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Showing papers on "Linear approximation published in 2013"


Journal ArticleDOI
TL;DR: In this article, a non-probabilistic reliability model is given for structures with convex model uncertainty, which is defined as a ratio of the multidimensional volume falling into the reliability domain to the one of the whole model.

174 citations


Journal ArticleDOI
TL;DR: It is shown that for certain classes of admissible inputs, the existence of an ISS-Lyapunov function implies the ISS of a system, and it is proved a linearization principle that allows a construction of a local ISS- Lyap unov function for a system.
Abstract: We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs, the existence of an ISS-Lyapunov function implies the ISS of a system. Then for the case of systems described by abstract equations in Banach spaces, we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system, the linear approximation of which is ISS. In order to study the interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov function for an entire interconnection, if ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate the theory on examples of linear and semilinear reaction-diffusion equations.

172 citations


Proceedings ArticleDOI
01 Dec 2013
TL;DR: A linear method for global camera pose registration from pair wise relative poses encoded in essential matrices that minimizes an approximate geometric error to enforce the triangular relationship in camera triplets and produces good accuracy, robustness, and outperforms some well-known systems on efficiency.
Abstract: We present a linear method for global camera pose registration from pair wise relative poses encoded in essential matrices. Our method minimizes an approximate geometric error to enforce the triangular relationship in camera triplets. This formulation does not suffer from the typical `unbalanced scale' problem in linear methods relying on pair wise translation direction constraints, i.e. an algebraic error, nor the system degeneracy from collinear motion. In the case of three cameras, our method provides a good linear approximation of the trifocal tensor. It can be directly scaled up to register multiple cameras. The results obtained are accurate for point triangulation and can serve as a good initialization for final bundle adjustment. We evaluate the algorithm performance with different types of data and demonstrate its effectiveness. Our system produces good accuracy, robustness, and outperforms some well-known systems on efficiency.

163 citations


Journal ArticleDOI
TL;DR: In this article, a multi-geometry and multi-physics model is developed for a Li-ion battery module which includes three cells connected in series by electrical busbars, and the model can be used to predict the 3D profiles of the electrical potentials and temperature in the battery.

87 citations


Journal ArticleDOI
TL;DR: A new hybrid algorithm is developed to use the optimal solution of a mixed-integer linear program as a starting point when solving a nonlinear formulation of the problem, and always finds a solution when the linear model is feasible while still taking into account the nonlinear nature of the Problem.
Abstract: The approach presented in this paper aims at finding a solution to the problem of conflict-free motion planning for multiple aircraft on the same flight level with trajectory recovery. One contribution of this work is to develop three consistent models, i.e., from a continuous-time representation to a discrete-time linear approximation. Each of these models guarantees separation at all times and trajectory recovery, but they are not equally difficult to solve. A new hybrid algorithm is thus developed to use the optimal solution of a mixed-integer linear program as a starting point when solving a nonlinear formulation of the problem. The significance of this process is that it always finds a solution when the linear model is feasible while still taking into account the nonlinear nature of the problem. A test bed containing numerous data sets is then generated from three virtual scenarios. A comparative analysis with three different initializations of nonlinear optimization validates the efficiency of the hybrid method.

57 citations


01 Jan 2013
TL;DR: In this paper, sampling variances of non-linear functions are determined as the variance of their first order Taylor series expansions, which is used to obtain sampling errors for estimates of heritabilities and correlations and these quantities can be computed with most software performing such analyses.
Abstract: Approximate lower bound sampling errors of maximum likelihood estimates of covariance components and their linear functions can be obtained from the inverse of the information matrix. For non-linear functions, sampling variances are commonly determined as the variance of their first order Taylor series expansions. This is used to obtain sampling errors for estimates of heritabilities and correlations, and these quantities can be computed with most software performing such analyses. In other instances, however, more complicated functions are of interest or the linear approximation is difficult or inadequate. A pragmatic alternative then is to evaluate sampling characteristics by repeated sampling of parameters from their asymptotic, multivariate normal distribution, calculating the function(s) of interest for each sample and inspecting the distribution across replicates. This paper demonstrates the use of this approach and examines the quality of approximation obtained.

