Least-squares Function Research Articles

First-Order System Least-Squares Methods for an Optimal Control Problem by the Stokes Flow

The least-squares approximations of an optimal control problem governed by the Stokes equations are considered, which leads to an unconstrained coupled optimization problem by the Lagrange multiplier method. The least-squares functionals for the two- and three-dimensional first-order coupled optimality systems are employed by modifying those functionals in [Z. Cai, T. A. Manteuffel, and S. F. McCormick, SIAM J. Numer. Anal., 34 (1997), pp. 1727–1741]. The established ellipticity and continuity in a product $H^1$ norm yield the optimal discretization error estimates in the finite element spaces. For numerical tests, we apply V-cycle multigrid methods to the whole discrete algebraic system.

Negative norm least-squares methods for the incompressible magnetohydrodynamic equations

The purpose of this article is to develop and analyze least-squares approximations for the incompressible magnetohydrodynamic equations. The major advantage of the least-squares finite element method is that it is not subjected to the so-called Ladyzhenskaya-Babuška-Brezzi (LBB) condition. The authors employ least-squares functionals which involve a discrete inner product which is related to the inner product in H -1(ω).

Topology optimization-based distribution design of actuation voltage in static shape control of plates

This paper investigates the optimal spatial distribution of single-channel actuation voltage in static structural shape control problem. It is pointed out that single-channel actuation voltage input for shape control applications is of practical importance due to its ease of implementation. The Mindlin plate theory is used in the finite element modeling of laminates with piezoelectric layers. The optimal distribution of the single-channel actuation voltage is formulated as a discrete optimization problem with tri-level design variables, where a least-square function measuring the structural shape error is to be minimized under a constraint on the control effort. Such a problem is then transformed into a continuous one by introducing element-wise artificial design variables defining the topological feature of the control voltage distribution. A power-law function relating the design variables and the applied voltages is proposed to penalize intermediate values of the design variables. Based on the sensitivity analysis, the problem is solved effectively using the MMA algorithm tailored for constrained least-square problems. Numerical examples presented demonstrate the validity of the proposed problem formulation and numerical approaches.

Rank-based variable selection

This note considers variable selection in the robust linear model via R-estimates. The proposed rank-based approach is a generalisation of the penalised least-squares estimators where we replace the least-squares loss function with Jaeckel's (1972) dispersion function. Our rank-based method is robust to outliers in the errors and has roots in traditional non-parametric statistics for simple location-shift problems. We establish the theoretical properties of our estimators which ensure desirable asymptotic behaviour of setting coefficient estimates to zero for unimportant variables and consistently estimating coefficients for important variables. Numerical studies indicate that the rank-based methods perform well for both light- and heavy-tailed error distributions.

Measurement of thermal radiative and conductive properties of semitransparent materials using a photothermal crenel method

This paper deals with a theoretical and an experimental study allowing the measurement of the radiative and the conductive properties of semitransparent materials. The method consists of applying a crenel heat flux on the front face of a semitransparent sample and recording the temperature at the rear face using an open thermocouple junction. Parameter identification is performed by the minimization of the ordinary least-squares function comparing the measured and the calculated temperatures. This later is obtained from the thermal model describing the heat transfer by conduction and radiation in the medium. This model is built by the thermal quadrupole formalism. Measurements are reported on commercial glasses and plexiglass samples, and the used iterative algorithm is based on the Gauss–Newton method.

Iterative regularization for elliptic inverse problems

Elliptic inverse problems can be formulated using coefficient-dependent energy least-squares functionals, resulting in a smooth, convex objective functional. A variational inequality emerges as a necessary and sufficient optimality condition. The principle of iterative regularization, when coupled with the auxiliary problem principle, results in a strongly convergent scheme for the solution of elliptic inverse problems.

Viscoplastic parameter estimation by high strain-rate experiments and inverse modelling – Speckle measurements and high-speed photography

A methodology based on inverse modelling for estimating viscoplastic material parameters at high strain-rate conditions is presented. The methodology is demonstrated for a mild steel exposed for compression loading in a split Hopkinson pressure bar arrangement. By using dog-bone shaped specimens nonhomogeneous states of deformation are obtained throughout the entire deformation process. The resulting nonhomogeneous deformation of the specimens is evaluated using digital speckle photography (DSP) to give in-plane point-wise displacement and strain fields. The photographs are captured with a high-speed camera of image converter type, which acquire time resolved images during the impact loading. The experiments are simulated using finite element analysis (FEA), where the material model suggested by Johnson–Cook for high-strain rate conditions are utilised. Experimental and FE-calculated field information are compared in order to estimate the viscoplastic parameter in the Johnson–Cook material model. The estimation is performed by minimising least-square functions that contain the differences in displacements and strains, respectively. The quality of the estimated parameters is studied from statistical point of view.

A simple and efficient design of variable fractional delay FIR filters

In this paper, a simple and efficient approach for designing one-dimensional variable fractional delay finite impulse response digital filters is proposed. Two matrix equations, based respectively on the weighted least-squares function of the optimum fixed fractional delay filter and the filter coefficient polynomial fitting, are formulated in tandem to form the design algorithm, which only has the computation complexity comparable with that of designing fixed finite impulse response digital filters. A design example is also given to justify the effectiveness and advantages of the proposed design method.

