Piecewise Quadratic Approximation Research Articles

Approximate dynamic programming for constrained linear systems: A piecewise quadratic approximation approach

Approximate dynamic programming (ADP) faces challenges in dealing with constraints in control problems. Model predictive control (MPC) is, in comparison, well-known for its accommodation of constraints and stability guarantees, although its computation is sometimes prohibitive. This paper introduces an approach combining the two methodologies to overcome their individual limitations. The predictive control law for constrained linear quadratic regulation (CLQR) problems has been proven to be piecewise affine (PWA) while the value function is piecewise quadratic. We exploit these formal results from MPC to design an ADP method for CLQR problems with a known model. A novel convex and piecewise quadratic neural network with a local–global architecture is proposed to provide an accurate approximation of the value function, which is used as the cost-to-go function in the online dynamic programming problem. An efficient decomposition algorithm is developed to generate the control policy and speed up the online computation. Rigorous stability analysis of the closed-loop system is conducted for the proposed control scheme under the condition that a good approximation of the value function is achieved. Comparative simulations are carried out to demonstrate the potential of the proposed method in terms of online computation and optimality.

Real-Time Implementation of a Frequency Shifter for Enhancement of Heart Sounds Perception on VLIW DSP Platform

Auscultation of heart sounds is important to perform cardiovascular assessment. External noises may limit heart sound perception. In addition, heart sound bandwidth is concentrated at very low frequencies, where the human ear has poor sensitivity. Therefore, the acoustic perception of the operator can be significantly improved by shifting the heart sound spectrum toward higher frequencies. This study proposes a real-time frequency shifter based on the Hilbert transform. Key system components are the Hilbert transformer implemented as a Finite Impulse Response (FIR) filter, and a Direct Digital Frequency Synthesizer (DDFS), which allows agile modification of the frequency shift. The frequency shifter has been implemented on a VLIW Digital Signal Processor (DSP) by devising a novel piecewise quadratic approximation technique for efficient DDFS implementation. The performance has been compared with other DDFS implementations both considering piecewise linear technique and sine/cosine standard library functions of the DSP. Piecewise techniques allow a more than 50% reduction in execution time compared to the DSP library. Piecewise quadratic technique also allows a more than 50% reduction in total required memory size in comparison to the piecewise linear. The theoretical analysis of the dynamic power dissipation exhibits a more than 20% reduction using piecewise techniques with respect to the DSP library. The real-time operation has been also verified on the DSK6713 rapid prototyping board by Texas Instruments C6713 DSP. Audiologic tests have also been performed to assess the actual improvement of heart sound perception. To this aim, heart sound recordings were corrupted by additive white Gaussian noise, crowded street noise, and helicopter noise, with different signal-to-noise ratios. All recordings were collected from public databases. Statistical analyses of the audiological test results confirm that the proposed approach provides a clear improvement in heartbeat perception in noisy environments.

A non-convex piecewise quadratic approximation of $$\ell _{0}$$ regularization: theory and accelerated algorithm

A non-convex piecewise quadratic approximation of $$\ell _{0}$$ regularization: theory and accelerated algorithm

Sparse and risk diversification portfolio selection.

Portfolio risk management has become more important since some unpredictable factors, such as the 2008 financial crisis and the recent COVID-19 crisis. Although the risk can be actively managed by risk diversification, the high transaction cost and managerial concerns ensue by over diversifying portfolio risk. In this paper, we jointly integrate risk diversification and sparse asset selection into mean-variance portfolio framework, and propose an optimal portfolio selection model labeled as JMV. The weighted piecewise quadratic approximation is considered as a penalty promoting sparsity for the asset selection. The variance associated with the marginal risk regard as another penalty term to diversify the risk. By exposing the feature of JMV, we prove that the KKT point of JMV is the local minimizer if the regularization parameter satisfies a mild condition. To solve this model, we introduce the accelerated proximal gradient (APG) algorithm [Wen in SIAM J. Optim 27:124–145, 2017], which is one of the most efficient first-order large-scale algorithm. Meanwhile, the APG algorithm is linearly convergent to a local minimizer of the JMV model. Furthermore, empirical analysis consistently demonstrate the theoretical results and the superiority of the JMV model.

