Hardware-Based Activation Function-Core for Neural Network Implementations

Today, embedded systems (ES) tend towards miniaturization and the carrying out of complex tasks in applications such as the Internet of Things, medical systems, telecommunications, among others. Currently, ES structures based on artificial intelligence using hardware neural networks (HNNs) are becoming more common. In the design of HNN, the activation function (AF) requires special attention due to its impact on the HNN performance. Therefore, implementing activation functions (AFs) with good performance, low power consumption, and reduced hardware resources is critical for HNNs. In light of this, this paper presents a hardware-based activation function-core (AFC) to implement an HNN. In addition, this work shows a design framework for the AFC that applies a piecewise polynomial approximation (PPA) technique. The designed AFC has a reconfigurable architecture with a wordlength-efficient decoder, i.e., reduced hardware resources are used to satisfy the desired accuracy. Experimental results show a better performance of the proposed AFC in terms of hardware resources and power consumption when it is compared with state of the art implementations. Finally, two case studies were implemented to corroborate the AFC performance in widely used ANN applications.

Bridging the Multiscale Hybrid-Mixed and Multiscale Hybrid High-Order methods

We establish the equivalence between the Multiscale Hybrid-Mixed (MHM) and the Multiscale Hybrid High-Order (MsHHO) methods for a variable diffusion problem with piecewise polynomial source term. Under the idealized assumption that the local problems defining the multiscale basis functions are exactly solved, we prove that the equivalence holds for general polytopal (coarse) meshes and arbitrary approximation orders. We also leverage the interchange of properties to perform a unified convergence analysis, as well as to improve on both methods.

Limit Cycles from Hopf Bifurcation in Nongeneric Quadratic Reversible Systems with Piecewise Perturbations

In this paper, we consider a nongeneric quadratic reversible system with piecewise polynomial perturbations. We use the expansion of the first order Melnikov function to obtain the maximal number of small-amplitude limit cycles produced by Hopf bifurcation in the perturbed systems.

Construction of a normalized basis of a univariate quadratic $C^1$ spline space and application to the quasi-interpolation

In this paper, we use the finite element method to construct a new normalized basis of a univariate quadratic $C^1$ spline space. We give a new representation of Hermite interpolant of any piecewise polynomial of class at least $C^1$ in terms of its polar form. We use this representation for constructing several superconvergent and super-superconvergent discrete quasi-interpolants which have an optimal approximation order. This approach is simple and provides an interesting approximation. Numerical results are given to illustrate the theoretical ones.

Stable broken 𝐻(𝑐𝑢𝑟𝑙) polynomial extensions and 𝑝-robust a posteriori error estimates by broken patchwise equilibration for the curl–curl problem

We study extensions of piecewise polynomial data prescribed in a patch of tetrahedra sharing an edge. We show stability in the sense that the minimizers over piecewise polynomial spaces with prescribed tangential component jumps across faces and prescribed piecewise curl in elements are subordinate in the broken energy norm to the minimizers over the broken H ( curl ) \boldsymbol H(\boldsymbol {\operatorname {curl}}) space with the same prescriptions. Our proofs are constructive and yield constants independent of the polynomial degree. We then detail the application of this result to the a posteriori error analysis of the curl–curl problem discretized with Nédélec finite elements of arbitrary order. The resulting estimators are reliable, locally efficient, polynomial-degree-robust, and inexpensive. They are constructed by a broken patchwise equilibration which, in particular, does not produce a globally H ( curl ) \boldsymbol H(\boldsymbol {\operatorname {curl}}) -conforming flux. The equilibration is only related to edge patches and can be realized without solutions of patch problems by a sweep through tetrahedra around every mesh edge. The error estimates become guaranteed when the regularity pick-up constant is explicitly known. Numerical experiments illustrate the theoretical findings.

Primary-space Adaptive Control Variates Using Piecewise-polynomial Approximations

We present an unbiased numerical integration algorithm that handles both low-frequency regions and high-frequency details of multidimensional integrals. It combines quadrature and Monte Carlo integration by using a quadrature-based approximation as a control variate of the signal. We adaptively build the control variate constructed as a piecewise polynomial, which can be analytically integrated, and accurately reconstructs the low-frequency regions of the integrand. We then recover the high-frequency details missed by the control variate by using Monte Carlo integration of the residual. Our work leverages importance sampling techniques by working in primary space, allowing the combination of multiple mappings; this enables multiple importance sampling in quadrature-based integration. Our algorithm is generic and can be applied to any complex multidimensional integral. We demonstrate its effectiveness with four applications with low dimensionality: transmittance estimation in heterogeneous participating media, low-order scattering in homogeneous media, direct illumination computation, and rendering of distribution effects. Finally, we show how our technique is extensible to integrands of higher dimensionality by computing the control variate on Monte Carlo estimates of the high-dimensional signal, and accounting for such additional dimensionality on the residual as well. In all cases, we show accurate results and faster convergence compared to previous approaches.

Number of Limit Cycles from a Class of Perturbed Piecewise Polynomial Systems

In this paper, we study Poincaré bifurcation of a class of piecewise polynomial systems, whose unperturbed system has a period annulus together with two invariant lines. The main concerns are the number of zeros of the first order Melnikov function and the estimation of the number of limit cycles which bifurcate from the period annulus under piecewise polynomial perturbations of degree [Formula: see text].