Piecewise Constant Interpolation Research Articles

Sobolev-Type Embeddings for Neural Network Approximation Spaces

We consider neural network approximation spaces that classify functions according to the rate at which they can be approximated (with error measured in L^p) by ReLU neural networks with an increasing number of coefficients, subject to bounds on the magnitude of the coefficients and the number of hidden layers. We prove embedding theorems between these spaces for different values of p. Furthermore, we derive sharp embeddings of these approximation spaces into Hölder spaces. We find that, analogous to the case of classical function spaces (such as Sobolev spaces, or Besov spaces) it is possible to trade “smoothness” (i.e., approximation rate) for increased integrability. Combined with our earlier results in Grohs and Voigtlaender (Proof of the theory-to-practice gap in deep learning via sampling complexity bounds for neural network approximation spaces, 2021. arXiv preprint arXiv:2104.02746), our embedding theorems imply a somewhat surprising fact related to “learning” functions from a given neural network space based on point samples: if accuracy is measured with respect to the uniform norm, then an optimal “learning” algorithm for reconstructing functions that are well approximable by ReLU neural networks is simply given by piecewise constant interpolation on a tensor product grid.

Three-dimensional numerical manifold formulation with continuous nodal gradients for dynamics of elasto-plastic porous media

A three-dimensional (3D) numerical manifold formulation with continuous nodal gradients (H8-CNS) is presented for dynamic analysis of saturated elasto-plastic porous media based on the three-variable (u−w−p) formulation. Using meshes of 8-node hexahedra as mathematical patches, the skeleton displacement (u) and fluid velocity (w) interpolations are constructed with a constrained and ortho-normalized least-squares (CO-LS) scheme and a piecewise constant interpolation for fluid pressure field is employed. One feature of H8-CNS is to overcome locking in both limiting cases of low permeability and rigid skeleton. Another feature of H8-CNS is to achieve higher accuracy in stress results for porous media problems involving both elastic and elasto-plastic skeleton behaviors. H8-CNS is also able to fully capture dynamic responses of porous media under high-frequency loading. The accuracy and stability of H8-CNS are verified by performing a variety of numerical simulations. Time integration of H8-CNS is shown to be stable and accurate using the energy balance condition.

Block-centered local refinement methods for the time-fractional equations

In this paper, we present and analyze two block-centered local refinement (BLR) methods for solving the time-fractional equations. The main difference between the two methods is that different approximation methods are used for the value of the pressure at the slave nodes. One adopts a piecewise constant interpolation approximation (PCIA) method, which is called simple block-centered local refinement (S-BLR) method. The other utilizes a piecewise linear interpolation approximation (PLIA) method, which called more accurate block-centered local refinement (MA-BLR) method. The stability analysis is proved carefully. It is estimated that the discrete L2 errors for the velocity and pressure are O(▵t2−α+h3/2) and O(▵t2−α+h3/2) in use of S-BLR and MA-BLR methods, respectively. Where Δt is the time step and h is the maximal mesh size. These error estimate results are all established on locally refinement composite grids. Finally, a numerical experiment is presented to show that the convergence rates are in agreement with the theoretical analysis.

Approximation and application of the Riesz-Caputo fractional derivative of variable order with fixed memory

In this paper, the Riesz-Caputo fractional derivative of variable order with fixed memory is considered. The studied non-integer differential operator is approximated by means of modified basic rules of numerical integration. The three proposed methods are based on polynomial interpolation: piecewise constant, piecewise linear, and piecewise quadratic interpolation. The errors generated by the described methods and the experimental rate of convergence are reported. Finally, an application of the Riesz-Caputo fractional derivative of space-dependent order in continuum mechanics is depicted.

Some applications of Sigmoid functions

In numerical analysis, the process of fitting a function via given data is called interpolation. Interpolation has many applications in engineering and science. There are several formal kinds of interpolation, including linear interpolation, polynomial interpolation, piecewise constant interpolation, trigonometric interpolation, and so on. In this article, by using Sigmoid functions, a new type of interpolation formula is presented. To illustrate the efficiency of the proposed new interpolation formulas, some ap plications in quadrature formulas (in both open and closed types), numerical integration for double integral, and numerical solution of an ordinary differential equation are included. The advantage of this new approach is shown in the numerical applications section.

