Piecewise Polynomial Approximation Research Articles

Lower bounds for piecewise polynomial approximations of oscillatory functions

We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when the polynomial degree is fixed. These lower bounds, for example, apply when approximating solutions to Helmholtz plane wave scattering problem.

Incorporating slow NMDA-type receptors with nonlinear voltage-dependent magnesium block in a next generation neural mass model: derivation and dynamics

We derive a next generation neural mass model of a population of quadratic-integrate-and-fire neurons, with slow adaptation, and conductance-based AMPAR, GABAR and nonlinear NMDAR synapses. We show that the Lorentzian ansatz assumption can be satisfied by introducing a piece-wise polynomial approximation of the nonlinear voltage-dependent magnesium block of NMDAR current. We study the dynamics of the resulting system for two example cases of excitatory cortical neurons and inhibitory striatal neurons. Bifurcation diagrams are presented comparing the different dynamical regimes as compared to the case of linear NMDAR currents, along with sample comparison simulation time series demonstrating different possible oscillatory solutions. The omission of the nonlinearity of NMDAR currents results in a shift in the range (and possible disappearance) of the constant high firing rate regime, along with a modulation in the amplitude and frequency power spectrum of oscillations. Moreover, nonlinear NMDAR action is seen to be state-dependent and can have opposite effects depending on the type of neurons involved and the level of input firing rate received. The presented model can serve as a computationally efficient building block in whole brain network models for investigating the differential modulation of different types of synapses under neuromodulatory influence or receptor specific malfunction.

Compact: Approximating Complex Activation Functions for Secure Computation

Secure multi-party computation (MPC) techniques can be used to provide data privacy when users query deep neural network (DNN) models hosted on a public cloud. State-of-the-art MPC techniques can be directly leveraged for DNN models that use simple activation functions (AFs) such as ReLU. However, these techniques are ineffective and/or inefficient for the complex and highly non-linear AFs used in cutting-edge DNN models. We present Compact, which produces piece-wise polynomial approximations of complex AFs to enable their efficient use with state-of-the-art MPC techniques. Compact neither requires nor imposes any restriction on model training and results in near-identical model accuracy. To achieve this, we design Compact with input density awareness, and use an application specific simulated annealing type optimization to generate computationally more efficient approximations of complex AFs. We extensively evaluate Compact on four different machine-learning tasks with DNN architectures that use popular complex AFs silu, gelu, and mish. Our experimental results show that Compact incurs negligible accuracy loss while being 2x-5x computationally more efficient than state-of-the-art approaches for DNN models with large number of hidden layers. Our work accelerates easy adoption of MPC techniques to provide user data privacy even when the queried DNN models consist of a number of hidden layers, and trained over complex AFs.

Optimal Piecewise Polynomial Approximation for Minimum Computing Cost by Using Constrained Least Squares.

In this paper, the optimal approximation algorithm is proposed to simplify non-linear functions and/or discrete data as piecewise polynomials by using the constrained least squares. In time-sensitive applications or in embedded systems with limited resources, the runtime of the approximate function is as crucial as its accuracy. The proposed algorithm searches for the optimal piecewise polynomial (OPP) with the minimum computational cost while ensuring that the error is below a specified threshold. This was accomplished by using smooth piecewise polynomials with optimal order and numbers of intervals. The computational cost only depended on polynomial complexity, i.e., the order and the number of intervals at runtime function call. In previous studies, the user had to decide one or all of the orders and the number of intervals. In contrast, the OPP approximation algorithm determines both of them. For the optimal approximation, computational costs for all the possible combinations of piecewise polynomials were calculated and tabulated in ascending order for the specific target CPU off-line. Each combination was optimized through constrained least squares and the random selection method for the given sample points. Afterward, whether the approximation error was below the predetermined value was examined. When the error was permissible, the combination was selected as the optimal approximation, or the next combination was examined. To verify the performance, several representative functions were examined and analyzed.

Minimax Regret Robust Screening with Moment Information

Problem definition: We study a robust screening problem where a seller attempts to sell a product to a buyer knowing only the moment and support information of the buyer’s valuation distribution. The objective is to maximize the competitive ratio relative to an optimal hindsight policy equipped with full valuation information. Methodology/results: We formulate the robust screening problem as a linear programming problem, which can be solved efficiently if the support of the buyer’s valuation is finite. When the support of the buyer’s valuation is continuous and the seller knows the mean and the upper and lower bounds of the support for the buyer’s valuation, we show that the optimal payment is a piecewise polynomial function of the valuation with a degree of at most two. Moreover, we derive the closed-form competitive ratio corresponding to the optimal mechanism. The optimal mechanism can be implemented by a randomized pricing mechanism whose price density function is a piecewise inverse function adjusted by a constant. When the mean and variance are known to the seller, we propose a feasible piecewise polynomial approximation of the optimal payment function with a degree of at most three. We also demonstrate that the optimal competitive ratio exhibits a logarithmic decay with respect to the coefficient of variation of the buyer’s valuation distribution. Managerial implications: Our general framework provides an approach to investigating the value of moment information in the robust screening problem. We establish that even a loose upper bound of support or a large variance can guarantee a good competitive ratio. Funding: The research of S. Wang is partially supported by the National Natural Science Foundation of China [Grant 72394395]. The research of S. Liu is partly supported by the National Natural Science Foundation of China [Grant NSFC-72072117]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0072 .

