Nonuniform Partitions Research Articles

Machine learning discovery of optimal quadrature rules for isogeometric analysis

We propose the use of machine learning techniques to find optimal quadrature rules for the construction of stiffness and mass matrices in isogeometric analysis (IGA). We initially consider 1D spline spaces of arbitrary degree spanned over uniform and non-uniform knot sequences, and then the generated optimal rules are used for integration over higher-dimensional spaces using tensor products. The quadrature rule search is posed as an optimization problem and solved by a machine learning strategy based on adaptive gradient-descent. However, since the optimization space is highly non-convex, the success of the search strongly depends on the number of quadrature points and the parameter initialization. Thus, we use a dynamic programming strategy that initializes the parameters from the optimal solution over the spline space with a lower number of knots. With this method, we found optimal quadrature rules for spline spaces when using IGA discretizations with up to 50 uniform elements and polynomial degrees up to 8, showing the generality of the approach in this scenario. For non-uniform partitions, the method also finds an optimal rule in a reasonable number of test cases. We also assess the generated optimal rules in two practical case studies, namely, the eigenvalue problem of the Laplace operator and the eigenfrequency analysis of freeform curved beams, where the latter problem shows the applicability of the method to curved geometries. In particular, the proposed method results in savings with respect to traditional Gaussian integration of up to 44% in 1D, 68% in 2D, and 82% in 3D spaces.

Non-Uniform Spline Quasi-Interpolation to Extract the Series Resistance in Resistive Switching Memristors for Compact Modeling Purposes

An advanced new methodology is presented to improve parameter extraction in resistive memories. The series resistance and some other parameters in resistive memories are obtained, making use of a two-stage algorithm, where the second one is based on quasi-interpolation on non-uniform partitions. The use of this latter advanced mathematical technique provides a numerically robust procedure, and in this manuscript, we focus on it. The series resistance, an essential parameter to characterize the circuit operation of resistive memories, is extracted from experimental curves measured in devices based on hafnium oxide as their dielectric layer. The experimental curves are highly non-linear, due to the underlying physics controlling the device operation, so that a stable numerical procedure is needed. The results also allow promising expectations in the massive extraction of new parameters that can help in the characterization of the electrical device behavior.

Effects of non-uniform porosity on thermochemical performance of solar driven methane reforming

As one of the effective methods for solving the greenhouse problem, methane reforming has received increasing attention. The performance of a reactor is strongly influenced by the foam structure with different parameters. In this study, the idea of non-uniform partition design of porosity (an expansion kind of hierarchical porous structure) in a porous media reactor was put forward and investigated. The heat and mass transfer phenomena and thermochemical properties of methane reforming under high convergence solar radiation were numerically studied. Finite volume method coupled with thermochemical kinetics is considered to solve this problem. Based on a constant pore size, the porosity of the porous medium is divided into two layers and combined in different forms in an axial direction. The effects of non-uniform porosity on thermochemical performance of solar driven methane reforming are analyzed and an optimization method for non-uniform porosity is proposed. The results show that methane conversion in a reactor with uniform porous media increases with growth up of the porosity. In addition, a reactor with non-uniform porous media exhibits better thermochemical performance and reforming performance compared to one with uniform porous media. Finally, different non-uniform partitions have an impact on methane conversion. When the porosity is constant, methane conversion decreases with an increase of the length of the first layer of porous media, and the maximum methane conversion (82.42%) was achieved at L1 = 0.01 m with φ1 = 0.75, φ2 = 0.9.

Preserving vorticity improves the accuracy of multiscale solvers

Within reservoir management, there is a growing demand for highly accurate and fast reservoir simulators. So-called multiscale methods are designed to preserve fine-scale information accurately in an accelerated coarse-scale computation of fluid flow. In many cases, this requires that coarse partitions are meticulously adapted to prominent geological features that determine flow paths. Partitions that seek to preserve the vorticity from a representative fine-scale flow simulation can be particularly effective to this end. We describe how to incorporate such partitions in the state-of-the-art multiscale restriction-smoothed basis (MsRSB) method, and present a series of two-phase test cases to validate and demonstrate the effectiveness of the resulting method. Our results show that using a nonuniform vorticity-based partition improves the accuracy of the MsRSB solver compared with uniform partitions and nonuniform partitions with a similar number of coarse blocks that adapt to permeability or velocity.

An Adaptive Triangular Partition Algorithm for Digital Images

The partition algorithm as a digital image processing technique is significant to many applications, such as data encryption, image denoising, and 3-D reconstruction. In order to achieve well partition that can availably reduce the distortion phenomenon, a novel approach named image adaptive triangular partition (IATP) is proposed, which considers the grayscale distribution of the image and removes the shared edges between the adjacent triangles in the partition mesh. The least-squares method is used to fit the sampled position-associated gray value of the image to determine whether further partition should be performed, that is, if the sum of squared residuals is bigger than the preselected control value, the current area will be divided into four separated sub-triangles by using the self-similar method, and then preparing the next fitting on each of them in recursion; otherwise, the terminal operation is reached. When the recursive partition of the image is done, the triangular partition mesh with the quaternary notations is obtained. The experimental results demonstrate that the performance of the IATP algorithm proposed in this paper is better than the existing state-of-the-art nonuniform partitions, and it solves the redundant coding problem and reduces the image quality losses. In addition, two applications–image steganography and information encryption–are selected to verify that the proposed algorithm has good feasibility and robustness.

