Least-squares Solver Research Articles

Three-dimensional freeform reflector design with a microfacet surface roughness model

This paper proposes a methodology for the design of freeform reflectors with scattering surfaces. We use microfacets, which are small, tilted mirrors superimposed on a smooth surface. We form a simple model of surface roughness and light scattering based on the orientations of the microfacets. Using a least-squares solver to compute the smooth reflector as a starting point, we can subsequently alter the surface using an optimization procedure to account for the scattering. After optimization, the resulting reflector surface produces the desired scattered light distribution. We verify the resulting reflector using raytracing. This study focuses on freeform systems with a collimated incident beam and a far-field target intensity.

Distributed Nonlinear Least-Squares Solver for Practical Network Determination

Distributed Nonlinear Least-Squares Solver for Practical Network Determination

Degradation of binary mixtures of water pollutants in a photocatalytic microreactor: Competitive kinetics with explicit radiation absorption effects

The photocatalytic degradation of 17-α-ethinylestradiol (EE2) and phenol in aqueous solutions was studied using a continuous flow, planar microreactor under simulated sunlight. A TiO2 thin film was employed as photocatalyst. The degradation rates of single compounds and binary mixtures were well described by the Langmuir-Hinshelwood equation, including competitive terms for the companion pollutant and the intermediate products, and the effect of photon absorption by the photocatalyst. The absorbed radiation in the microreactor was calculated from the optical properties of the TiO2 film and the application of the Net Radiation method. The estimation of the kinetic parameters was carried out by numerical integration of coupled mass balance equations for EE2 and phenol employing the Runge-Kutta method, and adjusting the kinetics parameters with a nonlinear least-squares solver. The objective of this routine was to minimize the sum of the squares of the differences between the simulated and experimental outlet concentrations of EE2 and phenol under different operating conditions. Using this approach, intrinsic kinetic parameters were obtained. The model was able to predict the degradation of both organic pollutants under different inlet concentrations, irradiation levels, and fluid flow rates. This methodology can be extended to assess the simultaneous degradation of multi-component systems.

Spatio-temporal LPV model of 2D workpiece temperature for Direct Laser Deposition

Additive manufacturing of metallic components presents challenges regarding quality assurance and consistency, owing to the workpiece's complex spatially-dependent temperature history and a lack of suitable models for closed-loop temperature control of the fabrication process. The selected case study is the laying of cladding material (workpiece) through Direct Laser Deposition. The paper proposes identifying a linear spatio-temporal parameter-varying model for the workpiece temperature for control-oriented applications. For this, the temperature distribution is reconstructed from data measured by an infrared camera. The model structure is derived from the inhomogeneous heat equation. Finally, the model parameters are identified using a nonlinear least-squares solver. The relationships between theoretical and estimated model parameters are presented.

Numerical Calculation of Pick-Up Coils Frequency Response as a Useful Tool for Local Magnetic Field Sensors Design

The aim of this article is to present a useful numerical procedure suitable to accurately model pick-up probes for design-oriented purposes, reducing the required prototyping activity and allowing more accurate results with respect to analytic estimations, taking into account both inductive and capacitive effects in complex geometries. The proposed methodology is characterized by the following steps: 1) a partial element equivalent circuit (PEEC) formulation, speeded up by a parametric model order reduction (pMOR) technique, is used to efficiently model the probe in the frequency domain taking into account both inductive and capacitive effects; 2) a nonlinear least-squares solver is used to identify an equivalent <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$RLC$ </tex-math></inline-formula> network describing the full behavior of the probe up to the first resonance frequency; and 3) a circuital model, considering the probe, the transmission line (TL), and the terminating impedance, is used to estimate the bandwidth of the complete system from the probe to the acquisition system. The experimental validation of the proposed method against the RFX-mod2 three-axis pick-up probe prototype is also presented, together with the application to the design of the magnetic sensors for the new Divertor tokamak test (DTT) facility.

Fresnel reflections in inverse freeform lens design.

In this paper we propose a method to design a freeform lens including the effect of Fresnel reflections on the transmitted intensity. This method is elaborated for a lens with one freeform surface shaping a far-field target from a point source or collimated input beam. It combines the optical mapping with the energy balance incorporating the loss due to Fresnel reflections, which leads to a generalized Monge-Ampère equation. We adapt a least-squares solver from previous research to solve the model numerically. This is then tested with a theoretical example and a test case related to road lighting.