Piecewise polynomial approximation of spatial curvilinear profiles using the Hough transform

Given a curvilinear profile P represented as a set of points in the space R3 and four families of low-degree polynomial curves that respectively depend on the parameters in the space R4, our goal is to identify the piecewise space polynomial curve best fitting the profile P. We use a parametric representation of the space curves and subdivide the profile into smaller portions that can be fitted with regular curves. We provide theoretical guarantees to the existence of such an approximation and an algorithm for the profile approximation. We take advantage of the implicit function theorem to locally project a space curve on at most two planes and to locally recognise it with a low-degree polynomial curve obtained by applying the Hough transform. Finally, we recombine the curve expressions on the two planes backwards in the space R3. The outcome of the algorithm is thus a piecewise polynomial curve approximating the profile. We validate our approach to approximate curvilinear profiles extracted from 3D point clouds representing real objects and to simplify and resample point clouds.

QPA: A Quantization-Aware Piecewise Polynomial Approximation Methodology for Hardware-Efficient Implementations

Piecewise polynomial approximation (PPA) on nonlinear functions plays an important role in high-precision computing. In this article, we proposed QPA, an integration of error-flattened quantization-aware PPA methods, to generate the optimized coefficients for efficient hardware implementations targeting any polynomial order. QPA incorporated four key features to minimize the fitting error and the hardware cost, including using the Remez algorithm to compute the minimax fitting polynomial, combining the fitting and quantization operations to get an error-flattened characteristic, assigning specific coefficient bit width to each multiplier to reduce the hardware cost, and fine-tuning the truncated coefficients to further reduce the fitting error. Experimental results showed that our methods consistently achieved the lowest fitting error compared with the state-of-the-art error-flattened piecewise approximation methods. We synthesized the proposed designs with 28-nm TSMC CMOS technology. The results showed that the proposed designs achieved up to 37.0% area reduction and 50.5% power consumption reduction compared to the state-of-the-art error-flattened piecewise linear (PWL) method, and up to 27.0% area reduction, 21.4% delay reduction, and 20.8% power consumption reduction compared to the state-of-the-art error-flattened piecewise quadratic (PWQ) method.

Higher-order time domain boundary elements for elastodynamics: graded meshes and hp versions

The solution to the elastodynamic equation in the exterior of a polyhedral domain or a screen exhibits singular behavior from the corners and edges. The detailed expansion of the singularities implies quasi-optimal estimates for piecewise polynomial approximations of the Dirichlet trace of the solution and the traction. The results are applied to hp and graded versions of the time domain boundary element method for the weakly singular and the hypersingular integral equations. Numerical examples confirm the theoretical results for the Dirichlet and Neumann problems for screens and for polygonal domains in 2d. They exhibit the expected quasi-optimal convergence rates and the singular behavior of the solutions.

Basic Concept of the Beam Wave based Element for Mid and High Frequency Analysis

This paper describes a Hermite beam wave based element of the steady-state dynamic response of a 1D structure system. This study focuses on the development of beam wave based elements. Compared with standard piecewise polynomial approximation, the wave basis is shown to give a considerable reduction in computational degrees of freedom. In practical terms, it is concluded that the degrees of freedom for which accurate results can be obtained, using these new techniques, can be up to half of that of the conventional finite-element method.

Accuracy Estimates for Bilinear Optimal Control Problems Governed by Ordinary Differential Equations

First- and second-order accuracy estimates for an optimal control problem governed by a system of ordinary differential equations with a bilinear control mechanism are presented. The numerical time discretization scheme under consideration is the finite element method with continuous piecewise linear functions. Central to this work is a first- and second-order analysis of optimality of the continuous and approximated optimal control problems. In the case of box constraints on the control, first-order error estimates for the control function are obtained assuming a piecewise constant approximation of the control, whereas second-order accuracy can be obtained in the case of a continuous, piecewise polynomial approximation. Numerical evidence is presented that supports the theoretical findings.

