Nonlinear Minimization Algorithm Research Articles

One-dimensional convolutional neural network for Jacobian in Diffuse Optical Tomography.

Non-linear least square minimization algorithms are often employed to solve Diffuse Optical Tomography (DOT) inverse problem. However, it is time-consuming to calculate the Jacobian matrix. This work has proposed a data-driven neural network method to improve computational efficiency. The singular value decomposition is employed to compute the updated Jacobian and a mapping from boundary measurements to the singular values based on a convolutional neural network (CNN) is learned to obtain the singular values. The method is validated with 3D numerical simulation data. We have demonstrated that the approach can save computation time compared to Adjoint method, and reconstructed absorption coefficient close to Adjoint method.Clinical Relevance- These results are not focused on clinical relevance currently, but in the future may be helpful to accelerant DOT reconstruction in clinic.

Statistical Analysis of the Measurement Noise in Dynamic Impedance Spectra.

Different strategies can be used to acquire dynamic impedance spectra during a cyclic voltammetry experiment. The spectra are then analyzed by fitting them with a model using a weighted non‐linear least‐squares minimization algorithm. The choice of the weighting factors is not trivial and influences the value of the extracted parameters. At variance with the classic electrochemical impedance spectroscopy, dynamic impedance measurements are performed under non‐stationary conditions, making them typically more prone to errors arising from the voltage and current analog‐to‐digital conversion. Under the assumption that the noise in the voltage and current signals have a constant variance along the measurement and that it is uncorrelated between distinct samples, we calculate an expression for the expected variance of the error of the resulting immittances, which considers the specific procedure used to extract the spectra under the time‐varying nature of the measurements. The calculated variance of the error is then used as a rigorous way to evaluate the weighting factors of the least‐squares minimization, assuming that the fitted model is ideally exact and that there are no systematic errors. By exploring two classical electrochemical systems and fitting the measured spectra with a transfer function measurement model, namely with the Padé approximants, we show that the variance evaluated with our method captures the frequency dependence of the resulting residuals and can be used for reliably performing the complex non‐linear least‐squares fitting procedure.

Optimization of the Wake Oscillator for Transversal VIV

Vibrations of slender structures associated with the external flow present a design challenge for the energy production systems placed in the marine environment. The current study explores the accuracy of the semi-empirical wake oscillator models for vortex-induced vibrations (VIV) based on the optimization of (a) the damping term and (b) empirical coefficients in the fluid equation. This work investigates the effect of ten fluid damping variations, from the classic van der Pol to more sophisticated fifth-order terms, and prediction of the simplified case of the VIV of transversally oscillating rigid structures provides an opportunity for an extended, comprehensive comparison of the performance of tuned models. A constrained nonlinear minimization algorithm in MATLAB is applied to calibrate considered models using the published experimental data, and the weighted objective function is formulated for three different mass ratios. Comparison with several sources of published experimental data for cross-flow oscillations confirms the model accuracy in the mass ratio range. The study indicates the advantageous performance of the models tuned with the medium mass ratio data and highlights some advantages of the Krenk–Nielsen wake oscillator.

A new logistic growth model applied to COVID-19 fatality data

Background:Recent work showed that the temporal growth of the novel coronavirus disease (COVID-19) follows a sub-exponential power-law scaling whenever effective control interventions are in place. Taking this into consideration, we present a new phenomenological logistic model that is well-suited for such power-law epidemic growth. Methods:We empirically develop the logistic growth model using simple scaling arguments, known boundary conditions and a comparison with available data from four countries, Belgium, China, Denmark and Germany, where (arguably) effective containment measures were put in place during the first wave of the pandemic. A non-linear least-squares minimization algorithm is used to map the parameter space and make optimal predictions. Results:Unlike other logistic growth models, our presented model is shown to consistently make accurate predictions of peak heights, peak locations and cumulative saturation values for incomplete epidemic growth curves. We further show that the power-law growth model also works reasonably well when containment and lock down strategies are not as stringent as they were during the first wave of infections in 2020. On the basis of this agreement, the model was used to forecast COVID-19 fatalities for the third wave in South Africa, which was in progress during the time of this work. Conclusion:We anticipate that our presented model will be useful for a similar forecasting of COVID-19 induced infections/deaths in other regions as well as other cases of infectious disease outbreaks, particularly when power-law scaling is observed.