Analysis and computation of least-squares methods for a compressible Stokes problem

We develop and analyze a negative norm least-squares method for the compressible Stokes equations with an inflow boundary condition. Least-squares principles are derived for a first-order form of the equations obtained by using ω = ∇×u and φ = ∇ · u as new dependent variables. The resulting problem is incompletely elliptic, i.e., it combines features of elliptic and hyperbolic equations. As a result, well-posedness of least-squares functionals cannot be established using the ADN elliptic theory and so we use direct approaches to prove their norm-equivalence. The article concludes with numerical examples that illustrate the theoretical convergence rates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

A least-squares preconditioner for radial basis functions collocation methods

Although meshless radial basis function (RBF) methods applied to partial differential equations (PDEs) are not only simple to implement and enjoy exponential convergence rates as compared to standard mesh-based schemes, the system of equations required to find the expansion coefficients are typically badly conditioned and expensive using the global Gaussian elimination (G-GE) method requiring \(\mathcal{O}(N^{3})\) flops. We present a simple preconditioning scheme that is based upon constructing least-squares approximate cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements. The ACBFs transforms a badly conditioned linear system into one that is very well conditioned, allowing us to solve for the expansion coefficients iteratively so we can reconstruct the unknown solution everywhere on the domain. Our preconditioner requires \(\mathcal{O}(mN^{2})\) flops to set up, and \(\mathcal{O}(mN)\) storage locations where m is a user define parameter of order of 10. For the 2D MQ-RBF with the shape parameter \(c\sim1/\sqrt{N}\) , the number of iterations required for convergence is of order of 10 for large values of N, making this a very attractive approach computationally. As the shape parameter increases, our preconditioner will eventually be affected by the ill conditioning and round-off errors, and thus becomes less effective. We tested our preconditioners on increasingly larger c and N. A more stable construction scheme is available with a higher set up cost.

Detection of Subsurface Bubbles with Discrete Electromagnetic Geotomography

Abstract In electromagnetic geotomography, the inversion is very ill-posed due to the limita- tion of an angular range of projections, and consequently the reconstructed image is different than the true image. Nevertheless, the reconstruction can be much im- proved if we are interested only in a piece of information on the interior volume of the examined object, e.g. detection of some anomalies (bubbles). We assume that the image can be represented by binary values, hence: Discrete Electromagnetic Geotomography (DEG). In this paper, we discuss two discrete image reconstruction algorithms. The first one uses the D.C. programming to find the integer solution to the penalized least-squares problem. The penalty term is based on the Gibbs distribution. In the second one, the mean field annealing is applied to maximize the Gibbs-Boltzmann distribution associated with the penalized least-squares function. The numerical results are presented for noise-free and noisy data.

A variational meshfree method for solving time-discrete diffusion equations

A meshfree method is developed for solving time-discrete diffusion equations that arise in models in brain research. Important criteria for a suitable method are flexibility with respect to domain geometry and the ability to work with very small moving sources requiring easy refinement possibilities. One part of the work concerns a meshfree discretization of the modified Helmholtz equation based on the related minimization problem and a local least-squares function approximation. In a second part, a node choosing algorithm is presented that moves around randomly distributed nodes for optimizing the node distribution and varying the node density as needed. The method is illustrated by two numerical tests.

On the equivalence between a commonly used correlation coefficient and a least-squares function

Many objective functions have been proposed in X-ray crystallography to solve the molecular replacement (MR) problem and other optimization problems. This paper establishes the equivalence of optimizing two of these target functions, a commonly used correlation coefficient and a least-squares function. This equivalence may exist only in the neighborhoods about the global optima or the entire MR variable space depending on whether the mean values of the observed and calculated data are subtracted from the data. In addition, an argument is presented that the correlation coefficient between structure-factor magnitudes is likely to perform better than the correlation coefficient between intensities, especially when low-resolution data are used. This prediction was tested during coarse grid searches at low resolution using the MR program SOMoRe.

Iterative absolute electroanalytical approach to characterization of bulk redox conducting systems.

A novel electroanalytical approach is proposed here, and it is demonstrated with the direct and simultaneous determination of two unknowns: the concentration of redox sites and the apparent diffusion coefficient for charge propagation in a single crystal of dodecatungstophosphoric acid. This Keggin-type polyoxometalate serves as a model bulk redox conducting inorganic material for solid-state voltammetry. The system has been investigated using an ultramicrodisk working electrode in the absence of external liquid supporting electrolyte. The analytical method requires numerical solution of the combination of two equations in which the first one describes current (or charge) in a well-defined (either spherical or linear) diffusional regime and the second general equation describes chronoamperometric (or normal pulse voltammetric current) under mixed (linear-spherical) conditions. The iterative approach is based on successive approximations through calculation and minimizing the least-squares error function. The method is fairly universal, and in principle, it can be extended to the investigation of other bulk systems including sol-gel processed materials, redox melts, and solutions on condition that they are electroactive and well behaved, they contain redox centers at sufficiently high level, and a number of electrons for the redox reaction considered is known.

