Affine Interpolation Research Articles

Neural Data–Enabled Predictive Control

Data–enabled predictive control (DeePC) for linear systems utilizes data matrices of recorded trajectories to directly predict new system trajectories, which is very appealing for real-life applications. In this paper we leverage the universal approximation properties of neural networks (NNs) to develop neural DeePC algorithms for nonlinear systems. Firstly, we point out that the outputs of the last hidden layer of a deep NN implicitly construct a basis in a so-called neural (feature) space, while the output linear layer performs affine interpolation in the neural space. As such, we can train of-line a deep NN using large data sets of trajectories to learn the neural basis and compute on-line a suitable affine interpolation using DeePC. Secondly, methods for guaranteeing consistency of neural DeePC and for reducing computational complexity are developed. Several neural DeePC formulations are illustrated on a nonlinear pendulum example.

Construction of Continuous and Piecewise Affine Feedback Stabilizers for Nonlinear Systems

In this article, two methods for constructing continuous and piecewise affine (CPA) feedback stabilizers for nonlinear systems are presented. First, a construction based on a piecewise affine interpolation of Sontag's “universal” formula is developed. Stability of the corresponding closed-loop system is verified a posteriori by means of a CPA control Lyapunov function and subsequently solving a feasibility problem. Second, we develop a procedure for computing CPA feedback stabilizers via linear programming, which allows for the optimization of a control-oriented criterion in the synthesis procedure. Stability conditions are a priori specified in the linear program, which removes the necessity for a posteriori verification of closed-loop stability. We illustrate the developed methods via two application-inspired examples considering the stabilization of an inverted pendulum and the stabilization of a healthy equilibrium of the hypothalamic-pituitary-adrenal axis.

A quasicontinuum theory for the nonlinear mechanical response of general periodic truss lattices

We present a framework for the efficient, yet accurate description of general periodic truss networks based on concepts of the quasicontinuum (QC) method. Previous research in coarse-grained truss models has focused either on simple bar trusses or on two-dimensional beam lattices undergoing small deformations. Here, we extend the truss QC methodology to nonlinear deformations, general periodic beam lattices, and three dimensions. We introduce geometric nonlinearity into the model by using a corotational beam description at the level of individual truss members. Coarse-graining is achieved by the introduction of representative unit cells and an affine interpolation analogous to traditional QC. General periodic lattices defined by the periodic assembly of a single unit cell are modeled by retaining all unique degrees of freedom of the unit cell (identified by a lattice decomposition into simple Bravais lattices) at each macroscopic point in the simulation, and interpolating each degree of freedom individually. We show that this interpolation scheme accurately captures the homogenized properties of periodic truss lattices for uniform deformations. In order to showcase the efficiency and accuracy of the method, we perform simulations to predict the brittle fracture toughness of multiple lattice architectures and compare them to results obtained from significantly more expensive discrete truss simulations. Finally, we demonstrate the applicability of the method for nonlinear elastic truss lattices undergoing finite deformations. Overall, the new technique shows convincing agreement with exact, discrete results for most lattice architectures, and offers opportunities to reduce computational expenses in structural lattice simulations and thus to efficiently extract the effective mechanical performance of discrete networks.

Compensated Convexity Methods for Approximations and Interpolations of Sampled Functions in Euclidean Spaces: Applications to Contour Lines, Sparse Data, and Inpainting

This paper is concerned with applications of the theory of approximation and interpolation based on compensated convex transforms developed in [K. Zhang, E. Crooks, and A. Orlando, SIAM J. Math. Anal., 48 (2016), pp. 4126--4154]. We apply our methods to (i) surface reconstruction starting from the knowledge of finitely many level sets (or “contour lines"); (ii) scattered data approximation; (iii) image inpainting. For (i) and (ii) our methods give interpolations. For the case of finite sets (scattered data), in particular, our approximations provide a natural triangulation and piecewise affine interpolation. Prototype examples of explicitly calculated approximations and inpainting results are presented for both finite and compact sets. We also show numerical experiments for applications of our methods to high density salt & pepper noise reduction in image processing, for image inpainting, and for approximation and interpolations of continuous functions sampled on finitely many level sets and on scattered points.

Approximation in Sobolev spaces by piecewise affine interpolation

Functions in a Sobolev space are approximated directly by piecewise affine interpolation in the norm of the space. The proof is based on estimates for interpolations and does not rely on the density of smooth functions.

