enduring efforts on constructing discrete frames and tight frames for improving the orientation selectivity of tensor product real-valued wavelet bases/frames. In this paper, we studied the general theory of discrete Gabor frames for finite signals, and constructed a class of discrete 2D Gabor frames with optimal orientation selectivity for sparse image approximation. Besides high orientation selectivity, the proposed multi-scale discrete 2D Gabor frames also allow us to simultaneously exploit sparsity prior of cartoon image regions in spatial domain and the sparsity prior of textural image regions in local frequency domain. Using a composite sparse image model, we showed the advantages of the proposed discrete Gabor frames over the existing wavelet frames in several image recovery experiments.

A variational multiscale method for steady natural convection problem based on two-grid discretization

In this paper, we propose and analyze a two-grid variational multiscale method for the steady natural convection problem based on two local Gauss integrations technique. This method possesses the best algorithmic characteristics of both variational multiscale method and two-grid discretization. The main idea is to first solve the nonlinear steady natural convection problem on the coarse grid, then to use the coarse grid solution to fix the nonlinear terms, and to solve a linear problem on the fine grid. Stability and optimal error estimates of the discrete solutions in both one-grid and two-grid variational multiscale formulations are established. Finally, some numerical examples are presented to verify the method’s promise and testify the theoretical predictions.

An adaptive algorithm based on the shifted inverse iteration for the Steklov eigenvalue problem

This paper proposes and analyzes an a posteriori error estimator for the finite element multi-scale discretization approximation of the Steklov eigenvalue problem. Based on the a posteriori error estimates, an adaptive algorithm of shifted inverse iteration type is designed. Finally, numerical experiments comparing the performances of three kinds of different adaptive algorithms are provided, which illustrate the efficiency of the adaptive algorithm proposed here.

A new adaptive mixed finite element method based on residual type a posterior error estimates for the Stokes eigenvalue problem

In this article, we combine mixed finite element method, multiscale discretization, and Rayleigh quotient iteration to propose a new adaptive algorithm based on residual type a posterior error estimates for the Stokes eigenvalue problem. Both reliability and efficiency of the error indicator are proved. The efficiency of the algorithm is also investigated using Chen's Innovation Finite Element Method (iFEM) package. Numerical results are satisfying.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 31–53, 2015

The adaptive finite element method based on multi-scale discretizations for eigenvalue problems

In this paper, adaptive finite element methods for differential operator eigenvalue problems are discussed. For multi-scale discretization schemes based on Rayleigh quotient iteration (see Scheme 3 in [Y. Yang, H. Bi, A two-grid discretization scheme based on shifted-inverse power method, SIAM J. Numer. Anal. 49 (2011) 1602–1624]), a reliable and efficient a posteriori error indicator is given, in addition, a new adaptive algorithm based on the multi-scale discretizations is proposed, and we apply the algorithm to the Schrödinger equation for hydrogen atoms. The algorithm is performed under the package of Chen, and satisfactory numerical results are obtained.

Multiscale Discrete Approximations of Fourier Integral Operators Associated with Canonical Transformations and Caustics

We develop an algorithm for the computation of general Fourier integral operators associated with canonical graphs. The algorithm is based on dyadic parabolic decomposition using wave packets and enables the discrete approximate evaluation of the action of such operators on data in the presence of caustics. The procedure consists in the construction of a universal operator representation through the introduction of locally singularity-resolving diffeomorphisms, enabling the application of wave packet driven computation, and in the construction of the associated pseudo-differential joint-partition of unity on the canonical graphs. We apply the method to a parametrix of the wave equation in the vicinity of a cusp singularity.

Local and Parallel Finite Element Discretizations for Eigenvalue Problems

Based on the work of Xu and Zhou [Math. Comp., 69 (2000), pp. 881--909], this paper combines the local defect-correction technique and the shifted-inverse power method to establish new local and parallel finite element three-scale schemes for a class of eigenvalue problems. It is proved that with these schemes, the solution of an eigenvalue problem on a fine grid $\pi_{h}$ is reduced to the solution of an eigenvalue problem on a much coarser grid $\pi_{H}$, the solution of a linear algebraic system on a globally mesoscopic grid $\pi_{w}$, and the solutions of linear systems on several locally fine grids in parallel. The principle to determine the diameters of three different scale grids is given. Especially, this paper devises a new local and parallel finite element multiscale discretization scheme. Theoretical analysis and numerical experiments show that the computational approach proposed in this paper is simple and easy to carry out and can be used to solve singular eigenvalue problems efficiently.