Adaptive method for sensorless temperature estimation over the lifetime of lithium-ion batteries

Over the last decade, several impedance-based temperature estimation methods for lithium-ion cells have been proposed in the literature. However, the influence of cell degradation on these methods is rarely considered. In this paper, we therefore investigate the influence of aging on the temperature estimation method, presented in our previous work, by tracking the capacity fade and resistance change of a 6s1p module over 200 cycles. Both capacity fade and resistance change were found to affect the accuracy of the temperature estimation method, leading to a root mean square error (RMSE) of up to 15 K without adaptation to cell aging. Fitting the reference used for temperature estimation with linear operations and a nonlinear least-squares solver (NLS) to the aging data proved to be a valid method of compensating for the effects of aging. Derived from the fitting results, an online applicable aging adjustment scheme based on the checkup values is proposed to maintain a stable temperature estimation over battery lifetime. Using a simple resistance offset correction and an accurate state of charge and health estimation, the temperature estimation error stabilizes at an average RMSE of below 2 K for each cell in the module over its entire lifetime. • Sensorless method for resistance-based temperature estimation for lithium-ion batteries. • Investigation of the effects of cell aging on temperature estimation in a battery module. • Aging compensation with offset correction and accurate estimation of state of charge. • Stable temperature estimation over battery lifetime with online adaptation scheme.

A Distributed Least-Squares Solver for Linear Systems of Algebraic Equations

This paper introduces a consensus-based continuous-time distributed algorithm to find the least-squares solution to overdetermined systems of linear algebraic equations over directed multi-agent networks. It is assumed that each agent has only access to a subsystem of the algebraic equations, and the underlying communication network is strongly connected. We show that, along the flow of the proposed algorithm, the local estimate of each agent converges exponentially to the exact least-squares solution, provided that the aggregate system of linear equations has full column rank, and each agent knows an upper bound on the total number of the participating agents in the network.

Improved Image Quality for Static BLADE Magnetic Resonance Imaging Using the Total-Variation Regularized Least Absolute Deviation Solver.

In order to improve the image quality of BLADE magnetic resonance imaging (MRI) using the index tensor solvers and to evaluate MRI image quality in a clinical setting, we implemented BLADE MRI reconstructions using two tensor solvers (the least-squares solver and the L1 total-variation regularized least absolute deviation (L1TV-LAD) solver) on a graphics processing unit (GPU). The BLADE raw data were prospectively acquired and presented in random order before being assessed by two independent radiologists. Evaluation scores were examined for consistency and then by repeated measures analysis of variance (ANOVA) to identify the superior algorithm. The simulation showed the structural similarity index (SSIM) of various tensor solvers ranged between 0.995 and 0.999. Inter-reader reliability was high (Intraclass correlation coefficient (ICC) = 0.845, 95% confidence interval: 0.817, 0.87). The image score of L1TV-LAD was significantly higher than that of vendor-provided image and the least-squares method. The image score of the least-squares method was significantly lower than that of the vendor-provided image. No significance was identified in L1TV-LAD with a regularization strength of 0.4–1.0. The L1TV-LAD with a regularization strength of 0.4–0.7 was found consistently better than least-squares and vendor-provided reconstruction in BLADE MRI with a SENSitivity Encoding (SENSE) factor of 2. This warrants further development of the integrated computing system with the scanner.

Solving nonlinear integral equations for laser pulse retrieval with Newton's method

We present an algorithm based on numerical techniques that have become standard for solving nonlinear integral equations: Newton's method, homotopy continuation, the multilevel method, and random projection to solve the inversion problem that appears when retrieving the electric field of an ultrashort laser pulse from a two-dimensional intensity map measured with frequency-resolved optical gating (FROG), dispersion-scan, or amplitude-swing experiments. Here we apply the solver to FROG and specify the necessary modifications for similar integrals. Unlike other approaches we transform the integral and work in time domain where the integral can be discretized as an overdetermined polynomial system and evaluated through list autocorrelations. The solution curve is partially continues and partially stochastic, consisting of small linked path segments and enables the computation of optimal solutions in the presents of noise. Interestingly, this is an alternative method to find real solutions of polynomial systems, which are notoriously difficult to find. We show how to implement adaptive Tikhonov-type regularization to smooth the solution when dealing with noisy data, and we compare the results for noisy test data with a least-squares solver and propose the L-curve method to fine-tune the regularization parameter.

