Newton Type Research Articles

Active and Reactive Power Metering in Non-Sinusoidal Conditions Using Newton Type Algorithm.

This paper presents a new two stage Newton Type numerical algorithm for active and reactive power measurement in electric power systems with sinusoidal, non-sinusoidal linear and non-sinusoidal nonlinear circuits. In order to estimate their spectra, in the first stage of the algorithm the current and voltage signals are separately processed, whereas in the second stage the power components are calculated. As the main advantage, this technique provides measurements not sensitive to frequency deviations. The algorithm performance is tested using computer simulated tests.

SHED: A Newton-type algorithm for federated learning based on incremental Hessian eigenvector sharing

There is a growing interest in the distributed optimization framework that goes under the name of Federated Learning (FL). In particular, much attention is being turned to FL scenarios where the network is strongly heterogeneous in terms of communication resources (e.g., bandwidth) and data distribution. In these cases, communication between local machines (agents) and the central server (Master) is a main consideration. In this work, we present SHED, an original communication-constrained Newton-type (NT) algorithm designed to accelerate FL in such scenarios. SHED is by design robust to non independent identically distributed (non i.i.d.) data distributions, handles heterogeneity of agents’ communication resources (CRs), only requires sporadic Hessian computations, and achieves global asymptotic super-linear convergence. This is possible thanks to an incremental strategy, based on eigendecomposition of the local Hessian matrices, which exploits (possibly) outdated second-order information. SHED is thoroughly validated on real datasets by assessing (i) the number of communication rounds required for convergence, (ii) the overall amount of data transmitted and (iii) the number of local Hessian computations. For all these metrics, SHED shows superior performance against state-of-the art techniques like BFGS, GIANT and FedNL.

Convergence of a smoothing algorithm for horizontal linear complementarity problem with a nonmonotone line search

In this paper, we present a nonmonotone smoothing Newton type algorithm for solving the horizontal linear complementarity problem. We show that the proposed algorithm is globally convergent under an assumption that has the column -property. Such an assumption is weaker than those required by most other smoothing Newton algorithms. The preliminary numerical results are reported.

Newton and interior-point methods for (constrained) nonconvex–nonconcave minmax optimization with stability and instability guarantees

We address the problem of finding a local solution to a nonconvex–nonconcave minmax optimization using Newton type methods, including primal-dual interior-point ones. The first step in our approach is to analyze the local convergence properties of Newton’s method in nonconvex minimization. It is well established that Newton’s method iterations are attracted to any point with a zero gradient, irrespective of it being a local minimum. From a dynamical system standpoint, this occurs because every point for which the gradient is zero is a locally asymptotically stable equilibrium point. We show that by adding a multiple of the identity such that the Hessian matrix is always positive definite, we can ensure that every non-local-minimum equilibrium point becomes unstable (meaning that the iterations are no longer attracted to such points), while local minima remain locally asymptotically stable. Building on this foundation, we develop Newton-type algorithms for minmax optimization, conceptualized as a sequence of local quadratic approximations for the minmax problem. Using a local quadratic approximation serves as a surrogate for guiding the modified Newton’s method toward a solution. For these local quadratic approximations to be well-defined, it is necessary to modify the Hessian matrix by adding a diagonal matrix. We demonstrate that, for an appropriate choice of this diagonal matrix, we can guarantee the instability of every non-local-minmax equilibrium point while maintaining stability for local minmax points. Using numerical examples, we illustrate the importance of guaranteeing the instability property. While our results are about local convergence, the numerical examples also indicate that our algorithm enjoys good global convergence properties.

