Constrained Least-squares Technique Research Articles

Constrained least-squares based adaptive-order finite-volume WENO scheme for the simulation of viscous compressible flows on unstructured grids

We develop a novel constrained least-squares technique for fourth-order weighted essentially non-oscillatory (WENO) polynomial reconstruction on general unstructured grids composed of curved-edged quadrilateral elements. Our technique relies on a segregation of the conditions that match the spatial averages of the candidate WENO polynomial and the reconstructed state variable over stencil-forming cells into two sets. The first set consists of exactly enforced linearly independent constraints that are applied only over the central cell and its edge-sharing neighbors. The second set consists of the remaining conditions and are enforced only approximately in a least-squares sense. We develop a specialized technique that reduces the computational burden associated with the numerical solution of the constrained least-squares system for the WENO interpolation coefficients. For a comparable computational expense, our constrained reconstruction approach yields pointwise estimates that are consistently more accurate than the conventional least-squares based estimates over a wide spectrum of spatial wavenumbers. This enhanced spectral resolution yields a significant, over two-fold increase in the accuracy with which unsteady smooth flow features, such as an advecting isentropic vortex, are captured over highly distorted curvilinear unstructured grids. Our constraint based reconstruction framework is generic, facilitates implementation of high-order boundary closures, and is amenable to large scale parallelization over several thousand CPU cores. Numerical tests over a range of inviscid and viscous flow configurations are presented to demonstrate the accuracy and robustness of our technique. Our constrained least-squares based adaptive WENO discretization faithfully captures intricate features that arise in strongly shocked, and highly unsteady and separated complex geometry flows. Our results suggest exact preservation of the match between the cell averages of the interpolant and the reconstructed state variable over the nearest neighbors to be quite an effective strategy that dramatically enhances the accuracy and spectral resolution of the high-order finite-volume reconstruction operation.

Obtaining Gradients of XCO2 in Atmosphere Using the Constrained Linear Least-Squares Technique and Multi-Wavelength IPDA LiDAR

Integrated-path differential absorption (IPDA) LiDAR is a promising means of measuring the global distributions of the column weighted xCO2 (dry-air mixing ratio of CO2) with adequate accuracy and precision. Most IPDA LiDARs are incapable of discerning the vertical information of CO2 diffusion, which is of great significance for studies on the carbon cycle and climate change. Hence, we developed an inversion method using the constrained linear least-squares technique for a pulsed direct-detection multi-wavelength IPDA LiDAR to obtain sliced xCO2. In the proposed inversion method, the atmosphere is sliced into three different layers, and the xCO2 of those layers is then retrieved using the constrained linear least-squares technique. Assuming complete knowledge of the water vapor content, the accuracy of the retrieved sliced xCO2 could be as high as 99.85% when the signal-to-noise ratio of central wavelength retrievals is higher than 25 (with a log scale). Further experiments demonstrated that different carbon characteristics can be identified by the sign of the carbon gradient of the retrieved xCO2 between the ABL (atmospheric boundary layer) and FT (free troposphere). These results highlight the potential applications of multiple wavelength IPDA LiDAR.

Constrained least-squares solution and regularization in inverse boundary element analysis of photoelastic models involving contact

Existing work by the present team has involved the development of an inverse boundary element approach for fitting unknown boundary conditions to full-field stress data obtained using photoelasticity, treating the displacements on the unknown region as the solution variables. However, the recovered boundary conditions sometimes show unrealistic features such as negative contact pressures or shear tractions which exceed realistic frictional values; noisy or incomplete input data make these problems worse and can make the results oscillatory. The work described here exploits available constrained least-squares techniques in order to express the contact conditions as linear inequality constraints upon the displacement solution. A variant of Tikhonov regularization is also used to penalize against oscillatory traction distributions on the unknown boundary. The chosen implementation, solved using the LSSOL algorithm, can handle the issue of rigid-body motion without the need for manual imposition of restraints. The technique is illustrated using simulated and real examples, demonstrating that the technique correctly reconstructs the individual principal stresses.

Simultaneous multivintage time-shift estimation

Time-shift estimation is a key step in seismic time-lapse processing as well as in many other signal-processing applications. We consider the time-shift problem in the setting of multiple repeat surveys that must be aligned consistently. We introduce an optimized least-squares method based on the Taylor expansion for estimating two-vintage time shifts and compare it to crosscorrelation. The superiority of the proposed algorithm is demonstrated with synthetic data and residual time-lapse matching on a U. K. continental shelf data set. We then discuss the shortcomings of cascaded time alignment in multiple repeat monitor surveys and propose an approach to estimate simultaneous multivintage time shifts that uses a constrained least-squares technique combined with elements of network theory. The resulting time shifts are consistent across all vintages in a least-squares sense, improving overall alignment when compared to the classical flow of alignment in a cascaded manner. The method surpasses the cascaded approach, as noted with sample synthetic and three-vintage U. K. continental shelf time-lapse data sets.