Machine learning a molecular Hamiltonian for predicting electron dynamics

We develop a computational method to learn a molecular Hamiltonian matrix from matrix-valued time series of the electron density. As we demonstrate for three small molecules, the resulting Hamiltonians can be used for electron density evolution, producing highly accurate results even when propagating 1,000 time steps beyond the training data. As a more rigorous test, we use the learned Hamiltonians to simulate electron dynamics in the presence of an applied electric field, extrapolating to a problem that is beyond the field-free training data. We find that the resulting electron dynamics predicted by our learned Hamiltonian are in close quantitative agreement with the ground truth. Our method relies on combining a reduced-dimensional, linear statistical model of the Hamiltonian with a time-discretization of the quantum Liouville equation within time-dependent Hartree Fock theory. We train the model using a least-squares solver, avoiding numerous, CPU-intensive optimization steps. For both field-free and field-on problems, we quantify training and propagation errors, highlighting areas for future development.

Multidimensional Path Tracking With Global Least Squares Solution

In this paper, a new method for model based optimal tracking control is presented. The special composition of the cost functional leads to design parameters for constraining the solution so as to ensure that machine limitations are not violated. By minimizing the cost functional with the calculus of variations, or more precisely the Euler-Lagrange equations, the state space representation of the system dynamics is transformed into an augmented state space representation describing the optimal tracking dynamics. The optimal control input is numerically calculated by solving the set of differential equations, given by the augmented state space system, globally with a specialized least-squares solver. The general control approach is demonstrated on an underactuated crane-like system with fixed load hoisting length operating in the horizontal plane. In this case the introduced design parameters determine the tradeoff between the cost of tracking accuracy and the cost of using large values of crane speed and acceleration. The potential of the proposed control scheme is proven by both simulation and experimental tests. The multibody simulation is carried out with the software Simscape MultibodyTM. For the experimental verification an industrial robot is used whose end effector only moves in a horizontal plane to imitate the trolley of an overhead crane.

Radio Interferometric Calibration Using a Complex Student’s t-distribution and Wirtinger Derivatives

Abstract Radio interferometric gain calibration can be biased by incomplete sky models and radio frequency interference, resulting in calibration artefacts that can restrict the dynamic range of the resulting images. It has been suggested that calibration algorithms employing heavy-tailed likelihood functions are less susceptible to this due to their robustness against outliers in the data. We present an algorithm based on a Student’s t-distribution which leverages the framework of complex optimisation and Wirtinger calculus for efficient and robust interferometric gain calibration. We integrate this algorithm as an option in the newly released calibration software package, CubiCal. We demonstrate that the algorithm can mitigate some of the biases introduced by incomplete sky models and radio frequency interference by applying it to both simulated and real data. Our results show significant improvements compared to a conventional least-squares solver which assumes a Gaussian likelihood function. Furthermore, we provide some insight into why the algorithm outperforms the conventional solver, and discuss specific scenarios (for both direction-independent and direction-dependent self-calibration) where this is expected to be the case.

A Bounded-Variable Least-Squares Solver Based on Stable QR Updates

In this paper, a numerically robust solver for least-square problems with bounded variables (BVLS) is presented for applications including, but not limited to, model predictive control (MPC). The proposed BVLS algorithm solves the problem efficiently by employing a recursive QR factorization method based on Gram–Schmidt orthogonalization. A reorthogonalization procedure that iteratively refines the QR factors provides numerical robustness for the described primal active-set method, which solves a system of linear equations in each of its iterations via recursive updates. The performance of the proposed BVLS solver, which is implemented in C without external software libraries, is compared in terms of computational efficiency against state-of-the-art quadratic programming solvers for small- to medium-sized random BVLS problems and a typical example of embedded linear MPC application. The numerical tests demonstrate that the solver performs very well even when solving ill-conditioned problems in single-precision floating-point arithmetic.

A Distributed Algorithm That Finds Almost Best Possible Estimate Under Non-Vanishing and Time-Varying Measurement Noise

In this letter, we review an existing distributed least-squares solver and share some new insights on it. Then, by the observation that an estimation of a constant vector under output noise can be translated into finding the least-squares solution, we present an algorithm for distributed estimation of the state of linear time-invariant systems under measurement noise. The proposed algorithm consists of a network of local observers, where each of them utilizes local measurements and information transmitted from the neighbors. It is proven that even under non-vanishing and time-varying measurement noise, we could obtain an almost best possible estimate with arbitrary precision. Some discussions regarding the plug-and-play operation are also given.