An accurate and stable numerical method for option hedge parameters

We propose an unconditionally stable numerical algorithm, which uses the Feynman-Kac formula of the Black-Scholes equation to obtain accurate option prices and hedge parameters. We discretize the asset and time using uniform grid points. We approximate the option values by piecewise quadratic polynomials for each time step and integrate them analytically over the log-normal distribution. The piecewise quadratic approximation gives the third-order convergence in the asset direction, and the analytic integration reduces truncation error in the time direction. The estimation errors are propagated backward in time following the convection and diffusion characteristics of the Black-Scholes equation, which assures the unconditional stability of our method. The vectorized code implementation reduces the time complexity. The convergence test shows that our approach outperforms the Crank-Nicolson scheme of the finite difference method in both time and asset directions, and the stability test verifies that our method is stable as the Crank-Nicolson. Furthermore, we show that our algorithm reduces the price errors and hedge parameter errors by more than 50% from the benchmark.

Piecewise Parabolic Approximate Computation Based on an Error-Flattened Segmenter and a Novel Quantizer

This paper proposes a novel Piecewise Parabolic Approximate Computation method for hardware function evaluation, which mainly incorporates an error-flattened segmenter and an implementation quantizer. Under a required software maximum absolute error (MAE), the segmenter adaptively selects a minimum number of parabolas to approximate the objective function. By completely imitating the circuit’s behavior before actual implementation, the quantizer calculates the minimum quantization bit width to ensure a non-redundant fixed-point hardware architecture with an MAE of 1 unit of least precision (ulp), eliminating the iterative design time for the circuits. The method causes the number of segments to reach the theoretical limit, and has great advantages in the number of segments and the size of the look-up table (LUT). To prove the superiority of the proposed method, six common functions were implemented by the proposed method under TSMC-90 nm technology. Compared to the state-of-the-art piecewise quadratic approximation methods, the proposed method has advantages in the area with roughly the same delay. Furthermore, a unified function-evaluation unit was also implemented under TSMC-90 nm technology.

Alternating direction method of multipliers for the optimal siting, sizing, and technology selection of Li-ion battery storage

The present paper addresses the problem of optimal siting, sizing, and technology selection of Li-ion battery storage, employing a mathematical programming approach. The methodology extends the state-of-the-art by considering a nonlinear degradation mechanism of Li-ion batteries from the state of charge, depth of discharge, and storage temperature. Particularly, Mixed-Integer Convex Programming (MICP) problem formulation has been proposed to consider a nonlinear degradation behaviour of Li-ion battery storage with a piecewise quadratic approximation to meet the convexity requirements. The main drawback of the piecewise approximation resides in the inclusion of integer variables what increases a computational burden significantly. For the particular problem, it may easily become intractable with the increase of network size and a number of storage technologies considered. To overcome the issue of tractability and scalability of the combinatorial problem, the MICP problem has been decomposed per energy storage unit and resolved using the Alternating Direction Method of Multipliers. Finally, a sensitivity analysis has been performed within the proposed framework to evaluate the performance value of second-life energy storage solutions.

Computing the L∞-Induced Norm of LTI Systems: Generalization of Piecewise Quadratic and Cubic Approximations

This paper is concerned with performance analysis for bounded persistent disturbances of continuous-time linear time-invariant (LTI) systems. Such an analysis can be done by computing the L ∞ -induced norm of continuous-time LTI systems since it corresponds to the worst maximum magnitude of the output for the worst persistent external input with a unit magnitude. In our preceding study, piecewise constant and linear approximation schemes for analyzing this norm have been developed through two alternative approximation approaches, one for the input and the other for the relevant kernel function, via the fast-lifting technique. The approximation errors in these approximation schemes have been shown to converge to 0 at the rates of 1/N and 1/N 2 , respectively, as the fast-lifting parameter N is increased. Along this line, this paper aims at developing generalized techniques that offer improved accuracy named the piecewise quadratic and cubic approximation schemes. The generalization and the associated accuracy improvement discussed in this paper are not limited to the increased orders of approximation but are extended further to taking advantage of the freedom in the point around which relevant functions are expanded to Taylor series. The approximation errors in the piecewise quadratic and cubic approximation schemes are shown to converge to 0 at the rates of 1/N 3 and 1/N 4 , respectively, regardless of the point at which the Taylor expansion is applied. Finally, effectiveness of the developed computation methods is confirmed through a numerical example.