Multigrid solvers for multipoint flux approximations of the Darcy problem on rough quadrilateral grids

In this work, an efficient blackbox-type multigrid method is proposed for solving multipoint flux approximations of the Darcy problem on logically rectangular grids. The approach is based on a cell-centered multigrid algorithm, which combines a piecewise constant interpolation and the restriction operator by Wesseling/Khalil with a line-wise relaxation procedure. A local Fourier analysis is performed for the case of a Cartesian uniform grid. The method shows a robust convergence for different full tensor coefficient problems and several rough quadrilateral grids.

Algebraically stabilized Lagrange multiplier method for frictional contact mechanics with hydraulically active fractures

Accurate numerical simulation of coupled fracture/fault deformation and fluid flow is crucial to the performance and safety assessment of many subsurface systems. In this work, we consider the discretization and enforcement of contact conditions at such surfaces. The bulk rock deformation is simulated using low-order continuous finite elements, while frictional contact conditions are imposed by means of a Lagrange multiplier method. We employ a cell-centered finite-volume scheme to solve the fracture fluid mass balance equation. From a modeling perspective, a convenient choice is to use a single grid for both mechanical and flow processes, with piecewise-constant interpolation of Lagrange multipliers, i.e., contact tractions and fluid pressure. Unfortunately, this combination of displacement and multiplier variables is not uniformly inf–sup stable, and therefore requires a stabilization technique. Starting from a macroelement analysis, we develop two algebraic stabilization approaches and compare them in terms of robustness and convergence rate. The proposed approaches are validated against challenging analytical two- and three-dimensional benchmarks to demonstrate accuracy and robustness. These benchmarks include both pure contact mechanics problems and well as problems with tightly-coupled fracture flow.

A mixed finite element for the nonlinear analysis of in‐plane loaded masonry walls

SummaryA mixed membrane eight‐node quadrilateral finite element for the analysis of masonry walls is presented. Assuming that a nonlinear and history‐dependent 2D stress‐strain constitutive law is used to model masonry material, the element derivation is based on a Hu‐Washizu variational statement, involving displacement, strain, and stress fields as primary variables. As the behavior of masonry structures is often characterized by strain localization phenomena, due to strain softening at material level, a discontinuous, piecewise constant interpolation of the strain field is considered at element level, to capture highly nonlinear strain spatial distributions also within finite elements. Newton's method of solution is adopted for the element state determination problem. For avoiding pathological sensitivity to the finite element mesh, a novel algorithm is proposed to perform an integral‐type nonlocal regularization of the constitutive equations in the present mixed formulation. By the comparison with competing serendipity displacement‐based formulation, numerical simulations prove high performances of the proposed finite element, especially when coarse meshes are adopted.

A highly efficient time-space-domain optimized method with Lax-Wendroff type time discretization for the scalar wave equation

In this paper, we propose an efficient time-space-domain optimized (OptTS) finite difference scheme to model 2D and 3D scalar wave propagation. It adopts piecewise constant interpolation coefficients for several consecutive Courant number ranges, which avoids the extra time costs caused by loading the coefficients consecutively according to different wave velocities in heterogeneous media. The new scheme is capable of suppressing numerical dispersion errors using a large time interval because we introduce a Lax-Wendroff type time discretization, which utilizes the wave-field information of O(M1M2) neighboring grid points in each step, with merely O(M1+M2) computation, where M1 and M2 are the stencil radii to generate the 2nd-order and 4th-order spatial derivatives. The improved computational efficiency and accuracy of OptTS are validated by substantial experiments using both theoretical analysis and numerical modeling. They show that in heterogeneous models, the maximal Courant numbers reach 1.0 and 0.85 in 2D and 3D cases respectively without severe temporal error, and the scheme is computationally nearly three times faster than its high-order counterpart, providing a powerful tool for large-scale modeling and high-resolution imaging.