Designing Anisotropic, Inhomogeneous Metamaterial Devices Through Optimization

A method for designing multi-input, multi-output metamaterial devices is presented. The devices are 2-D anisotropic, inhomogeneous media. The design technique consists of a custom finite-element method solver coupled to a constrained, nonlinear minimization algorithm. The design method has advantages over existing metamaterial design methods such as transformation optics, in that material constraints can be imposed and multi-input, multi-output functionality allowed. Transformation optics devices only achieve multi-input, multi-output functionality through geometrical symmetry. The gradient-based optimization method employed in the metamaterial design process is fast given that an analytic expression for the gradient of the cost function can be written. The proposed technique is demonstrated through the design of two beamforming devices. A commercial full-wave solver is used to validate the design approach. The constitutive parameters of these 2-D designs can be implemented using printed-circuit board metamaterials, such as tensor transmission-line metamaterials.

On the Importance of Choosing the Best Minimization Algorithm for the Determination of Ternary Diffusion Coefficients by the Taylor Dispersion Method.

Taylor dispersion method is a common technique for the determination of diffusion coefficients in the case of multicomponent systems. One of the main problems related to the parameter estimation analysis of the collected results is the choice of the best minimization algorithm that allows finding the real minimum of the objective function. Usually, researchers use the Levenberg–Marquardt algorithm, averaging the parameters obtained by different estimation analyses. In this paper, some nonlinear minimization algorithms included in MATLAB R2016a have been tested, and the results are compared in terms of best fit on the experimental data collected for sodium dodecyl sulfate (SDS) + sodium octanoate (SOC) + water system.

A Hybridizable Discontinuous Galerkin Method for the $p$-Laplacian

We propose the first hybridizable discontinuous Galerkin method for the $p$-Laplacian equation. When using polynomials of degree $k \geq 0$ for the approximation spaces of $u$, $\boldsymbol{\nabla} u$, and $|\boldsymbol{\nabla} u|^{p-2}\boldsymbol{\nabla} u$, the method exhibits optimal $k+1$ order of convergence for all variables in $L^1$- and $L^p$-norms in our numerical experiments. For $k \geq 1$, an elementwise computation allows us to obtain a new approximation $u_h^*$ that converges to $u$ with order $k+2$. We rewrite the scheme as discrete minimization problems in order to solve them with nonlinear minimization algorithms. The unknown of the first problem is the approximation of $u$ on the skeleton of the mesh but requires solving nonlinear local problems. The second problem has the approximation on the elements as an additional unknown but it only requires solving linear local problems. We present numerical results displaying the convergence properties of the methods, demonstrating the utility of...

A note on “A multiple-vendor single-buyer integrated inventory model with a variable number of vendors”

In this technical note, an alternative solution procedure to the model that Glock (2011) developed is presented. In particular, we take into account the same assumptions, but the total cost function is reformulated, i.e., the step-type discontinuity is replaced by a logistic approximation and the total costs of the system are computed on the whole set of preselected suppliers. Subsequently, we show that the total cost function possesses some properties in terms of convexity and continuity that allow the exploitation of standard constrained nonlinear minimization algorithms. Finally, tests conducted on the same set of problems that Glock (2011) originally considered illustrate the good performances of the solution procedure developed in this note.

Non-linear redundancy calibration

For radio interferometric arrays with a sufficient number of redundant spacings the multiplicity of measurements of the same sky visibility can be used to determine both the antenna gains as well as the true visibilities. Many of the earlier approaches to this problem focused on linearized versions of the relation between the measured and the true visibilities. Here we propose to use a standard non-linear minimization algorithm to solve for both the antenna gains as well as the true visibilities. We show through simulations done in the context of the ongoing upgrade to the Ooty Radio Telescope that the non-linear minimization algorithm is fast compared to the earlier approaches. Further, unlike the most straightforward linearized approach, which works with the logarithms of the visibilities and the gains, the non-linear minimization algorithm leads to unbiased solutions. Finally we present error estimates for the estimated gains and visibilities. Monte-Carlo simulations establish that the estimator is indeed statistically efficient, achieving the Cramer-Rao bound.