Efficient Parallel Optimization for Approximating CAD Curves Featuring Super-convergence

We present an efficient, parallel, constrained optimization technique for approximating CAD curves with super-convergent rates. The optimization function is a disparity measure in terms of a piece-wise polynomial approximation and a curve re-parametrization. The constrained problem solves the disparity functional fixing the mesh element interfaces. We have numerical evidence that when the solver attains a minimum, the constrained disparity preserves the original super-convergence: 2p order for planar curves and ⌊32(p−1)⌋+2 for 3D curves, p being the mesh polynomial degree. Our optimization scheme consists of a globalized Newton method with a nonmonotone line search, and a log barrier function preventing element inversion in the curve re-parametrization. Moreover, we introduce a Julia interface to the EGADS geometry kernel and a parallel optimization algorithm. We test the potential of our curve mesh generation tool on a computer cluster using several aircraft CAD models. We conclude that the solver is well-suited for parallel computing, producing super-convergent approximations to CAD curves.

An energy-based discontinuous Galerkin method with tame CFL numbers for the wave equation

We extend and analyze the energy-based discontinuous Galerkin method for second order wave equations on staggered and structured meshes. By combining spatial staggering with local time-stepping near boundaries, the method overcomes the typical numerical stiffness associated with high order piecewise polynomial approximations. In one space dimension with periodic boundary conditions and suitably chosen numerical fluxes, we prove bounds on the spatial operators that establish stability for CFL numbers \(c \frac{\Delta t}{h} < C\) independent of order when stability-enhanced explicit time-stepping schemes of matching order are used. For problems on bounded domains and in higher dimensions we demonstrate numerically that one can march explicitly with large time steps at high order temporal and spatial accuracy.

Dual least‐squares finite element method with stabilization

AbstractThe LL*‐method is a least‐squares finite element approach producing an approximation by solving dual problem corresponding to the given partial differential equations. Due to the unique structure of LL* approximation, it has advantages if the problem has low regularities and when L2‐approximation needs to be established. As a drawback, piecewise polynomial type approximation often generates artifacts such as spurious oscillations near where shocks or discontinuities occur in solution. Allowing discontinuous piecewise polynomial approximation in LL* seems to exacerbate this trouble. This paper presents a stabilized LL*‐method that is designed to effectively reduce these oscillatory behavior. The consistency and error convergence of proposed method are analyzed and numerical examinations are performed.

COMPUTATIONAL FEATURES OF CALCULAT-ING EXTENDED CONDUCTOR ANTENNAS BASED ON THE INTEGRAL EQUATIONS METHOD

The numerical solution of integral equations or a system of integral equations (for a system of conductors) consists in their discretization and reduction to a system of linear algebraic equations for the desired function. For discretization, it is possible to use projection methods, for example, the Galerkin method or the collocation method. A system of linear algebraic equations for the class of problems under consideration is characterized by a complete (filled) complex matrix. For conductor antenna structures that are quite arbitrary in geometry and size, systems of linear algebraic equations turn out to be of a high order, and special computational algorithms are required to solve them. The purpose of the work is to study adequate mathematical models for wire antennas with a subsequent description of uniform numerical algorithms based on methods of sampling and approximation of antenna current. With this method, the error in solving integral equations is determined by the error in calculating the elements of the matrix of a system of linear algebraic equations for a given piecewise polynomial approximation of the solution at step h. If the quadrature formula for numerical integration is chosen, then there are two ways to reduce the solution error. An increase in the number of collocation points leads to a rapid increase in the volume of calculations and, consequently, to a rapid increase in the amount of occupied computer memory. Therefore, the question arises of choosing the most profitable method of piecewise polynomial interpolation and discretization step h from the point of view of using computer resources, ensuring an acceptable number of solutions.

Model-Based Reinforcement Learning via Stochastic Hybrid Models

Optimal control of general nonlinear systems is a central challenge in automation. Enabled by powerful function approximators, data-driven approaches to control have recently successfully tackled challenging applications. However, such methods often obscure the structure of dynamics and control behind black-box over-parameterized representations, thus limiting our ability to understand closed-loop behavior. This article adopts a hybrid-system view of nonlinear modeling and control that lends an explicit hierarchical structure to the problem and breaks down complex dynamics into simpler localized units. We consider a sequence modeling paradigm that captures the temporal structure of the data and derive an expectation-maximization (EM) algorithm that automatically decomposes nonlinear dynamics into stochastic piecewise affine models with nonlinear transition boundaries. Furthermore, we show that these time-series models naturally admit a closed-loop extension that we use to extract local polynomial feedback controllers from nonlinear experts via behavioral cloning. Finally, we introduce a novel hybrid relative entropy policy search (Hb-REPS) technique that incorporates the hierarchical nature of hybrid models and optimizes a set of time-invariant piecewise feedback controllers derived from a piecewise polynomial approximation of a global state-value function.