An optimized time-space-domain finite difference method with piecewise constant interpolation coefficients for scalar wave propagation

An optimized time-space-domain finite difference method with piecewise constant interpolation coefficients for scalar wave propagation

Block-centred finite difference methods on rectangular composite grids with refinement in space for parabolic equation

ABSTRACTTwo block-centred finite difference schemes with local refinement in space are introduced and analysed to solve parabolic equation. One scheme is simple approximation scheme, in which piecewise constant interpolation approximation is used for slave nodes. The other is more accurate approximation scheme, with piecewise linear interpolation approximation along the interface. Error estimates in the discrete norm are derived of order for simple approximation scheme and of order for more accurate approximation scheme. Numerical experiments using these two schemes show that the convergence rates are in agreement with the theoretical analysis.

Numerical Method for Fractional Advection-Diffusion Equation with Heredity

We propose a method of construction of difference schemes for fractional partial differential equations with delay in time. For the fractional equation with two-sided diffusion, fractional transfer in time, and a functional aftereffect, we construct an implicit difference scheme. We use the shifted Grunwald–Letnikov formulas for the approximation of fractional derivatives with respect to spatial variables and the L1-algorithm for the approximation of fractional derivatives in time. Also we use piecewise constant interpolation and extrapolation by extending the discrete prehistory of the model in time. The algorithm is a fractional analog of a purely implicit method; on each time step, it is reduced to the solution of linear algebraic systems. We prove the stability of the method and find its order of convergence.

14T Full Adder in 125nm CMOS technology for FFT applications using piecewise constant interpolation f(x) = 2.0

14T Full Adder in 125nm CMOS technology for FFT applications using piecewise constant interpolation f(x) = 2.0

Numerical discretization for fractional differential equations with feedback control

Abstract In this paper, we introduce a numerical scheme for fractional differential equations with feedback control. Due to the possibility of dealing with a feedback control as a functional delay, we construct a numerical method based on Euler method accompanied with piecewise constant interpolation. The method is based on the idea of separating the current state and the prehistory function. The convergence of the method is stated and proved. Numerical experiments are given to clarify the good agreement between numerical and theoretical results.

Error expansion of piecewise constant interpolation rule for certain two-dimensional Cauchy principal value integrals

The classical composite rectangle (midpoint) rule for the computation of two dimensional singular integrals is discussed, with the error functional of rectangle rule for computing two dimensional singular integrals, the local coordinate of certain point and the convergence results O(h2) are obtained. When the local coordinate is coincided with certain priori known coordinates, we get the convergence rate the same as the Riemann integral at certain point. At last, numerical examples are presented to illustrate our theoretical analysis which agree with it very well.

Methods of approximation influence aquatic ecosystem metabolism estimates

Aquatic ecologists have recently employed dynamic models to estimate aquatic ecosystem metabolism. All approaches involve numerically solving a differential equation describing dissolved oxygen (DO) dynamics. Although the DO differential equation can be solved accurately with linear multistep or Runge–Kutta methods, less accurate methods, such as the Euler method, have been applied. The methods also differ in how discrete temperature and light measurements are used to drive DO dynamics. Here, we used a representative stream DO data set to compare the metabolism estimates generated by multiple Euler based methods and an accurate numerical method. We also compared metabolism estimates using linear, piecewise constant and smoothing spline interpolation of light and temperature. Using observed DO to calculate DO saturation deficit in the Euler method results in a substantial difference in metabolism estimates compared to all other methods. If modeled DO is used to calculate DO saturation deficit, the Euler method introduces smaller error in metabolism estimates, which diminishes as logging interval decreases. Linear and smoothing spline interpolation result in similar metabolism estimates, but differ from estimates based on piecewise constant interpolation. We demonstrate how different computational methods imply distinct assumptions about process and observation error, and conclude that under the assumption of observation error, the best practice is to use the accurate numerical method of solving differential equation with a continuous interpolation of light and temperature. The Euler method will introduce minimal error if it is paired with frequently logged data and DO saturation deficit is computed using modeled DO.