Securing RSA against power analysis attacks through non‐uniform exponent partitioning with randomisation

This study presents an approach to compute randomised modular exponentiation through non-uniform exponent partitioning. The exponent has been first partitioned into multiple parts and then shuffled by Fisher Yates method. Thereafter, every partition randomly computes modular exponentiation followed by a final modulo operation to generate the desired result. The shuffling has been introduced to randomise the execution order of individual modular exponentiation. This work is implemented in Rivest-Shamir-Adleman (RSA) and Chinese remainder theorem RSA as they are modular exponentiation based public key cryptosystems. The results have been analysed during decryption with different key sizes. The results indicate that the proposed work can generate non-uniform partitions of the exponent which could not be easily anticipated even in multiple iterations. Also, the shuffling method could completely randomise the execution order of modular exponentiation operations. With non-uniform exponent partitions and randomised modular exponentiation, the proposed work could challenge all the variances of power analysis attacks.

Nonuniform abstractions, refinement and controller synthesis with novel BDD encodings

This paper presents a control synthesis algorithm for dynamical systems to satisfy specifications given in a fragment of linear temporal logic. It is based on an abstraction-refinement scheme with nonuniform partitions of the state space. A novel encoding of the resulting transition system is proposed that uses binary decision diagrams for efficiency. We discuss several factors affecting scalability and present some benchmark results demonstrating the effectiveness of the new encodings. These ideas are also being implemented on a publicly available prototype tool, ARCS, that we briefly introduce in the paper.

Second order approximations for kinetic and potential energies in Maxwell's wave equations

In this paper we propose a numerical scheme for wave type equations with damping and space variable coefficients. Relevant equations of this kind arise for instance in the context of Maxwell's equations, namely, the electric potential equation and the electric field equation. The main motivation to study such class of equations is the crucial role played by the electric potential or the electric field in enhanced drug delivery applications. Our numerical method is based on piecewise linear finite element approximation and it can be regarded as a finite difference method based on non-uniform partitions of the spatial domain. We show that the proposed method leads to second order convergence, in time and space, for the kinetic and potential energies with respect to a discrete L2-norm.

Alternating direction implicit orthogonal spline collocation on some non-rectangular regions with inconsistent partitions

The alternating direction implicit (ADI) method is a highly efficient technique for solving multi-dimensional dependent initial-boundary value problems on rectangles. Earlier we have used the ADI technique in conjunction with orthogonal spline collocation (OSC) for discretization in space to solve parabolic problems on rectangles and rectangular polygons. Recently, we extended applications of ADI OSC schemes to the solution of parabolic problems on some non-rectangular regions that allow for consistent nonuniform partitions. However, for many regions, it is impossible to construct such partitions. Therefore, in this paper, we show how to extend our approach further to solve parabolic problems on some non-rectangular regions using inconsistent uniform partitions. Numerical results are presented using piecewise Hermite cubic polynomials for spatial discretizations and our ADI OSC scheme for parabolic problems to demonstrate its performance on several regions.

Automatic Domain Partitioning of Piecewise-Affine Simplicial Functions Implementing Model Predictive Controllers

This brief is concerned with the problem of mitigating the “curse of dimensionality” in the piecewise-affine simplicial approximation of multivariate model predictive control (MPC) functions by resorting to nonuniform domain partitions, in view of their implementation in high-speed embedded digital architectures. Two dual strategies are used to reduce the number of simplices with respect to uniform partitions, without reducing approximation accuracy. Both circuit implementation purposes and the MPC technique impose additional constraints, with respect to other techniques already available for multiresolution domain partitioning. The algorithms are tested on a benchmark example concerned with adaptive cruise control.

Time-dependent diffusion in tubes with periodic partitions

The presence of obstacles leads to a slowdown of diffusion. We study the slowdown when diffusion occurs in a tube, and obstacles are periodically spaced identical partitions with circular apertures of arbitrary radius in their centers. The mean squared displacement of a particle diffusing in such a system at large times is given by ⟨Δx2(t)⟩=2Defft, t→∞, where Deff is the effective diffusion coefficient, which is smaller than the particle diffusion coefficient in the tube with no partitions, D0. The latter characterizes the short-time behavior of the mean squared displacement, ⟨Δx2(t)⟩=2D0t, t→0. Thus, the particle diffusion coefficient decreases from D0 to Deff as time goes from zero to infinity. We derive analytical solutions for the Laplace transforms of the time-dependent diffusion coefficient and the mean squared displacement that show how these functions depend on the geometric parameters of the tube. To obtain these solutions we replace nonuniform partitions with apertures by effective partitions that are uniformly permeable for diffusing particles. Our choice of the partition permeability is based on the recent result for the corresponding effective trapping rate obtained by means of boundary homogenization. To establish the range of applicability of our approximate theory we compare its predictions with the results found in Brownian dynamics simulations. Comparison shows excellent agreement between the two at arbitrary value of the aperture radius when the tube radius does not exceed the interpartition distance.