Superconvergence and the Numerical Flux: a Study Using the Upwind-Biased Flux in Discontinuous Galerkin Methods

One of the beneficial properties of the discontinuous Galerkin method is the accurate wave propagation properties. That is, the semi-discrete error has dissipation errors of order 2k+1 (le Ch^{2k+1}) and order 2k+2 for dispersion (le Ch^{2k+2}). Previous studies have concentrated on the order of accuracy, and neglected the important role that the error constant, C, plays in these estimates. In this article, we show the important role of the error constant in the dispersion and dissipation error for discontinuous Galerkin approximation of polynomial degree k, where k=0,1,2,3. This gives insight into why one may want a more centred flux for a piecewise constant or quadratic approximation than for a piecewise linear or cubic approximation. We provide an explicit formula for these error constants. This is illustrated through one particular flux, the upwind-biased flux introduced by Meng et al., as it is a convex combination of the upwind and downwind fluxes. The studies of wave propagation are typically done through a Fourier ansatz. This higher order Fourier information can be extracted using the smoothness-increasing accuracy-conserving (SIAC) filter. The SIAC filter ties the higher order Fourier information to the negative-order norm in physical space. We show that both the proofs of the ability of the SIAC filter to extract extra accuracy and numerical results are unaffected by the choice of flux.

A new piecewise quadratic approximation approach for L0 norm minimization problem

In this paper, we consider the problem of finding sparse solutions for underdetermined systems of linear equations, which can be formulated as a class of $L_{0}$ norm minimization problem. By using the least absolute residual approximation, we propose a new piecewise quadratic function to approximate the $L_{0}$ norm. Then, we develop a piecewise quadratic approximation (PQA) model where the objective function is given by the summation of a smooth non-convex component and a non-smooth convex component. To solve the (PQA) model, we present an algorithm based on the idea of the iterative thresholding algorithm and derive the convergence and the convergence rate. Finally, we carry out a series of numerical experiments to demonstrate the performance of the proposed algorithm for (PQA). We also conduct a phase diagram analysis to further show the superiority of (PQA) over $L_{1}$ and $L_{1/2}$ regularizations.

Highly accurate computation of Carson formulas based on exponential approximation

In this paper, analytical expressions for a highly accurate computation of Carson formulas are developed. Highly accurate analytical expressions for per-unit-length (pul) self and mutual impedance corrections are now reduced to two linear combinations of 50 functions. Coefficients of these linear combinations are computed using exponential approximation of integrals’ kernel function. Proposed algorithm has simplified computation in comparison with other highly accurate numerical algorithms. Results computed by proposed algorithm and by two approximation methods are compared with 7‐digit accurate results computed by piecewise quadratic approximation for large frequency range.

Education

This issue contains two articles in the Education section. The first one is “An Introduction to Trajectory Optimization: How to Do Your Own Direct Collocation,” by Matthew Kelly. The author presents a tutorial on discretization ideas for optimal control problems with ordinary differential equations. The article starts with a simple example, which is used to introduce some basic notions and a general formulation of an optimal control problem, where the dynamics is described by an ordinary differential equation. That formulation includes general state and control constraints. The main focus is put on methods, which construct a polynomial approximation of the state and control functions. The state and control spaces are discretized, and the infinite-dimensional control problem becomes a finite-dimensional nonlinear optimization problem. Piecewise linear and quadratic approximations of the state and the control are explained and illustrated by figures. Attention is brought to the error accumulated in the numerical calculation of integrals according to several methods. The author offers some practical suggestions for the initialization of the numerical optimization algorithm, as well as for the interpretation of the solution provided by the optimization method. The optimization models obtained as a result of the discretization procedure typically lack convexity properties; this entails that the optimization procedure would provide a local solution, which would not be globally optimal, in general. Some ideas about how to look for a better solution or confirm the quality of the one at hand are provided. Further discussion includes suggestions for smoothing of nondifferentiable functions that may occur in the problem and a comparison of open loop vs. closed loop solutions. The last section of the paper briefly surveys methods based on discretization and numerical treatment of the conditions of optimality. The author has made his software---a general-purpose MATLAB library for solving trajectory optimization problems, complete with documentation---publicly available at https://GitHub.com/MatthewPeterKelly/OptimTraj. The electronic supplement is described in Appendix A. In addition to the optimization software, it includes all example problems, which are presented in the paper. The second article is “How to Mitigate Sloshing,” authored by H. Ockendon and J. R. Ockendon. The authors start with an example from everyday life familiar to most of us: carrying a mug of coffee can often lead to spillage. How should we reduce this danger? The authors devise a model of this dynamical system under certain simplifying assumptions. A point (representing the hand carrying the coffee) is oscillating horizontally with a given frequency and amplitude (representing the walk) about a fixed mean position. A spring connects the hand to the coffee mug, which can slide freely on a smooth table. The effect of that spring on the motion of the liquid is the focus of attention. It is assumed additionally that the mug is rectangular and that no motion perpendicular to the direction of the action of the spring is present. The goal becomes to model and analyze the two-dimensional motion of liquid in a rectangular container of given width and depth. To this end, the authors introduce and analyze a mathematical model based on partial differential equations. Although the article is compact, the authors make a number of suggestions for future investigation and extensions.