Least-Squares Methods for Incompressible Newtonian Fluid Flow: Linear Stationary Problems

This paper develops and analyzes two least-squares methods for the numerical so- lution of linear, stationary incompressible Newtonian fluid flow in two and three dimensions. Both approaches use the L 2 norm to define least-squares functionals. One is based on the stress-velocity formulation (see section 3.2), and it applies to general boundary conditions. The other is based on an equivalent formulation for the pseudostress and velocity (see section 4.2), and it applies to pure velocity Dirichlet boundary conditions. The velocity gradient and vorticity can be obtained alge- braically from this new tensor variable. It is shown that the homogeneous least-squares functionals are elliptic and continuous in the H(div; Ω) d × H 1 (Ω) d norm. This immediately implies optimal error estimates for conforming finite element approximations. As well, it admits optimal multigrid solution methods if Raviart-Thomas finite element spaces are used to approximate the stress or the pseudostress tensor.

Least‐squares spectral collocation method for the Stokes equations

AbstractFirst‐order system least‐squares spectral collocation methods are presented for the Stokes equations by adopting the first‐order system and modifying the least‐squares functionals in 2. Then homogeneous Legendre and Chebyshev (continuous and discrete) functionals are shown to be elliptic and continuous with respect to appropriate product weighted norms. The spectral convergence is analyzed for the proposed methods with some numerical experiments. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 128–139, 2004

Sequential annealing–gradient Gamma-Knife radiosurgery optimization

Simulated annealing and gradient methods are commonly employed for inverse planning of radiotherapy delivery schemes. Annealing is effective in finding an approximation of the global solution, suffering from slow late convergence and in some cases poor dose homogeneity. Gradient methods converge well but not necessarily to the global minimum. We explored simulated annealing followed by gradient optimization to improve on either method alone, using radiosurgery as the model system. Simulated annealing and gradient inverse planning programs using the same objective function were adapted for radiosurgical optimization. The objective function chosen is a least-squares dose-matching function, with differential weighting of tissues. A simple test target allowing local minima in the objective function was evaluated. Two hundred trials using the gradient method were done. The gradient method approximated the global solution only 12% of the time, commonly finding a local minimum. The annealing–gradient technique converged to the global minimum in 78 out of 80 trials, more efficiently than annealing alone. Dose homogeneity was improved. In conclusion, sequential annealing–gradient optimization can improve on either method alone. The technique may be extensible to radiotherapy inverse planning in general, with benefit expected for problems characterized by slow gradient method convergence and local minima.

Pseudospectral Least-Squares Method for the Second-Order Elliptic Boundary Value Problem

The least-squares Legendre and Chebyshev pseudospectral methods are presented for a first-order system equivalent to a second-order elliptic partial differential equation. Continuous and discrete homogeneous least-squares functionals using Legendre and Chebyshev weights are shown to be equivalent to the H1(\Omega)$ norm and Chebyshev-weighted Div-Curl norm over appropriate polynomial spaces, respectively. The spectral error estimates are derived. The block diagonal finite element preconditioner is developed for the both cases. Several numerical tests are demonstrated on the spectral discretization errors and on performances of the finite element preconditioner.

Thermal characterization of poorly conducting solids by the flash method

Estimation of the thermal diffusivity of a poorly conducting solid placed between two metallic layers by the flash method is described. The method consists of applying a brief heat pulse to the front face of the three-layer composite and measuring the temperature rise versus time of the rear face. The thermal contact resistance between the layers and the heat transfer coefficient are obtained simultaneously with the thermal diffusivity of the poorly conducting solid. These parameters are identified by minimisation of the least-squares function comparing the measured and the calculated temperatures. Three models are developed using the thermal quadrupole method. An optimal design of the experiment is also presented and the choice of thermal model for parameter estimation is discussed. The experimental results are in good agreement with the theoretical analysis.

Uniaxial creep behavior of V–4Cr–4Ti alloy

We are undertaking a systematic study at Argonne National Laboratory to evaluate the uniaxial creep behavior of V–Cr–Ti alloys in a vacuum environment as a function of temperature in the range of 650–800 °C and at applied stress levels of 75–380 MPa. Creep strain in the specimens is measured by a linear-variable-differential transducer, which is attached between the fixed and movable pull rods of the creep assembly. Strain is measured at sufficiently frequent intervals during testing to define the creep strain/time curve. A linear least-squares analysis function is used to ensure consistent extraction of minimum creep rate, onset of tertiary creep and creep strain at the onset of tertiary creep. Creep test data, obtained at 650, 700, 725 and 800 °C, showed power-law creep behavior. Extensive analysis of the tested specimens is conducted to establish hardness profiles, oxygen content and microstructural characteristics. The data are also quantified by the Larson–Miller approach, and correlations are developed to relate time to rupture, onset of tertiary creep, times for 1% and 2% strain, exposure temperature and applied stress.