57 citations


Proceedings ArticleDOI
23 Jun 2013
TL;DR: A Fast Trust Region (FTR) approach for optimization of segmentation energies with non-linear regional terms, which are known to be challenging for existing algorithms, and is 1-2 orders of magnitude faster than the existing state-of-the-art methods while converging to comparable or better solutions.
Abstract: Trust region is a well-known general iterative approach to optimization which offers many advantages over standard gradient descent techniques. In particular, it allows more accurate nonlinear approximation models. In each iteration this approach computes a global optimum of a suitable approximation model within a fixed radius around the current solution, a.k.a. trust region. In general, this approach can be used only when some efficient constrained optimization algorithm is available for the selected non-linear (more accurate) approximation model. In this paper we propose a Fast Trust Region (FTR) approach for optimization of segmentation energies with non-linear regional terms, which are known to be challenging for existing algorithms. These energies include, but are not limited to, KL divergence and Bhattacharyya distance between the observed and the target appearance distributions, volume constraint on segment size, and shape prior constraint in a form of L2 distance from target shape moments. Our method is 1-2 orders of magnitude faster than the existing state-of-the-art methods while converging to comparable or better solutions.

52 citations


Journal ArticleDOI
TL;DR: In this paper, a generalization of the concept of Lyapunov exponents for discrete linear systems is proposed, which may be used in the case of unbounded coefficients.
Abstract: In this article we propose a generalization of the concept of Lyapunov exponents for discrete linear systems which may be used in the case of unbounded coefficients. We show some simplest properties of this generalization and apply it to define a generalization of regular system. Finally, we discuss the problem of stability by linear approximation. † This article was originally published with an error. This version has been corrected. Please see Corrigendum (http://dx.doi.org/10.1080/14689367.2012.756700)

51 citations


Journal ArticleDOI
TL;DR: An algorithm introduced recently by Kolda, Lewis, and Torczon for linearly constrained derivative-free optimization is employed for this purpose and under usual assumptions, convergence to stationary points is proved.
Abstract: A new method is introduced for solving constrained optimization problems in which the derivatives of the constraints are available but the derivatives of the objective function are not. The method is based on the inexact restoration framework, by means of which each iteration is divided in two phases. In the first phase one considers only the constraints, in order to improve feasibility. In the second phase one minimizes a suitable objective function subject to a linear approximation of the constraints. The second phase must be solved using derivative-free methods. An algorithm introduced recently by Kolda, Lewis, and Torczon for linearly constrained derivative-free optimization is employed for this purpose. Under usual assumptions, convergence to stationary points is proved. A computer implementation is described and numerical experiments are presented.

43 citations


Journal ArticleDOI
TL;DR: A solution to the optimization problem having a significantly reduced complexity with respect to existing techniques is provided and numerical results show the merits of the proposed approach for typical smart microgrid scenarios.
Abstract: In a smart microgrid currents injected by distributed energy resources (DERs) and by the point of common coupling can be adapted to minimize the energy cost. Design and quality constraints usually make the problem grow fast with the number of nodes in the network. In this paper we provide a solution to the optimization problem having a significantly reduced complexity with respect to existing techniques. The efficiency of the proposed solution stems by modeling the smart microgrid as a linear network where loads are approximated as impedances. This simplification allows avoiding explicit use of power flow equations, and having a number of equation proportional to the number of DERs rather than to the total number of nodes (loads and DERs). The optimal power flow problem is then solved by a semidefinite programming (SDP) relaxation, which provides the initial point for the search of a feasible solution by a sequential convex programming procedure based on a local linear approximation of non-convex constraints. Numerical results show the merits of the proposed approach for typical smart microgrid scenarios.

42 citations


Journal ArticleDOI
TL;DR: In this paper, the authors propose a nonlinear infinite moving average as an alternative to the standard state space policy function for solving nonlinear DSGE models and derive the third order approximation explicitly, examine the accuracy of the method using Euler equation tests and compare with state space approximations.