Optimal apparent damping as a function of the bandwidth of an array of vibration absorbers

The transient response of a resonant structure can be altered by the attachment of one or more substantially smaller resonators. Considered here is a coupled array of damped harmonic oscillators whose resonant frequencies are distributed across a frequency band that encompasses the natural frequency of the primary structure. Vibration energy introduced to the primary structure, which has little to no intrinsic damping, is transferred into and trapped by the attached array. It is shown that, when the properties of the array are optimized to reduce the settling time of the primary structure's transient response, the apparent damping is approximately proportional to the bandwidth of the array (the span of resonant frequencies of the attached oscillators). Numerical simulations were conducted using an unconstrained nonlinear minimization algorithm to find system parameters that result in the fastest settling time. This minimization was conducted for a range of system characteristics including the overall bandwidth of the array, the ratio of the total array mass to that of the primary structure, and the distributions of mass, stiffness, and damping among the array elements. This paper reports optimal values of these parameters and demonstrates that the resulting minimum settling time decreases with increasing bandwidth.

Modeling of Dough Mixing Profile Under Thermal and Nonthermal Constraint for Evaluation of Breadmaking Quality of Hard Spring Wheat Flour

ABSTRACTThis research was initiated to investigate associations between flour breadmaking traits and mixing and empirical dough rheological properties under thermal stress. Thirty hard spring wheat flour samples were analyzed by a Mixolab standard procedure. Mixolab profiles were divided into six different stages, and torque measurements of individual stages were modeled by nonlinear curve fitting using a compound of two solution searching procedures, multidimensional unconstrained nonlinear minimization and genetic algorithm. Mixing patterns followed exponential equations. Dough torque patterns under heat constraint, specifically dough thermal weakening and pasting profiles, were described by a sigmoid logistic equation as a function of time. Dough stability during heating appeared important for bread loaf volume increase from significant correlations between bread loaf volume and parameters generated from models of a dough thermal weakening stage. Multivariate continuum regression was employed to calibrate prediction models of baking traits using Mixolab parameters. Coefficients of determination estimated from prediction models and cross‐validation were greater than 0.98 for bake water absorption, mixing time, and bread loaf volume, indicating that the Mixolab parameters have a potential to enhance evaluation of flour breadmaking quality.

On the inverse doping profile problem

We obtain new analytic results for the problem of the recovery of a doped region $D$ in semiconductor devices from the total flux of electrons/holes through a part of the boundary for various applied potentials on some complementary part of the boundary. We consider the stationary two-dimensional case and we use the index of the gradient of solutions of the linear elliptic equation modeling a unipolar device. Under mild assumptions we prove local uniqueness of smooth $D$ and global uniqueness of polygonal $D$ satisfying some geometrical (star-shapednedness or convexity in some direction) assumptions. We design a nonlinear minimization algorithm for numerical solution and we demonstrate its effectiveness on some basic examples. An essential ingredient of this algorithm is a numerical solution of the direct problem by using single layer potentials.

Prediction of the thermodynamic properties of {ammonia + water} using cubic equations of state with the SOF cohesion function

The SOF cohesion function for cubic equations of state is based on the behavior of the residual energy of pure fluids. It contains two adjustable parameters for each component, which have been obtained for over 800 substances by regression of pure-fluid saturation pressures, and correlated in terms of a four-parameter corresponding states principle. In the present work, we compare the performance of this function and of the original Soave cohesion function with the Redlich–Kwong and Peng–Robinson equations of state in the prediction of vapor–liquid equilibria and enthalpy–composition diagrams for the polar system {ammonia + water}. We use simple van der Waals one-fluid mixing rules, linear for the covolume and quadratic for the cohesion parameter with one (symmetric) and two (asymmetric) binary interaction parameters. The non-linear least squares minimization algorithm lsqnonlin, in Matlab ®, is used to adjust the interaction parameters to phase equilibrium and enthalpy data taken from the IAPWS fundamental formulation. Upper and lower bounds of the optimized interaction parameters are obtained using Matlab ®bootstrap with 95% confidence of a normal distribution sampling. The validity of the parameters as functions of temperature is between the triple point of water and the critical point of ammonia. At lower temperatures, a rapid increase of statistical uncertainties is observed that can be attributed to the scarcity of phase equilibrium data. The two-parameter SOF cohesion function and the cubic equations of state are shown to give accurate predictions of the VLE and enthalpies of {ammonia + water}. Both equations of state give very similar results. Statistical analysis of the interaction parameters shows that their values (within the range of validity mentioned above) are effectively the same for both cohesion functions. At higher temperatures, however, extrapolation of the two cohesion functions gives different results, and correspondingly requires different interaction parameters.