Near-Best Univariate Spline Discrete Quasi-Interpolants on Nonuniform Partitions

The univariate spline quasi-interpolants (QIs) studied in this paper are approximation operators using B-spline expansions with coefficients that are linear combinations of discrete values of the function to be approximated. When working with nonuniform partitions, the main challenge is to find QIs that have both good approximation orders and uniform norms which are bounded independently of the given partition. Near-best QIs are obtained by minimizing an upper bound for the infinity norm of QIs depending on a certain number of free parameters, thus reducing this norm. This paper is devoted to the study of some families of near-best discrete quasi-interpolants (dQIs) of approximation order 3.

A functional calculus approach for the rational approximation with nonuniform partitions

By introducing $M$-functional calculus we generalize some results of Brenner and Thomée on the stability and convergence of rational approximation schemes of bounded semigroups for nonuniform time steps. We give also the rate of convergence for the approximation of the time derivative of these semigroups.

A quadrature formula associated with a univariate spline quasi interpolant

We study a new simple quadrature rule based on integrating a C 1 quadratic spline quasi-interpolant on a bounded interval. We give nodes and weights for uniform and non-uniform partitions. We also give error estimates for smooth functions and we show that our formula is a useful companion to Simpson’s rule.

Recovery type superconvergence and a posteriori error estimates for control problems governed by Stokes equations

In this paper, we derive recovery type superconvergence analysis and a posteriori error estimates for the finite element approximation of the distributed optimal control governed by Stokes equations. We obtain superconvergence results and asymptotically exact a posteriori error estimates by applying two recovery methods, which are the patch recovery technique and the least-squares surface fitting method. Our results are based on some regularity assumption for the Stokes control problems and are applicable to the first order conforming finite element method with regular but nonuniform partitions.

Optimized Local Trigonometric Bases with Nonuniform Partitions

The authors provide optimized local trigonometric bases with nonuniform partitions which efficiently compress trigonometric functions. Numerical examples demonstrate that in many cases the proposed bases provide better compression than the optimized bases with uniform partitions obtained by Matviyenko.

Optimal Quadratic and Cubic Spline Collocation on Nonuniform Partitions

We develop optimal quadratic and cubic spline collocation methods for solving linear second-order two-point boundary value problems on non-uniform partitions. To develop optimal nonuniform partition methods, we use a mapping function from uniform to nonuniform partitions and develop expansions of the error at the nonuniform collocation points of some appropriately defined spline interpolants. The existence and uniqueness of the spline collocation approximations are shown, under some conditions. Optimal global and local orders of convergence of the spline collocation approximations and derivatives are derived, similar to those of the respective methods for uniform partitions. Numerical results on a variety of problems, including a boundary-layer problem, and a nonlinear problem, verify the optimal convergence of the methods, even under more relaxed conditions than those assumed by theory.

Remarks on the rational approximation of holomorphic semigroups with nonuniform partitions

Some convergence and stability results for rational approximations of holomorphic semigroups are presented. The approximations are related to time discrete schemes with nonuniform stepsize to a linear evolution equation of parabolic type in a Banach space. Our theorems cover not only the backward Euler but also the Crank-Nicolson and forward Euler schemes.

A Hybrid Collocation Method for Volterra Integral Equations with Weakly Singular Kernels

The commonly used graded piecewise polynomial collocation method for weakly singular Volterra integral equations may cause serious round-off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time-dependent equations to round-off errors. The singularity preserving (nonpolynomial) collocation method is known to have only local convergence. To overcome the shortcoming of these well-known methods, we introduce a hybrid collocation method for solving Volterra integral equations of the second kind with weakly singular kernels. In this hybrid method we combine a singularity preserving (nonpolynomial) collocation method used near the singular point of the derivative of the solution and a graded piecewise polynomial collocation method used for the rest of the domain. We prove the optimal order of global convergence for this method. The convergence analysis of this method is based on a singularity expansion of the exact solution of the equations. We prove that the solutions of such equations can be decomposed into two parts, with one part being a linear combination of some known singular functions which reflect the singularity of the solutions and the other part being a smooth function. A numerical example is presented to demonstrate the effectiveness of the proposed method and to compare it to the graded collocation method.

Superconvergence of nonconforming finite element method for the Stokes equations

AbstractThis article derives a general superconvergence result for nonconforming finite element approximations of the Stokes problem by using a least‐squares surface fitting method proposed and analyzed recently by Wang for the standard Galerkin method. The superconvergence result is based on some regularity assumption for the Stokes problem and is applicable to any nonconforming stable finite elements with regular but nonuniform partitions. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 143–154, 2002; DOI 10.1002/num.1036