Journal ArticleDOI
TL;DR: In this article, a semi-analytical method for milling stability prediction based on linear acceleration approximation is presented, and the results verify the validity of the proposed approach and the presented method is proved to be computationally highly efficient.
Abstract: Chatter stability predictions catch much attention during machining operations in modern automotive and aerospace industry. This paper presents a novel time domain semi-analytical method for milling stability prediction based on linear acceleration approximation. Firstly, the milling dynamics considering the regenerative effect is presented as a linear time-delay system with periodic coefficients. The second step is to equally discretize the time duration of the forced vibration of the tooth passing period into small intervals where acceleration of the flexible cutter is approximated by linearly interpolating between the two boundary values, while the free vibration is analytically solved. Then, recursive formulas with constant recursive matrices are found for the presentation of relations between initial and final cutter motions (including position, velocity and acceleration) of each small time interval. Employing the method of weighted residuals over each time interval, discrete maps are constructed which relate motions of a period to the corresponding values one period earlier. Finally, the eigenvalues of the transition matrix are used to determine stability based on Floquet theory. By using the benchmark examples in literatures, the convergence and computational time of the proposed method are compared with those of the semi-discretization methods (SDMs), full-discretization method (FDM) and numerical integration method (NIM). The results verify the validity of the proposed approach, and the presented method is proved to be computationally highly efficient.

Proceedings ArticleDOI
21 Jul 2013
TL;DR: In this paper, an effective piecewise linear (PWL) approximation technique is introduced which shows promising performance in linearizing the nonlinear functions.
Abstract: Nonlinear functions are often encountered in power system optimizations. In this paper, an effective piecewise linear (PWL) approximation technique is introduced which shows promising performance in linearizing the nonlinear functions. This method uses a series of linear functions, called max-affine functions, to linearize a multivariate function over a bounded domain. The important advantage of this method is its ability to decide on the size of the subspaces, which other methods are not capable of. It is also shown that using the PWL approximation, significant efficiency is achievable in computation burden of most power system optimizations, such as unit commitment.

Book ChapterDOI
01 Jan 2013

Journal ArticleDOI
TL;DR: This article provides an almost characterization of the approximation classes appearing when using adaptive finite elements of Lagrange type of any fixed polynomial degree in terms of Besov regularity.
Abstract: We provide an almost characterization of the approximation classes appearing when using adaptive finite elements of Lagrange type of any fixed polynomial degree. The characterization is stated in terms of Besov regularity, and requires the approximation within spaces with integrability indices below one. This article generalizes to higher order finite elements the results presented for linear finite elements by Binev et. al. [BDDP 2002].

Journal ArticleDOI
TL;DR: In this article, the authors analyzed the stability of extended objects of a $Q$-ball type with piecewise parabolic potential in ($3+1$)- and ($1+ 1$)-dimensional space-times.
Abstract: Explicit solutions for extended objects of a $Q$-ball type were found analytically in a model describing complex scalar field with piecewise parabolic potential in ($3+1$)- and ($1+1$)-dimensional space-times. Such a potential provides a variety of solutions which were thoroughly examined. It was shown that, depending on the values of the parameters of the model and according to the known stability criteria, there exist stable and unstable solutions. The classical stability of solutions in ($1+1$)-dimensional space-time was examined in the linear approximation and it was shown explicitly that the spectrum of linear perturbations around some solutions contains exponentially growing modes while it is not so for other solutions.

Journal ArticleDOI
TL;DR: In this article, a one-phase supercooled Stefan problem, with a nonlinear relation between the phase change temperature and front velocity, is analyzed and the results show that for large supercooling the linear model may be highly inaccurate and even qualitatively incorrect.

Journal ArticleDOI
TL;DR: In this article, the controllability problem of a nonlinear fractional order discrete-time system is solved using the Riemann-Liouville, Caputo and Grünwald-Letnikov fractional-order difference operators.
Abstract: The Riemann-Liouville, Caputo and Grünwald-Letnikov fractional order difference operators are discussed and used to state and solve the controllability problem of a nonlinear fractional order discrete-time system. It is shown that independently of the type of fractional order difference, such a system is locally controllable in q steps if its linear approximation is globally controllable in q steps.