Optimal apodization design for medical ultrasound using constrained least squares part II simulation results

In the first part of this work, we introduced a novel general ultrasound apodization design method using constrained least squares (CLS). The technique allows for the design of system spatial impulse responses with narrow mainlobes and low sidelobes. In the linear constrained least squares (LCLS) formulation, the energy of the point spread function (PSF) outside a certain mainlobe boundary was minimized while maintaining a peak gain at the focus. In the quadratic constrained least squares (QCLS) formulation, the energy of the PSF outside a certain boundary was minimized, and the energy of the PSF inside the boundary was held constant. In this paper, we present simulation results that demonstrate the application of the CLS methods to obtain optimal system responses. We investigate the stability of the CLS apodization design methods with respect to errors in the assumed wave propagation speed. We also present simulation results that implement the CLS design techniques to improve cystic resolution. According to novel performance metrics, our apodization profiles improve cystic resolution by 3 dB to 10 dB over conventional apodizations such as the flat, Hamming, and Nuttall windows. We also show results using the CLS techniques to improve conventional depth of field (DOF).

Optimal apodization design for medical ultrasound using constrained least squares part I: theory

Aperture weighting functions are critical design parameters in the development of ultrasound systems because beam characteristics affect the contrast and point resolution of the final output image. In previous work by our group, we developed a metric that quantifies a broadband imaging system's contrast resolution performance. We now use this metric to formulate a novel general ultrasound beamformer design method. In our algorithm, we use constrained least squares (CLS) techniques and a linear algebra formulation to describe the system point spread function (PSF) as a function of the aperture weightings. In one approach, we minimize the energy of the PSF outside a certain boundary and impose a linear constraint on the aperture weights. In a second approach, we minimize the energy of the PSF outside a certain boundary while imposing a quadratic constraint on the energy of the PSF inside the boundary. We present detailed analysis for an arbitrary ultrasound imaging system and discuss several possible applications of the CLS techniques, such as designing aperture weightings to maximize contrast resolution and improve the system depth of field.

Application of constrained least-squares techniques to IMRT treatment planning

Purpose: The purpose of this work was to apply the method of constrained least-squares to inverse treatment planning and to explore its potential for providing a fast interactive planning environment for intensity-modulated radiation therapy (IMRT). Methods and Materials: The description of the dose inside a patient is a linear matrix transformation of beamlet weights. The constrained least-squares method adds additional matrix operators and produces beamlet weights by a direct linear transformation. These matrix operators contain a priori knowledge about the radiation distribution. The constrained least-squares technique was applied to obtain IMRT plans for prostate and paraspinal cancer patients and compared with the corresponding plans optimized using the CORVUS inverse planning system. Results: It was demonstrated that a constrained least-squares technique is suitable for IMRT plan optimization with significantly increased computing speed. For the two cases we have tested, the constrained least-squares method was an order of magnitude faster than conventional iterative techniques because of the avoidance of the iterative calculations. We also found that the constrained least-squares method is capable of generating clinically acceptable treatment plans with less trial-and-error adjustments of system variables, and with improved target volume coverage as well as sensitive structure sparing in comparison with that obtained using CORVUS. Conclusions: The constrained least-squares method has the advantage that it does not require iterative calculation and thus significantly speeds up the therapeutic plan optimization process. Besides shedding important insight into the inverse planning problem, the technique has strong potential to provide a fast and interactive environment for IMRT treatment planning.

Deblurring of x-ray spectra acquired with a Nal-photomultiplier detector by constrained least-squares deconvolution.

A constrained least-squares technique to correct diagnostic x-ray tube energy spectra for inherent blurring by scintillation detectors was developed. The measured detector response function to monoenergetic sources was used to construct a matrix that modeled the energy broadening in the crystal. This blurring operator, along with an estimate of statistical noise in the count data, comprised the a priori system knowledge required for application of the method. Tungsten anode spectra up to 90 kVp were acquired with a Nal-photomultiplier detector system at a source-to-detector distance of 30 cm. X-ray tube output was collimated at the detector by a 0.5 mm diameter pinhole collimator. Measured Nal spectra were compared to both published reference data and to spectra acquired in our laboratory with a Ge detector system. Application of the constrained least-squares technique involved first defining a criterion function that combined an assessment of the goodness of fit with a weighted measure of the smoothness of the solution. Minimization of this function resulted in the corrected spectrum. While it is not possible to recover the characteristic tungsten peaks, the success of our method in deconvolving the measured spectra was demonstrated by a significant improvement in agreement with reference data. To provide a measure of this agreement, a histogram of the differences between the two curves was generated. The full width at half maximum (FWHM) of the Gaussian distribution fit to the histogram was used to quantify the similarity between the spectra and the reference data, both before and after correction. As spectral agreement improves, the FWHM becomes smaller. We show that application of the constrained least-squares technique improved spectral matching by 20%-60%.