Two‐level multiscale enrichment finite volume method based on the Newton iteration for the stationary incompressible magnetohydrodynamics flow

In this paper, we propose a stabilized finite volume method based on the Newton's type iteration for the steady incompressible magnetohydrodynamics (MHD) problem. In order to reduce the computational complexity, the lowest order mixed finite element pair ( ‐ ‐ ) is adopted to approximate the velocity, pressure and magnetic fields, and the multiscale enrichment method is introduced to overcome the restriction of discrete inf‐sup (LBB) condition. We firstly solve the incompressible MHD equations by the Newton iterations on a coarse mesh with the mesh size , stability and convergence results of numerical solutions are provided. Secondly, the combination of the two‐level method with the Newton iteration is used to approximate the considered problem, a coupled nonlinear problem is solved on the coarse mesh , then the Stokes and Maxwell equations are considered on a fine mesh with the mesh size , the uniform stability and optimal error estimates of two‐level Newton iterative method are provided. Theoretical findings show that the two‐level method has the same order as the one‐level method in ‐norm as long as the mesh sizes satisfy . However, the two‐level method involves much less work than the one‐level method. Finally, some numerical examples are presented to demonstrate the effectiveness of the considered numerical schemes.

Bilevel inverse problems in neuromorphic imaging

Event or neuromorphic cameras are novel biologically inspired sensors that record data based on the change in light intensity at each pixel asynchronously. They have a temporal resolution of microseconds. This is useful for scenes with fast moving objects that can cause motion blur in traditional cameras, which record the average light intensity over an exposure time for each pixel synchronously. This paper presents a bilevel inverse problem framework for neuromorphic imaging. Existence of solution to the inverse problem is established. Second order sufficient conditions are derived under special situations for this nonconvex problem. A second order Newton type solver is derived to solve the problem. The efficacy of the approach is shown on several examples.

Recovering source location, polarization, and shape of obstacle from elastic scattering data

We consider an inverse elastic scattering problem of simultaneously reconstructing a rigid obstacle and the excitation sources using near-field measurements. Specifically, we are concerned with the inverse elastic scattering of a bounded obstacle embedded in an unbounded, homogeneous, and isotropic medium. The obstacle is illuminated by the time-harmonic incident field generated by each point source with a single given frequency, and the time-harmonic measurements are the total field along an entire circular curve enclosing the obstacle. A two-phase numerical method is proposed to achieve the co-inversion of multiple targets. In the first phase, we develop several indicator functionals to determine the source locations and the polarizations from the total field data, and then we manage to obtain the approximate scattered field. In this phase, only the inner products of the total field with the fundamental solutions are involved in the computation, and thus it is direct and computationally efficient. In the second phase, we propose an iteration method of Newton's type to reconstruct the shape of the obstacle from the approximate scattered field. Using the layer potential representations on an auxiliary curve inside the obstacle, the scattered field together with its derivative on each iteration surface can be easily derived. Theoretically, we establish the uniqueness of the co-inversion problem and analyze the indicating behavior of the sampling-type scheme. An explicit derivative is provided for the Newton-type method. Numerical results are presented to corroborate the effectiveness and efficiency of the proposed method.

Newton type methods for solving a Hasegawa–Mima plasma model

In Karakazian and Nassif (2019), the non-linear space-time Hasegawa–Mima plasma equation is formulated as a coupled system of two linear PDEs, a solution of which is a pair (u,w), with w=(I−Δ)u. The first equation is of hyperbolic type and the second of elliptic type. Variational frames for obtaining weak solutions to the initial value Hasegawa–Mima problem with periodic boundary conditions were also derived. In a more recent work Karakazian et al. (2022), a numerical approach consisting of a finite element space-domain combined with an Euler-implicit time scheme was used to discretize the coupled variational Hasegawa–Mima model. A semi-linear version of this implicit nonlinear scheme was tested for several types of initial conditions. This semi-linear scheme proved to lack efficiency for long time, which necessitates imposing a cap on the magnitude of the solution. To circumvent this difficulty, in this paper, we use Newton-type methods (Newton, Chord and an introduced Modified Newton method) to solve numerically the fully-implicit non-linear scheme. Testing these methods in FreeFEM++ indicates significant improvements as no cap needs to be imposed for long time. In the sequel, we demonstrate the validity of these methods by proving several results, in particular the convergence of the implemented methods.