Journal ArticleDOI
TL;DR: In this article, the authors proposed a reduced adjoint approach to variational data assimilation based on proper orthogonal decomposition (POD) which avoids the implementation of the adjoint of the tangent linear approximation of the original nonlinear model.

Journal ArticleDOI
TL;DR: The primary objectives of this article are to highlight the challenges in the estimation of the model components, to compare two approximations to the logistic regression function, linear and exponential, and to discuss their advantages and limitations.
Abstract: We provide insights into new methodology for the analysis of multilevel binary data observed longitudinally, when the repeated longitudinal measurements are correlated. The proposed model is logistic functional regression conditioned on three latent processes describing the within- and between-variability, and describing the cross-dependence of the repeated longitudinal measurements. We estimate the model components without employing mixed-effects modeling but assuming an approximation to the logistic link function. The primary objectives of this article are to highlight the challenges in the estimation of the model components, to compare two approximations to the logistic regression function, linear and exponential, and to discuss their advantages and limitations. The linear approximation is computationally efficient whereas the exponential approximation applies for rare events functional data. Our methods are inspired by and applied to a scientific experiment on spectral backscatter from long range infrared light detection and ranging (LIDAR) data. The models are general and relevant to many new binary functional data sets, with or without dependence between repeated functional measurements.

Proceedings ArticleDOI
12 Jun 2013
TL;DR: A linear controller in an outer loop provides the global stability, showing good regulation and disturbance rejection performance and the effectiveness of the design is shown.
Abstract: Underactuated systems are usually represented by nonlinear models and their control requires the design of nonlinear controllers. The flywheel inverted pendulum is a special case, also nonlinear, but admitting a linear approximation around the unstable equilibrium point in the upper position. In this paper, a linear controller is designed in two steps. First, the output is controlled with a simple PID controller, leading to an internally unstable plant. Then, a second linear controller in an outer loop provides the global stability, showing good regulation and disturbance rejection performance. Experimental results show the effectiveness of the design.

Journal ArticleDOI
TL;DR: An adaptive LASSO penalized least squares approach to estimating the index parameter and the unknown function in single-index models for continuous outcome and results indicate that the proposed methods perform well under a variety of circumstances and that an assumption of monotonicity, when appropriate, noticeably improves performance.
Abstract: We consider the problem of variable selection for monotone single-index models. A single-index model assumes that the expectation of the outcome is an unknown function of a linear combination of covariates. Assuming monotonicity of the unknown function is often reasonable and allows for more straightforward inference. We present an adaptive LASSO penalized least squares approach to estimating the index parameter and the unknown function in these models for continuous outcome. Monotone function estimates are achieved using the pooled adjacent violators algorithm, followed by kernel regression. In the iterative estimation process, a linear approximation to the unknown function is used, therefore reducing the situation to that of linear regression and allowing for the use of standard LASSO algorithms, such as coordinate descent. Results of a simulation study indicate that the proposed methods perform well under a variety of circumstances and that an assumption of monotonicity, when appropriate, noticeably improves performance. The proposed methods are applied to data from a randomized clinical trial for the treatment of a critical illness in the intensive care unit.

Journal ArticleDOI
TL;DR: A systematic optimization methodology for the Control Structure Selection Problem (CSSP) is presented that improves the accuracy of calculations and reduces computational time and effort necessary.

Journal ArticleDOI
TL;DR: An efficient approximation to the nonlinear phase diversity (PD) method for wavefront reconstruction and correction from intensity measurements with potential of being used in real-time applications is proposed.
Abstract: We propose an efficient approximation to the nonlinear phase diversity (PD) method for wavefront reconstruction and correction from intensity measurements with potential of being used in real-time applications. The new iterative linear phase diversity (ILPD) method assumes that the residual phase aberration is small and makes use of a first-order Taylor expansion of the point spread function (PSF), which allows for arbitrary (large) diversities in order to optimize the phase retrieval. For static disturbances, at each step, the residual phase aberration is estimated based on one defocused image by solving a linear least squares problem, and compensated for with a deformable mirror. Due to the fact that the linear approximation does not have to be updated with each correction step, the computational complexity of the method is reduced to that of a matrix-vector multiplication. The convergence of the ILPD correction steps has been investigated and numerically verified. The comparative study that we make demonstrates the improved performance in computational time with no decrease in accuracy with respect to existing methods that also linearize the PSF.