Colored and linear inversions to relative acoustic impedance

Acoustic impedance (AI) is a widely used seismic attribute in stratigraphic interpretation. Because of the frequency-band-limited nature of seismic data, seismic amplitude inversion cannot determine AI itself, but it can only provide an estimate of its variations, the relative AI (RAI). We have revisited and compared two alternative methods to transform stacked seismic data into RAI. One is colored inversion (CI), which requires well-log information, and the other is linear inversion (LI), which requires knowledge of the seismic source wavelet. We start by formulating the two approaches in a theoretically comparable manner. This allows us to conclude that both procedures are theoretically equivalent. We proceed to check whether the use of the CI results as the initial solution for LI can improve the RAI estimation. In our experiments, combining CI and LI cannot provide superior RAI results to those produced by each approach applied individually. Then, we analyze the LI performance with two distinct solvers for the associated linear system. Moreover, we investigate the sensitivity of both methods regarding the frequency content present in synthetic data. The numerical tests using the Marmousi2 model demonstrate that the CI and LI techniques can provide an RAI estimate of similar accuracy. A field-data example confirms the analysis using synthetic-data experiments. Our investigations confirm the theoretical and practical similarities of CI and LI regardless of the numerical strategy used in LI. An important result of our tests is that an increase in the low-frequency gap in the data leads to slightly deteriorated CI quality. In this case, LI required more iterations for the conjugate-gradient least-squares solver, but the final results were not much affected. Both methodologies provided interesting RAI profiles compared with well-log data, at low computational cost and with a simple parameterization.

Data-driven polynomial chaos expansions: A weighted least-square approximation

In this work, we combine the idea of data-driven polynomial chaos expansions with the weighted least-square approach to solve uncertainty quantification (UQ) problems. The idea of data-driven polynomial chaos is to use statistical moments of the input random variables to develop an arbitrary polynomial chaos expansion, and then use such data-driven bases to perform UQ computations. Here we adopt the bases construction procedure by following (Ahlfeld et al. (2016), [1]), where the bases are computed by using matrix operations on the Hankel matrix of moments. Different from previous works, in the postprocessing part, we propose a weighted least-squares approach to solve UQ problems. This approach includes a sampling strategy and a least-squares solver. The main features of our approach are two folds: On one hand, our sampling strategy is independent of the random input. More precisely, we propose to sampling with the equilibrium measure, and this measure is also independent of the data-driven bases. Thus, this procedure can be done in prior (or in a off-line manner). On the other hand, we propose to solve a Christoffel function weighted least-square problem, and this strategy is quasi-linearly stable – the required number of PDE solvers depends linearly (up to a logarithmic factor) on the number of (data-driven) bases. This new approach is thus promising in dealing with a class of problems with epistemic uncertainties. A number of numerical tests are presented to show the effectiveness of our approach.

Iterative least-squares solver for generalized maximum-likelihood ptychography.

Although ptychography does not require a precise knowledge of the illumination wavefront, common implementations rely upon assumptions such as accurate knowledge of the scan positions and constant illumination. Limited validity of these assumptions results in deterioration of the reconstruction quality. We present a generalized ptychography method that optimizes the reconstruction along multiple directions. In our manuscript, we demonstrate that the additional flexibility not only helps to amend imprecisions of the ptychography model in a self-consistent way but additionally leads to faster convergence without a significant increase of the computational cost per iteration.

Initial system design method for non-rotationally symmetric systems based on Gaussian brackets and Nodal aberration theory.

Freeform surfaces play important roles in improving the imaging performance of off-axis optical systems. However, for some systems with high requirements in specifications, the structure of the freeform surfaces could be very complicated and the number of freeform surfaces could be large. That brings challenges in fabrication and increases the cost. Therefore, to achieve a good initial system with minimum aberrations and reasonable structure before implementing freeform surfaces is essential for optical designers. The already existing initial system design methods are limited to certain types of systems. A universal tool or method to achieve a good initial system efficiently is very important. In this paper, based on the Nodal aberration theory and the system design method using Gaussian Brackets, the initial system design method is extended from rotationally symmetric systems to general non-rotationally symmetric systems. The design steps are introduced and on this basis, two off-axis three-mirror systems are pre-designed using spherical shape surfaces. The primary aberrations are minimized using the nonlinear least-squares solver. This work provides insight and guidance for initial system design of off-axis mirror systems.

A randomized least squares solver for terabyte-sized dense overdetermined systems

We present a fast randomized least-squares solver for distributed-memory platforms. Our solver is based on the Blendenpik algorithm, but employs multiple random projection schemes to construct a sketch of the input matrix. These random projection sketching schemes, and in particular the use of the randomized Discrete Cosine Transform, enable our algorithm to scale the distributed memory vanilla implementation of Blendenpik to terabyte-sized matrices and provide up to ×7.5 speedup over a state-of-the-art scalable least-squares solver based on the classic QR algorithm. Experimental evaluations on terabyte scale matrices demonstrate excellent speedups on up to 16,384 cores on a Blue Gene/Q supercomputer.