Extremum seeking for distributed delays

We develop extremum seeking schemes for static maps whose inputs are subject to distributed delays. With distributed delays, even generating a probing signal which, when passed through the distributed delay operator, acts as a sinusoid in the static map, is nontrivial. After constructing such a probing signal, using Artstein’s reduction approach, we design a delay-compensating update law for the input into the map and prove the convergence of the estimation error to a neighborhood of the origin with a combination of the averaging theory and a Lyapunov functional. The single-input design is followed by multi-parameter schemes of the gradient and Newton types. The effectiveness of the proposed scheme is confirmed by a numerical simulation.

Candida-associated denture stomatitis: clinical, epidemiological, and microbiological features.

The identification of Candida spp. in denture stomatitis, the clinical manifestations, and the antifungal susceptibility profile lead to a correct and individualized therapeutic management of the patients. This study is aimed at investigating the clinical manifestations and epidemiological and microbiological characteristics of Candida-associated denture stomatitis. The samples were obtained by swabbing the oral mucosa of the subjects and then seeded onto Sabouraud Dextrose Agar and onto CHROMagar® Candida plates. The identification at the species level was confirmed byMatrix Assisted Laser Desorption Time of Flight Mass Spectrometry. Clinical classification was performed according to the criteria proposed by Newton (1962): (i) pinpoint hyperemia, (ii) diffuse hyperemia, and (iii) granular hyperemia. For carrying out the antifungal susceptibility testing, we adopted the CLSI M27-S4 protocol. C. albicans was the most prevalent species in our study. Regarding non-albicans Candida species, C. glabrata was the most common species isolated from the oral mucosa (n = 4, 14.8%), while in the prosthesis, it was C. tropicalis (n = 4, 14.8%). The most prevalent clinical manifestation was pinpoint hyperemia and diffuse hyperemia. Candida albicans, C. glabrata, and C. parapsilosis were susceptible to all the tested antifungals. Concerning fluconazole and micafungin, only two strains showed dose-dependent sensitivity (minimum inhibitory concentration (MIC), 1μg/mL) and intermediate sensitivity (MIC, 0.25μg/mL). One C. tropicalis strain was resistant to voriconazole (MIC, 8μg/mL). C. albicans was the most common species found in oral mucosa and prosthesis. The tested antifungal drugs showed great activity against most isolates. The most prevalent clinical manifestations were Newton's type I and type II.

Coupled thermo‐viscoplastic fracture model for ductile‐brittle failure of amorphous glassy polymers with phase‐field approach

AbstractAmorphous glassy polymers have an extensive use in the industrial sectors including micro‐electronics, medical industry and aerospace, therefore their design and usage have become a significant task nowadays. The fracture response of these polymers may vary from ductile to brittle depending on several factors such as entanglement density, temperature level and external loading rate. The ductile response is driven by diffuse shear zones exhibiting volume–preserving inelastic deformations while the brittle response is manifested by very small crack‐like defects composed of a sequence of fibrillar bridges separated by micro‐voids, thereby connotating void formation consisting of nucleation and propagation steps. The presents study is focused on the description of shear yielding and crazing phenomenon in terms of their respective evolution equations. In addition, an extension towards the modelling of the fracture is employed via the crack phase–field approach, considering ductile and brittle failure simultaneously. This is provided by the novel failure criterion that features a critical amount of plastic strain and void volume fraction. Since the proposed approach unitedly models the macroscopic crack initiation and propagation for ductile or brittle failure, it is asserted to be more physically grounded compared to present models in the literature. Constitutive formulations for shear yielding, crazing, and void volume fraction are derived with their specific forms starting with the local and conductive component of the dissipation inequality. The performance of model is evaluated after developing the local and global Newton–type update algorithms for the dissipative internal and primary field variables, respectively and it has been analysed by fitting of several experimental data of homogeneous and inhomogeneous tests. The findings reveal the remarkable temperature dependency on the type of failure as well as the interaction between loading rate and temperature change owing to dissipative heating in the solid.