An improved delay-dependent bounded real lemma (BRL) andH∞controller synthesis for linear neutral systems

In this paper, a novel delay-dependent bounded real criterion and an improved sufficient condition are derived for the design of an H∞ state-feedback controller for linear neutral time-delay systems. On the basis of an augmented Lyapunov-Krasovskii functional, a new bounded real lemma is introduced in terms of a convex linear matrix inequality (LMI) condition that can be solved using interior point algorithms. The bounded real lemma is extended to obtain a sufficient condition for the existence of a delay-dependent H∞ memoryless state-feedback controller. Neither any model transformation nor bounding of any of the cross terms are utilized while deriving the bounded real lemma. Moreover, the use of any free slack matrix variable approach is avoided to a certain extent in order not to increase the complexity of the synthesis problem. A cone complementary nonlinear minimization algorithm is employed to achieve a feasible solution set for the synthesis conditions. Finally, seven numerical examples are given to illustrate the effectiveness of the proposed method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

A Metamodeling Technique for Variable Geometry Trusses Design via Equivalent Parametric Macroelements

A metamodeling methodology is applied to reduce the large computational cost required in the design of variable geometry trusses (VGTs). Using this methodology, submodels of finite elements within a complete FE model of the VGT are substituted by groups of fewer elements called equivalent parametric macroelements (EPMs). The EPM optimum parameters are obtained using equivalence criteria based on elastic energy and inertial properties. The optimization process is performed using nonlinear least square minimization and genetic algorithms.

Chemical kinetics of Ru-catalyzed ammonia borane hydrolysis

Ammonia borane (AB) is a candidate material for on-board hydrogen storage, and hydrolysis is one of the potential processes by which the hydrogen may be released. This paper presents hydrogen generation measurements from the hydrolysis of dilute AB aqueous solutions catalyzed by ruthenium supported on carbon. Reaction kinetics necessary for the design of hydrolysis reactors were derived from the measurements. The hydrolysis had reaction orders greater than zero but less than unity in the temperature range from 16 °C to 55 °C. A Langmuir–Hinshelwood kinetic model was adopted to interpret the data with parameters determined by a non-linear conjugate-gradient minimization algorithm. The ruthenium-catalyzed AB hydrolysis was found to have activation energy of 76 ± 0.1 kJ mol −1 and adsorption energy of −42.3 ± 0.33 kJ mol −1. The observed hydrogen release rates were 843 ml H 2 min −1 (g catalyst) −1 and 8327 ml H 2 min −1 (g catalyst) −1 at 25 °C and 55 °C, respectively. The hydrogen release from AB catalyzed by ruthenium supported on carbon is significantly faster than that catalyzed by cobalt supported on alumina. Finally, the kinetic rate of hydrogen release by AB hydrolysis is much faster than that of hydrogen release by base-stabilized sodium borohydride hydrolysis.

SU‐GG‐T‐311: A Distributed‐Computing Platform for Monte Carlo Dose Calculations and Beam‐Weight Optimization

Purpose: Monte carlo (MC) algorithms remain the gold standard in dose calculation routines. However, their long calculation times generally make them infeasible for clinical implementation, especially for IMRT optimization which requires rapid pencil‐beam dose computation. In a preliminary study, we are investigating how high‐throughput computing may be employed to perform MC dose calculations and rudimentary optimization using non‐commercial software. Method and Materials: Two non‐commercial software packages were used for this project, VMC++ and CERR. VMC++ is an MC dose calculation package which is more efficient than general‐purpose MC routines due to its use of variance reduction techniques. A combination of particle splitting and Russian Roulette methods is employed in VMC++ for photon transport modeling, which enhances the efficiency and speed of the calculation, making it approximately 50 times faster than the BEAMnrc code. CERR is an open‐source environment for radiotherapy calculations. We used the data‐importation and beam‐shaping features of CERR in conjunction with the dose‐calculation routines of VMC++ to perform dose computation on a 4‐field 3D CRT prostate case. We then optimized the beam weights using DVH‐based constraints and the Nelder‐Mead simplex technique, a multidimensional unconstrained nonlinear minimization algorithm. Collaborating with members of an academic supercomputing facility enabled us to perform the calculations on a subset of a cluster of 1000 dual‐core Linux‐based computers. Results: The dose calculation and optimization, involving several million particle histories, required only a few minutes to complete. The resulting plan was optimal based on the constraints provided to the system, and indicated that additional constraints may be necessary for clinical feasibility. Conclusion: We have shown that, using a distributed‐computing platform, an MC routine may be employed for treatment planning in a reasonable amount of time. This work will be extended to IMRT and beam‐angle optimization by combining a variety of Techniques.