Trajectory piecewise quadratic reduced-order model for subsurface flow, with application to PDE-constrained optimization

A new reduced-order model based on trajectory piecewise quadratic (TPWQ) approximations and proper orthogonal decomposition (POD) is introduced and applied for subsurface oil–water flow simulation. The method extends existing techniques based on trajectory piecewise linear (TPWL) approximations by incorporating second-derivative terms into the reduced-order treatment. Both the linear and quadratic reduced-order methods, referred to as POD-TPWL and POD-TPWQ, entail the representation of new solutions as expansions around previously simulated high-fidelity (full-order) training solutions, along with POD-based projection into a low-dimensional space. POD-TPWQ entails significantly more offline preprocessing than POD-TPWL as it requires generating and projecting several third-order (Hessian-type) terms. The POD-TPWQ method is implemented for two-dimensional systems. Extensive numerical results demonstrate that it provides consistently better accuracy than POD-TPWL, with speedups of about two orders of magnitude relative to high-fidelity simulations for the problems considered. We demonstrate that POD-TPWQ can be used as an error estimator for POD-TPWL, which motivates the development of a trust-region-based optimization framework. This procedure uses POD-TPWL for fast function evaluations and a POD-TPWQ error estimator to determine when retraining, which entails a high-fidelity simulation, is required. Optimization results for an oil–water problem demonstrate the substantial speedups that can be achieved relative to optimizations based on high-fidelity simulation.

Piece-wise quadratic approximations of arbitrary error functions for fast and robust machine learning

Most of machine learning approaches have stemmed from the application of minimizing the mean squared distance principle, based on the computationally efficient quadratic optimization methods. However, when faced with high-dimensional and noisy data, the quadratic error functionals demonstrated many weaknesses including high sensitivity to contaminating factors and dimensionality curse. Therefore, a lot of recent applications in machine learning exploited properties of non-quadratic error functionals based on L1 norm or even sub-linear potentials corresponding to quasinorms Lp (0<p<1). The back side of these approaches is increase in computational cost for optimization. Till so far, no approaches have been suggested to deal with arbitrary error functionals, in a flexible and computationally efficient framework. In this paper, we develop a theory and basic universal data approximation algorithms (k-means, principal components, principal manifolds and graphs, regularized and sparse regression), based on piece-wise quadratic error potentials of subquadratic growth (PQSQ potentials). We develop a new and universal framework to minimize arbitrary sub-quadratic error potentials using an algorithm with guaranteed fast convergence to the local or global error minimum. The theory of PQSQ potentials is based on the notion of the cone of minorant functions, and represents a natural approximation formalism based on the application of min-plus algebra. The approach can be applied in most of existing machine learning methods, including methods of data approximation and regularized and sparse regression, leading to the improvement in the computational cost/accuracy trade-off. We demonstrate that on synthetic and real-life datasets PQSQ-based machine learning methods achieve orders of magnitude faster computational performance than the corresponding state-of-the-art methods, having similar or better approximation accuracy.

Retinal oxygen distribution and the role of neuroglobin.

The retina is the tissue layer at the back of the eye that is responsible for light detection. Whilst equipped with a rich supply of oxygen, it has one of the highest oxygen demands of any tissue in the body and, as such, supply and demand are finely balanced. It has been suggested that the protein neuroglobin (Ngb), which is found in high concentrations within the retina, may help to maintain an adequate supply of oxygen via the processes of transport and storage. We construct mathematical models, formulated as systems of reaction-diffusion equations in one-dimension, to test this hypothesis. Numerical simulations show that Ngb may play an important role in oxygen transport, but not in storage. Our models predict that the retina is most susceptible to hypoxia in the regions of the photoreceptor inner segment and inner plexiform layers, where Ngb has the potential to prevent hypoxia and increase oxygen uptake by 30-40%. Analysis of a simplified model confirms the utility of Ngb in transport and shows that its oxygen affinity ([Formula: see text] value) is near optimal for this process. Lastly, asymptotic analysis enables us to identify conditions under which the piecewise linear and quadratic approximations to the retinal oxygen profile, used in the literature, are valid.