Dissertation
01 May 2013
TL;DR: This thesis considers topology optimization for structural mechanics problems, where the underlying PDE is derived from linear elasticity, and considers the formulation of the SIMP method as a mathematical optimization problem.
Abstract: Topology optimization is a tool for finding a domain in which material is placed that optimizes a certain objective function subject to constraints. This thesis considers topology optimization for structural mechanics problems, where the underlying PDE is derived from linear elasticity. There are two main approaches for solving topology optimization: Solid Isotropic Material with Penalisation (SIMP) and Evolutionary Structural Optimization (ESO). SIMP is a continuous relaxation of the problem solved using a mathematical programming technique and so inherits the convergence properties of the optimization method. By contrast, ESO is based on engineering heuristics and has no proof of optimality. This thesis considers the formulation of the SIMP method as a mathematical optimization problem. Including the linear elasticity state equations is considered and found to be substantially less reliable and less efficient than excluding them from the formulation and solving the state equations separately. The convergence of the SIMP method under a regularising filter is investigated and shown to impede convergence. A robust criterion to stop filtering is proposed and demonstrated to work well in high-resolution problems (O(10^6)). The ESO method is investigated to fully explain its non-monotonic convergence behaviour. Through a series of analytic examples, the steps taken by the ESO algorithm are shown to differ arbitrarily from a linear approximation. It is this difference between the linear approximation and the actual value taken which causes ESO to occasionally take non-descent steps. A mesh refinement technique has been introduced with the sole intention of reducing the ESO step size and thereby ensuring descent of the algorithm. This is shown to work on numerous examples. Extending the classical topology optimization problem to included a global buckling constraint is considered. This poses multiple computational challenges, including the introduction of numerically driven spurious localised buckling modes and ill-defined gradients in the case of non-simple eigenvalues. To counter such issues that arise in a continuous relaxation approach, a method for solving the problem that enforces the binary constraints is proposed. The method is designed specifically to reduce the number of derivative calculations made, which is by far the most computationally expensive step in optimization involving buckling. This method is tested on multiple problems and shown to work on problems of size O(10^5).

Journal ArticleDOI
TL;DR: The best linear approximation in least squares sense is discussed, which is user friendly and well understood and often approximated with linear systems.
Abstract: The engineers and scientists want mathematical models of the observed system for understanding, design and control. Most of these systems are nonlinear. There is not a unique solution because of the many different types of nonlinear systems with different behaviors and so the modeling is very involved and universally usable design tools are not available. For these reasons the nonlinear systems are often approximated with linear systems, because this theory is user friendly and well understood. In this paper we will discuss the best linear approximation in least squares sense.

Journal ArticleDOI
TL;DR: In this paper, the use of digital image correlation (DIC) in order to detect the development of local non-linearities corresponding to damage (with no other nonlinearities) and is limited to small deformation.