Estimation of Permeability and Permeability Anisotropy From Straddle-Packer Formation-Tester Measurements Based on the Physics of Two-Phase Immiscible Flow and Invasion

Summary We describe the successful application of a new method to estimate permeability and permeability anisotropy from transient measurements of pressure acquired with a wireline straddle-packer formation tester. Unlike standard algorithms used for the interpretation of formation-tester measurements, the method developed in this paper incorporates the physics of two-phase immiscible flow as well as the processes of mudcake buildup and invasion. An efficient 2D (cylindrical coordinates) implicit-pressure explicit-saturation finite-difference algorithm is used to simulate both the process of invasion and the pressure measurements acquired with the straddle-packer formation tester. Initial conditions for the simulation of formation-tester measurements are determined by the spatial distributions of pressure and fluid saturation resulting from mud-filtrate invasion. Inversion is performed with a Levenberg-Marquardt nonlinear minimization algorithm. Sensitivity analyses are conducted to assess nonuniqueness and the impact of explicit assumptions made about fluid viscosity, capillary pressure, relative permeability, mudcake growth, and time of invasion on the estimated values of permeability and permeability anisotropy. Applications of the inversion method to noisy synthetic measurements include homogeneous, anisotropic, single and multilayer formations for cases of low- and high-permeability rocks. We also study the effect of unaccounted impermeable bed boundaries on inverted formation properties. For cases where a priori information can be sufficiently constrained, our inversion methodology provides reliable and accurate estimates of permeability and permeability anisotropy. In addition, we show that estimation errors of permeability inversion procedures that neglect the physics of two-phase immiscible fluid flow and mud-filtrate invasion can be as high as 100%.

Strain accumulation across the Gazikoy–Saros segment of the North Anatolian Fault inferred from Persistent Scatterer Interferometry and GPS measurements

We use a combination of Persistent Scatterer Interferometry (PSI) and GPS observations to study interseismic crustal deformation on the Gazikoy–Saros segment (Ganos fault) of the North Anatolian Fault (NAF) zone, northwestern Turkey. The data include 44 C-band radar images collected by the ERS1 and ERS2 satellites in descending orbits between 1992 and 2003 over the study area, and campaign GPS horizontal velocities from 7 stations. The resultant secular velocity field is inverted using a nonlinear minimization algorithm to estimate parameters of two interseismic deformation models: aseismic deep slip in a purely elastic earth model (elastic half-space rheology), and viscoelastic flow in an elastic–viscoelastic earth model (viscoelastic-coupling rheology). The following conclusions are drawn: (1) The fault locking depth is estimated in the range of ∼ 8–17 km (95% confidence interval) regardless of the rheological model. (2) The elastic half-space model implies an upper bound of 20–27 mm/yr for the slip rate on the Ganos fault. (3) Models incorporating viscoelastic rheology and seismic cycle effects suggest a lower slip rate of 18–24 mm/yr, which agrees more closely with geological estimates. However, these values are slightly sensitive to the assumed earthquake recurrence interval. (4) A bootstrap analysis of deformation data yields average crust-upper mantle viscosities of 1.3 × 10 19–3.6 × 10 20 Pa s for the Ganos area.

Avoiding Spurious Submovement Decompositions II: A Scattershot Algorithm

Evidence for the existence of discrete submovements underlying continuous human movement has motivated many attempts to "extract" them. Although they produce visually convincing results, all of the methodologies that have been employed are prone to produce spurious decompositions. In previous work, a branch-and-bound algorithm for submovement extraction, capable of global nonlinear minimization, and hence, capable of avoiding spurious decompositions, was presented [Rohrer and Hogan (Biol Cybern 39:190-199, 2003)]. Here, we present a scattershot-type global nonlinear minimization algorithm that requires approximately four orders of magnitude less time to compute. A sensitivity analysis reveals that the scattershot algorithm can reliably detect changes in submovement parameters over time, e.g., over the course of neuromotor recovery.