Discrete-time chaotic-map truly random number generators: design, implementation, and variability analysis of the zigzag map

In this paper, we introduce a novel discrete chaotic map named zigzag map that demonstrates excellent chaotic behaviors and can be utilized in truly random number generators (TRNGs). We comprehensively investigate the map and explore its critical chaotic characteristics and parameters. We further present two circuit implementations for the zigzag map based on the switched current technique as well as the current-mode affine interpolation of the breakpoints. In practice, implementation variations can deteriorate the quality of the output sequence as a result of variation of the chaotic map parameters. In order to quantify the impact of variations on the map performance, we model the variations using a combination of theoretical analysis and Monte-Carlo simulations on the circuits. We demonstrate that even in the presence of the map variations, a TRNG based on the zigzag map passes all of the NIST 800-22 statistical randomness tests using simple post processing of the output data.

On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality

This work is intended to provide a convenient tool for the mathematical analysis of a particular kind of finite volume approximation which can be used, for instance, in the context of nonlinear and/or anisotropic diffusion operators in 3D. Following the approach developed by F. Hermeline and by K.~Domelevo and P. Omnes in 2D, we consider a ``double'' covering $\Tau$ of a three-dimensional domain by a rather general primal mesh and by a well-chosen ``dual'' mesh. The associated discrete divergence operator $\div^{\ptTau}$ is obtained by the standard finite volume approach. A simple and consistent discrete gradient operator $\grad^\ptTau$ is defined by local affine interpolation that takes into account the geometry of the double mesh. Under mild geometrical constraints on the choice of the dual volumes, we show that $-\div^{\ptTau}$, $\grad^\ptTau$ are linked by the ``discrete duality property'', which is an analogue of the integration-by-parts formula. The primal mesh need not be conformal, and its interfaces can be general polygons. We give several numerical examples for anisotropic linear diffusion problems; good convergence properties are observed. The sequel [3] of this paper will summarize some key discrete functional analysis tools for DDFV schemes and give applications to proving convergence of DDFV schemes for several nonlinear degenerate parabolic PDEs.

The two-well problem with surface energy

Let Ω be a bounded Lipschitz domain in R2, let H be a 2 × 2 diagonal matrix with det(H) = 1. Let ε > 0 and consider the functional over AF ∩ W2,1(Ω), where AF is the class of functions from Ω satisfying affine boundary condition F. It can be shown by convex integration that there exists F ∉ SO(2) ∪ SO(2)H and u ∈ AF with I0(u) = 0. Let 0 < ζ1 < 1 < ζ2 < ∞, .In this paper we begin the study of the asymptotics of mε ≔ infBF∩W2,1Iε for such F. This is one of the simplest minimization problems involving surface energy for which we can hope to see the effects of convex integration solutions. The only known lower bounds are lim infε→0mε/ε = ∞.We link the behaviour of mε to the minimum of I0 over a suitable class of piecewise affine functions. Let {τi} be a triangulation of Ω by triangles of diameter less than h and let denote the class of continuous functions that are piecewise affine on a triangulation {τi}. For the function u ∈ BF let be the interpolant, i.e. the function we obtain by defining ũ⌊τi to be the affine interpolation of u on the corners of τi. We show that if for some small ω > 0 there exists u ∈ BF ∩ W2,1 with then, for h = ε(1+6399ω)/3201, the interpolant satisfies I0(ũ) ≤ h1−cω.Note that it is trivial that , so we reduce the problem of non-trivial (scaling) lower bounds on mε/ε to the problem of non-trivial lower bounds on .

A numerical approach to variational problems subject to convexity constraint

We describe an algorithm to approximate the minimizer of an elliptic functional in the form \(\int_\Omega j(x, u, \nabla u)\) on the set \({\cal C}\) of convex functions u in an appropriate functional space X. Such problems arise for instance in mathematical economics [4]. A special case gives the convex envelope \(u_0^{**}\) of a given function \(u_0\). Let \((T_n)\) be any quasiuniform sequence of meshes whose diameter goes to zero, and \(I_n\) the corresponding affine interpolation operators. We prove that the minimizer over \({\cal C}\) is the limit of the sequence \((u_n)\), where \(u_n\) minimizes the functional over \(I_n({\cal C})\). We give an implementable characterization of \(I_n({\cal C})\). Then the finite dimensional problem turns out to be a minimization problem with linear constraints.