Journal ArticleDOI
TL;DR: This work is the first to implement the numerical entropy production as a refinement indicator into adaptive finite volume methods used to solve the shallow water equations.
Abstract: Water flows can be modelled mathematically and one available model is the shallow water equations. This thesis studies solutions to the shallow water equations analytically and numerically. The study is separated into three parts. The first part is about well-balanced finite volume methods to solve steady and unsteady state problems. A method is said to be well-balanced if it preserves an unperturbed steady state at the discrete level~\cite{NPPN2006}. We implement hydrostatic reconstructions proposed by Audusse et al.~\cite{ABBKP2004} for the well-balanced methods with respect to the steady state of a lake at rest. Four combinations of quantity reconstructions are tested. Our results indicate an appropriate combination of quantity reconstructions for dealing with steady and unsteady state problems~\cite{MR2010}. The second part presents some new analytical solutions to debris avalanche problems~\cite{MR2011DA,MR2012PAAG} and reviews the implicit Carrier--Greenspan periodic solution for flows on a sloping beach~\cite{MR2012CG}. The analytical solutions to debris avalanche problems are derived using characteristics and a variable transformation technique. The analytical solutions are used as benchmarks to test the performance of numerical solutions. For the Carrier--Greenspan periodic solution, we show that the linear approximation of the Carrier--Greenspan periodic solution may result in large errors in some cases. If an explicit approximation of the Carrier--Greenspan periodic solution is needed, higher order approximations should be considered. We propose second order approximations of the Carrier--Greenspan periodic solution and present a way to get higher order approximations. The third part discusses refinement indicators used in adaptive finite volume methods to detect smooth and nonsmooth regions. In the adaptive finite volume methods, smooth regions are coarsened to reduce the computational costs and nonsmooth regions are refined to get more accurate solutions. We consider the numerical entropy production~\cite{Puppo2004} and weak local residuals~\cite{KKP2002} as refinement indicators. Regarding the numerical entropy production, our work is the first to implement the numerical entropy production as a refinement indicator into adaptive finite volume methods used to solve the shallow water equations. Regarding weak local residuals, we propose formulations to compute weak local residuals on nonuniform meshes. Our numerical experiments show that both the numerical entropy production and weak local residuals are successful as refinement indicators. Some publications corresponding to this thesis are listed in the References~\cite{MR2010,MR2011DA,MR2012PAAG,MR2012CG,MR2011NEP,MR2013NEP,MR2011IEEE}. 10.1017/S0004972713000750

Journal ArticleDOI
TL;DR: In this article, a surface finite-volume method is defined to approximate numerically parabolic partial differential equations on closed surfaces, namely on a sphere, ellipsoid or the Earth's surface.
Abstract: The paper deals with data filtering on closed surfaces using linear and nonlinear diffusion equations. We define a surface finite-volume method to approximate numerically parabolic partial differential equations on closed surfaces, namely on a sphere, ellipsoid or the Earth’s surface. The closed surface as a computational domain is approximated by a polyhedral surface created by planar triangles and we construct a dual co-volume grid. On the co-volumes we define a weak formulation of the problem by applying Green’s theorem to the Laplace–Beltrami operator. Then the finite-volume method is applied to discretize the weak formulation. Weak forms of elliptic operators are expressed through surface gradients. In our numerical scheme we use a piece-wise linear approximation of a solution in space and the backward Euler time discretization. Furthermore, we extend a linear diffusion on surface to the regularized surface Perona–Malik model. It represents a nonlinear diffusion equation, which at the same time reduces noise and preserves main edges and other details important for a correct interpretation of the real data. We present four numerical experiments. The first one has an illustrative character showing how an additive noise is filtered out from an artificial function defined on a sphere. Other three examples deal with the real geodetic data on the Earth’s surface, namely (i) we reduce a stripping noise from the GOCE satellite only geopotential model up to degree 240, (ii) we filter noise from the real GOCE measurements (the component $$T_{zz})$$ , and (iii) we reduce a stripping noise from the satellite only mean dynamic topography at oceans. In all experiments we focus on a comparison of the results obtained by both the linear and nonlinear models presenting advantages of the nonlinear diffusion.

Proceedings ArticleDOI
07 Apr 2013
TL;DR: The truncated approximate logarithm simultaneously improves the efficiency and precision of Mitchell's approximation while remaining simple to implement.
Abstract: The speed and levels of integration of modern devices have risen to the point that arithmetic can be performed very fast and with high precision. Precise arithmetic comes at a hidden cost-by computing results past the precision they require, systems inefficiently utilize their resources. Numerous designs over the past fifty years have demonstrated scalable efficiency by utilizing approximate logarithms. Many such designs are based off of a linear approximation algorithm developed by Mitchell. This paper evaluates a truncated form of binary logarithm as a replacement for Mitchell's algorithm. The truncated approximate logarithm simultaneously improves the efficiency and precision of Mitchell's approximation while remaining simple to implement.