On some Simpson's and Newton's type of inequalities in multiplicative calculus with applications

<abstract><p>In this paper, we establish an integral equality involving a multiplicative differentiable function for the multiplicative integral. Then, we use the newly established equality to prove some new Simpson's and Newton's inequalities for multiplicative differentiable functions. Finally, we give some mathematical examples to show the validation of newly established inequalities.</p></abstract>

A fully finite-element based model-space algorithm for three-dimensional inversion of magnetotelluric data

SUMMARY We present a fully finite-element based inversion methodology for imaging 3-D magnetotelluric impedance data on unstructured meshes. The inverse problem is formulated using a minimum-structure Gauss–Newton type optimization scheme that minimizes an objective function with respect to the model perturbation. By introducing a rigorous regularization scheme, we derived a Ritz-type variational formulation of the model objective function and designed a face-based finite-element basis function to discretize the model gradient across tetrahedron’s inter-element boundaries. The forward modelling engine of our optimization scheme is based on a finite-element solution of the E-field Helmholtz equation that is enforced for the magnetotelluric simulation problem using the appropriate edge-based basis functions and 3D boundary conditions. The optimization algorithm developed here utilizes a message passing interface scheme and uses a direct solver to factorize and store both the regularization matrix and the forward modelling coefficient matrix on the processes working in parallel. Having to do this only once within each Gauss–Newton optimization cycle facilitates both the calculation of the dot product of the model regularization terms with the evolving model perturbation, and computing implicitly the sensitivity-vector products. We validated the methodology and the correctness of the developed algorithm for two test examples (COMMEMI 3Ds) from the literature. Also, by comparing the performance between classes of iterative solvers we demonstrated the superior performance of generalized minimum residual solver in reducing the residual norm of the iterative solver during model updates. Using the algorithm in a geologically realistic scenario, we imaged the anticipated geometry of the Lalor volcanogenic massive sulphide deposit in Canada. The feasibility of the imaging methodology is further evaluated with the survey data, for which, again the algorithm converged to the anticipated model solution reproducing the lithostratigraphic sequence of the ore deposit.

Generalized damped Newton algorithms in nonsmooth optimization via second-order subdifferentials

The paper proposes and develops new globally convergent algorithms of the generalized damped Newton type for solving important classes of nonsmooth optimization problems. These algorithms are based on the theory and calculations of second-order subdifferentials of nonsmooth functions with employing the machinery of second-order variational analysis and generalized differentiation. First we develop a globally superlinearly convergent damped Newton-type algorithm for the class of continuously differentiable functions with Lipschitzian gradients, which are nonsmooth of second order. Then we design such a globally convergent algorithm to solve a structured class of nonsmooth quadratic composite problems with extended-real-valued cost functions, which typically arise in machine learning and statistics. Finally, we present the results of numerical experiments and compare the performance of our main algorithm applied to an important class of Lasso problems with those achieved by other first-order and second-order optimization algorithms.

A review of higher order Newton type methods and the effect of numerical damping for the solution of an advanced coupled Lemaitre damage model

In this paper, several Newton-type methods of convergence order 2 or higher were tested on various nonlinear systems of equations and on an advanced material law implemented in a finite-element code. The computational speed, numerical efficiency, and robustness of each method were evaluated for each studied case. The effect of numerical damping was also studied. The results were then compared to put in light the strengths and weaknesses of each method. The most efficient and robust method for the material law in the finite-element code was identified as the Newton method with a selective numerical damping. • Several Newton-type methods were evaluated for solving various numerical problems. • The methods were compared based on computational speed, efficiency, and robustness. • Classic numerical damping can improve the robustness of some, but not all, methods. • Newton method with selective numerical damping is the most efficient and robust.