Floating Point High Performance Low Area SFU

Objectives: Designing a highly accurate high speed low area Special Function Unit (SFU) is the objective of this work. 32 bit IEEE-754 floating point data format is supported in this system. Methods: The SFU implements elementary functions like inverse, exponential, inverse square root, square root and logarithm accurately. The unit can be utilized in programmable graphics processors where high performance and high accuracy evaluation is needed. The coefficients of the elementary functions are optimized by Genetic algorithm. The simulations were carried out in Xilinx EDA tool and MATLAB. Synthesis reports were taken from Cadence RTL compiler. Findings: Coefficient optimization and extraction is done using genetic algorithm by doing curve fitting with a second degree polynomial. There is a significant reduction around 40% in the area when constraint piecewise quadratic genetic approximation scheme is used. The number of iterations performed in optimization algorithm is 104. The percentage of error is 0.2578 %. The circuits operated at a frequency of 228MHz and the power dissipation was found to be 3.94 mW. This results in a highly accurate SFU. Conclusion: A significant advantage in area when compared to other previous techniques is obtained. The SFU can be utilized in programmable graphics engine.

Fast-rolling shutter compensation based on piecewise quadratic approximation of a camera trajectory

Rolling shutter effect commonly exists in a video camera or a mobile phone equipped with a complementary metal-oxide semiconductor sensor, caused by a row-by-row exposure mechanism. As video resolution in both spatial and temporal domains increases dramatically, removing rolling shutter effect fast and effectively becomes a challenging problem, especially for devices with limited hardware resources. We propose a fast method to compensate rolling shutter effect, which uses a piecewise quadratic function to approximate a camera trajectory. The duration of a quadratic function in each segment is equal to one frame (or half-frame), and each quadratic function is described by an initial velocity and a constant acceleration. The velocity and acceleration of each segment are estimated using only a few global (or semiglobal) motion vectors, which can be simply predicted from fast motion estimation algorithms. Then geometric image distortion at each scanline is inferred from the predicted camera trajectory for compensation. Experimental results on mobile phones with full-HD video demonstrate that our method can not only be implemented in real time, but also achieve satisfactory visual quality.

Isoparametric finite element approximation of Ricci curvature

The surface finite element method can be used to approximate curvatures on embedded hypersurfaces and to discretize geometric partial differential equations. In this paper, we present a definition of discrete Ricci curvature on polyhedral hypersurfaces of arbitrary dimension based on the discretization of a weak formulation with isoparametric finite elements. We prove that for a piecewise quadratic approximation of a two- or three-dimensional hypersurface Γ ⊂ ℝn+1, this definition approximates the Ricci curvature of Γ with a linear order of convergence in the L2 (Γ) norm. By using a smoothing scheme in the case of a piecewise linear approximation of Γ, we still get a convergence of order ⅔ in the L2 (Γ) norm and of order ⅓ in the W1, 2 (Γ) norm.

A BEM for transient thermoelastic analysis of a functionally graded layer on a homogeneous substrate under thermal shock

AbstractTransient thermoelastic analysis of isotropic and linear thermoelastic bimaterials, which are constituted by a functionally graded (FG) layer attached to a homogeneous substrate, subjected to thermal shock is presented in this paper. For this purpose, a boundary element method for transient linear coupled thermoelasticity is developed. The material properties of the FG layer are assumed to be continuous functions of the spatial coordinates. The boundary‐domain integral equations are derived by using the fundamental solutions of linear coupled thermoelasticity for the corresponding isotropic, homogeneous and linear thermoelastic solids in the Laplace‐transformed domain. For the numerical solution, a collocation method with piecewise quadratic approximation is implemented. Numerical results for the dynamic stress intensity factors are presented and discussed. (© 2012 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Piecewise-quadratic Approximations in Convex Numerical Optimization

We present a bundle method for convex nondifferentiable minimization where the model is a piecewise-quadratic convex approximation of the objective function. Unlike standard bundle approaches, the model only needs to support the objective function from below at a properly chosen (small) subset of points, as opposed to everywhere. We provide the convergence analysis for the algorithm, with a general form of master problem which combines features of trust region stabilization and proximal stabilization, taking care of all the important practical aspects such as proper handling of the proximity parameters and the bundle of information. Numerical results are also reported.