Positron Lifetime Analysis from Generalized Constrained Least-Squares Technique

The generalized least-squares method with non-negativity constraint is developed to analyze positron annihilation lifetime (PAL) spectra to obtain lifetimes and intensities. The method for obtaining the instrumental function from a single-component PAL spectrum is demonstrated. The present code is tested using computer simulated lifetime data.

An efficient neural network algorithm for reflector surface error compensation

A neural network algorithm for electromagnetic compensation of reflector surface error effects is formulated. Sets of trained neural networks are used to compute the compensation excitations for array feeds. The networks were trained using data generated with the constrained least squares (CLS) compensation method. Once trained, the calculation of the excitations is accomplished in significantly less time than required by the original constrained least squares algorithm. The surface error profile for a distorted reflector antenna is expanded using bivariate surface basis functions. Each of the trained networks corresponds to one of the expansion functions. Excitations computed using the neural networks are superposed to produce composite compensation excitations for the distorted reflector. The compensation results for a distorted reflector are presented, and the neural network algorithm performance is compared to the original CLS technique.

Constrained least-squares restoration and renogram deconvolution: a comparison with other techniques

It has previously been shown that an iterative constrained least-squares (CLSR) technique using a noise-based constraint may be superior to other methods of renogram deconvolution analysis. To test this hypothesis on real data, renography was performed on 70 patients with established diagnoses of normal, insufficient or acutely obstructed kidneys. Standard renography parameters were obtained from the time activity curves which were then deconvolved using three techniques. One kidney per patient was chosen at random for analysis resulting in a total of 43 normal and 27 diseased kidneys. The ability of each of the analytical techniques to discriminate between normal and diseased kidneys was assessed using logistic regression. CLSR proved to be robust and provide the best discrimination of the deconvolution techniques. However, the best overall discrimination was provided by a model based on the renography parameters excretion ratio, rate of uptake and time to peak activity which correctly classified 86% of the kidneys.

Radioactivity measurement of drum package waste by a computed-tomography technique

A method of evaluating the radionuclide content of low-level drum package waste using emission computed tomography (ECT) and transmission computed tomography (TCT) has been investigated. This method requires accurate correction for the γ-ray attenuation in the drum. The TCT and ECT analyses are performed according to the constrained least-squares technique. Satisfactory results were obtained in preliminary experiments carried out using test equipment with 20 NaI(Tl) scintillation detectors and sample 200-L drums which contained sealed γ-ray point sources of 60Co and 137Cs, and pipes and concrete blocks. The radioactive content of one or two 60Co sources (about 3.5 MBq) in a drum of about 200 kg was evaluated in 6 × 6 × 9 voxel with an uncertainty of about ±15% for an ECT measurement time of about 12.5 min.

Atomic physicochemical parameters for three-dimensional-structure-directed quantitative structure-activity relationships. 2. Modeling dispersive and hydrophobic interactions.

In an earlier paper (Ghose A. K.; Crippen, G. M. J. Comput. Chem. 1986, 7, 565) the need of atomic physicochemical properties for three-dimensional-structure-directed quantitative structure-activity relationships was demonstrated, and it was shown how atomic parameters can be developed to successfully evaluate the molecular water-1-octanol partition coefficient, which is a measure of hydrophobicity. In the present work the atomic values of molar refractivity are reported. Carbon, hydrogen, oxygen, nitrogen, sulfur, and halogens are divided into 110 atom types of which 93 atomic values are evaluated from 504 molecules by using a constrained least-squares technique. These values gave a standard deviation of 1.269 and a correlation coefficient of 0.994. The parameters were used to predict the molar refractivities of 78 compounds. The predicted values have a standard deviation of 1.614 and a correlation coefficient of 0.994. The degree of closeness of the linear relationship between the atomic water-1-octanol partition coefficients and molar refractivities has been checked by the correlation coefficient of 89 atom types used for both the properties. The correlation coefficient has been found to be 0.322. The low value suggests that both parameters can be used to model the intermolecular interaction. The origin of these physicochemical properties and the types of interaction that can be modeled by these properties have been critically analyzed.

First-kind Fredholm integral equation of liver kinetics: numerical solutions by constrained least squares

A model for the enzymatic elimination of substrates from blood flowing through the liver has recently been developed, in which the liver is represented by an ensemble of elements acting in parallel. Taking into consideration statistical distributions of biochemical and hemodynamic properties of the elements, and the mixing of the outflows from the elements, it has been possible to formulate a first-kind Fredholm integral equation relating input and output concentrations of the substrate. The kernel of the integral equation is given as a solution of a transcendental equation. The present work treats the inverse problem of recovering information about the distributions of properties of the elements, given a set of input and output concentration data. Constrained least-squares techniques are used in conjunction with a quadratic spline approximation to the unknown distribution. Results of applying the method to synthetic data and prospects for the analysis of real data are discussed.