MLE Evolution Equation for Fractional Diffusions and Berry-Esseen Inequality of Stochastic Gradient Descent Algorithm for American Option

We study recursive parameter estimation in fractional diffusion processes. First, stability and asymptotic properties of the global maximum likelihood estimator (MLE) of the drift parameter are obtained under some regularity conditions. Then we obtain an evolution equation for the MLE of the drift parameter in nonhomogeneous fractional stochastic differential equation (fSDE) driven by fractional Brownian motion. This equation is then modified to yield an algorithm which is consistent, asymptotically efficient and converges to the MLE. The gradient and Newton type algorithm are firstorder approximations. Finally we study the Berry-Esseen inequality for stochastic gradient descent in continuous time (SGDCT) algorithm for American option. We compare it with Longstaff-Schwartz regression based method.

Hessian Informed Mirror Descent

Inspired by the recent paper (L. Ying, Journal of Scientific Computing, 84, 1–14 (2020), we explore the relationship between the mirror descent and the variable metric method. When the metric in the mirror decent is induced by a convex function, whose Hessian is close to the Hessian of the objective function, this method enjoys both robustness from the mirror descent and superlinear convergence for Newton type methods. When applied to a linearly constrained minimization problem, we prove the global and local convergence, both in the continuous and discrete settings. As applications, we compute the Wasserstein gradient flows and Cahn-Hillard equation with degenerate mobility. When formulating these problems using a minimizing movement scheme with respect to a variable metric, our mirror descent algorithm offers a fast convergence speed for the underlying optimization problem while maintaining the total mass and bounds of the solution.

Iteratively regularized Gauss–Newton type methods for approximating quasi–solutions of irregular nonlinear operator equations in Hilbert space with an application to COVID–19 epidemic dynamics

We investigate a class of iteratively regularized methods for finding a quasi–solution of a noisy nonlinear irregular operator equation in Hilbert space. The iteration uses an a priori stopping rule involving the error level in input data. In assumptions that the Frechet derivative of the problem operator at the desired quasi–solution has a closed range, and that the quasi–solution fulfills the standard source condition, we establish for the obtained approximation an accuracy estimate linear with respect to the error level. The proposed iterative process is applied to the parameter identification problem for a SEIR–like model of the COVID–19 pandemic.

Dualization and Automatic Distributed Parameter Selection of Total Generalized Variation via Bilevel Optimization

Total Generalized Variation (TGV) regularization in image reconstruction relies on an infimal convolution type combination of generalized first- and second-order derivatives. This helps to avoid the staircasing effect of Total Variation (TV) regularization, while still preserving sharp contrasts in images. The associated regularization effect crucially hinges on two parameters whose proper adjustment represents a challenging task. In this work, a bilevel optimization framework with a suitable statistics-based upper level objective is proposed in order to automatically select these parameters. The framework allows for spatially varying parameters, thus enabling better recovery in high-detail image areas. A rigorous dualization framework is established, and for the numerical solution, a Newton type method for the solution of the lower level problem, i.e. the image reconstruction problem, and a bilevel TGV algorithm are introduced. Denoising tests confirm that automatically selected distributed regularization parameters lead in general to improved reconstructions when compared to results for scalar parameters.

2D Local identification of the elasticity parameters of heterogeneous materials by the finite element model updating method

Abstract This work is subjected to an identification of elasticity properties which are the Young’s modulus E and the Poisson’s ratio v of a heterogeneous material. We have created the experimental displacements by solving the direct problem. A re-calibration of the experimental and the simulated displacements is done based on the principle of the finite element model updating (FEMU). The minimization of the cost function obtained by the FEMU method in the inverse problem is done by Newton type algorithms. A complete study has been done by studying the sensitivity of the identified values with respect to the refinement of the mesh and with respect to the level of disturbance.