A composite mixed finite element model for plane structural problems

The paper proposes a mixed finite element model and experiments its capability in the analysis of elastic and plastic collapse plane problems. The model is easy to formulate and implement because it is based on simple assumptions for the unknown fields. A composite triangular mesh is assumed over the domain. Within each triangular element the displacement field is described by a quadratic interpolation, while the stress field is represented by a piece-wise constant description by introducing a subdivision of the element into three triangular regions. The elastic solution is constructed through the stationarity condition of the mixed Hellinger–Reissner functional. The plastic collapse analysis is formulated as a mathematical programming problem and is accomplished by an Interior Point algorithm which furnishes both the collapse multiplier and the collapse mechanism. A series of numerical experiments shows that the proposed model performs well in elastic problems and achieves the favorite context in plastic analysis, where it takes advantage of the absence of volumetric locking and the piecewise constant interpolation of the stress field within the element.

A computational framework for polyconvex large strain elasticity

This paper presents a novel computational formulation for large strain polyconvex elasticity. The formulation, based on the original ideas introduced by Schröder et al. (2011), introduces the deformation gradient (the fibre map), its adjoint (the area map) and its determinant (the volume map) as independent kinematic variables of a convex strain energy function. Compatibility relationships between these variables and the deformed geometry are enforced by means of a multi-field variational principle with additional constraints. This process allows the use of different approximation spaces for each variable. The paper extends the ideas presented in Schröder et al. (2011) by introducing conjugate stresses to these kinematic variables which can be used to define a generalised convex complementary energy function and a corresponding complementary energy principle of the Hellinger–Reissner type, where the new conjugate stresses are primary variables together with the deformed geometry. Both compressible and incompressible or nearly incompressible elastic models are considered. A key element to the developments presented in the paper is the new use of a tensor cross product, presented for the first time by de Boer (1982), page 76, which facilitates the algebra associated with the adjoint of the deformation gradient. For the numerical examples, quadratic interpolation of the displacements, piecewise linear interpolation of strain and stress fields and piecewise constant interpolation of the Jacobian and its stress conjugate are considered for compressible cases. In the case of incompressible materials two formulations are presented. First, continuous quadratic interpolation for the displacement together with piecewise constant interpolation for the pressure and second, linear continuous interpolation for both displacement and pressure stabilised via a Petrov–Galerkin technique.

A GPU accelerated aggregation algebraic multigrid method

We present an efficient, robust and fully GPU-accelerated aggregation-based algebraic multigrid preconditioning technique for the solution of large sparse linear systems. These linear systems arise from the discretization of elliptic PDEs. The method involves two stages, setup and solve. In the setup stage, hierarchical coarse grids are constructed through aggregation of the fine grid nodes. These aggregations are obtained using a set of maximal independent nodes from the fine grid nodes. We use a “fine-grain” parallel algorithm for finding a maximal independent set from a graph of strong negative connections. The aggregations are combined with a piece-wise constant (unsmooth) interpolation from the coarse grid solution to the fine grid solution, ensuring low setup and interpolation cost. The grid independent convergence is achieved by using recursive Krylov iterations (K-cycles) in the solve stage. An efficient combination of K-cycles and standard multigrid V-cycles is used as a preconditioner for Krylov iterative solvers such as generalized minimal residual and conjugate gradient. We compare the solver performance with other solvers based on smooth aggregation and classical algebraic multigrid methods.

P 0-approximation of $${\Delta(H_0^2 \cap W^{3,\infty})}$$ Δ ( H 0 2 ∩ W 3 , ∞ ) on square grids based on interior squares

We will approximate a function in \({\Delta(H_0^2 \cap W^{3,\infty})}\) with a piecewise constant function on the square grids, whose degrees of freedom are based on the interior squares only. In spite of the slight lack of degrees of freedom, the approximation error is analyzed to be O(h) in L2-norm, which is the optimal order for the piecewise-constant interpolation. The result in the paper is relevant to the finite element approximation for the divergence-free velocity in the Stokes problem with the locally divergence-free P1-nonconforming space on the square grids.