Immersed boundary Mindlin-Reissner 3D shell element for modeling isotropic and laminated composite shells

An immersed boundary 3D shell element is presented here that is based on Mindlin-Reissner shell theory assumptions and uses quadratic B-spline approximation for the solution. It enables mesh independent analysis wherein the surface representing the shell geometry is defined independently and not by the mesh. The shell geometry is immersed in a uniform background mesh whose elements are 3D B-spline shell elements. By making the geometric model independent of the mesh, the typical difficulties associated with mesh generation can be avoided. The quadratic 3D B-spline shell elements in the mesh can represent the displacement field as a tangent continuous piece-wise polynomial approximation. The nodes of the element have three translational and no rotational degrees of freedom. The element is formulated under small-deformation and plane stress assumptions. Constructing the element based on shell theory enables the use of effective properties to model thin fiber reinforced laminated composite shell-like structures. The element formulation and the mesh independent approach are validated against a series of common shell element benchmark problems. • Immersed boundary 3D B-spline element based on Mindlin-Reissner shell theory assumptions. • Mesh independent analysis where geometry is embedded in a uniform background mesh. • The iso-parametric quadratic B-spline shell elements are used for C 1 continuous approximation. • Step boundary method (SBM) is used for applying boundary conditions. • Immersed boundary FEM for shells is validated for isotropic and composite shells.

Efficient function approximation on general bounded domains using splines on a Cartesian grid

Functions on a bounded domain in scientific computing are often approximated using piecewise polynomial approximations on meshes that adapt to the shape of the geometry. We study the problem of function approximation using splines on a regular but oversampled grid that is defined on a bounding box. This approach allows the use of high-order and highly structured splines as a basis for piecewise polynomials. The methodology is analogous to that of Fourier extensions, using Fourier series on a bounding box, which leads to spectral accuracy for smooth functions. However, Fourier extension approximations involve solving a highly ill-conditioned linear system, and this is an expensive step. The computational complexity of recent algorithms is $\mathcal {O}\left (N\log ^{2}\left (N\right )\right )$ in 1-D and $\mathcal {O}\left (N^{2}\log ^{2}\left (N\right )\right )$ in 2-D. We show that, compared to Fourier extension, the compact support of B-splines enables improved complexity for multivariate approximations, namely $\mathcal {O}(N)$ in 1-D, $\mathcal {O}\left (N^{3/2}\right )$ in 2-D and more generally $\mathcal {O}\left (N^{3(d-1)/d}\right )$ in d-D with d > 1. By using a direct sparse QR solver for a related linear system, we also observe that the computational complexity can be nearly linear in practice. This comes at the cost of achieving only algebraic rates of convergence. Our statements are corroborated with numerical experiments and Julia code is available.

Dual Channel Multiplier for First-Order Piecewise Approximation for GPU

This paper demonstrates an energy-efficient implementation of piecewise polynomial approximation to evaluate the complex functions of graphical processing units (GPUs). A novel approach is employed to implement the first-order piecewise approximation serially which leads to high savings in power and area. Dual channel multiplier (DCM) scheme is proposed to simplify the hardware architecture of the piecewise polynomial approximation. The proposed methodology is implemented with 90[Formula: see text]nm CMOS technology and it can perform different complex functions using a simple multiplier hardware structure. DCM achieves improvement in energy saving by up to 81% at a penalty cost of 10 clock cycles. Simulation results confirm that this work attains at least 83%, and 55% saving in power and area as compared to the traditional techniques, respectively. The proposed methodology can implement any degree of the piecewise polynomial approximation utilizing the DCM.

Formal Verification of Transcendental Fixed- and Floating-point Algorithms using an Automatic Theorem Prover

We present a method for formal verification of transcendental hardware and software algorithms that scales to higher precision without suffering an exponential growth in runtimes. A class of implementations using piecewise polynomial approximation to compute the result is verified using MetiTarski, an automated theorem prover, which verifies a range of inputs for each call. The method was applied to commercial implementations from Cadence Design Systems with significant runtime gains over exhaustive testing methods and was successful in proving that the expected accuracy of one implementation was overly optimistic. Reproducing the verification of a sine implementation in software, previously done using an alternative theorem-proving technique, demonstrates that the MetiTarski approach is a viable competitor. Verification of a 52-bit implementation of the square root function highlights the method’s high-precision capabilities.

Regularity results and numerical solution by the discontinuous Galerkin method to semilinear parabolic initial boundary value problems with nonlinear Newton boundary conditions in a polygonal space-time cylinder

Abstract In this note we consider a parabolic evolution equation in a polygonal space-time cylinder. We show, that the elliptic part is given by a m-accretive mapping from L q (Ω) → L q (Ω). Therefore we can apply the theory of nonlinear semigroups in Banach spaces in order to get regularity results in time and space. The second part of the paper deals with the numerical solution of the problem. It is dedicated to the application of the space-time discontinuous Galerkin method (STDGM). It means that both in space as well as in time discontinuous piecewise polynomial approximations of the solution are used. We concentrate to the theoretical analysis of the error estimation.