Damped Newton Algorithm Research Articles

Metasurfaces and Optimal transport

This paper provides a theoretical and numerical approach to show existence, uniqueness, and the numerical determination of metalenses refracting radiation with energy patterns. The theoretical part uses ideas from optimal transport and for the numerical solution we study and implement a damped Newton algorithm to solve the semi discrete problem. A detailed analysis is carried out to solve the near field one source refraction problem and extensions to the far field are also mentioned.

The Aero-Engine Component-Level Modelling Research Based on NSDE Hybrid Damping Newton Method

Advanced aero-engine component-level models are characterized by strong nonlinearity and multivariate, and traditional iterative algorithms cannot meet the requirements of convergence, real-time, and accuracy at the same time. To improve the convergence and alleviate the initial value dependence, a hybrid damped Newton algorithm based on the neighborhood based speciation differential evolution (NSDE) is proposed in this paper for solving the aero-engine component-level model. The computational efficiency and convergence of the hybrid damped Newton algorithm and NSDE hybrid damped Newton algorithm under four typical steady-state operating point conditions are analyzed, and then, the accuracy of the model is verified. It is demonstrated that the hybrid damped Newton method has the advantage of low initial value sensitivity and high computational efficiency under large deviation conditions. The hybrid damped Newton method is more efficient than the Broyden algorithm in terms of iterative efficiency, faster than the traditional N-R algorithm in terms of computation speed, and has the highest computational convergence rate under the four typical operating conditions, but it cannot eliminate the initial value dependence. The NSDE hybrid damped Newton method offers high simulation accuracy and greatly increases computational real-time performance under large deviation conditions, and the maximum error between the numerical simulation results and the experimental reference value is 8.1%. This study provides advanced theoretical support for component-level modelling and has certain engineering application value.

Accurate and fast reconstruction for bioluminescence tomography based on adaptive Newton hard thresholding pursuit algorithm.

As a promising noninvasive medical imaging technique, bioluminescence tomography (BLT) dynamically offers three-dimensional visualization of tumor distribution in living animals. However, due to the high ill-posedness caused by the strong scattering property of biological tissues and the limited boundary measurements with noise, BLT reconstruction still cannot meet actual preliminary clinical application requirements. In our research, to recover 3D tumor distribution quickly and precisely, an adaptive Newton hard thresholding pursuit (ANHTP) algorithm is proposed to improve the performance of BLT. The ANHTP algorithm fully combines the advantages of sparsity constrained optimization and convex optimization to guarantee global convergence. More precisely, an adaptive sparsity adjustment strategy was developed to obtain the support set of the inverse system matrix. Based on the strong Wolfe line search criterion, a modified damped Newton algorithm was constructed to obtain optimal source distribution information. A series of numerical simulations and phantom and in vivo experiments show that ANHTP has high reconstruction accuracy, fast reconstruction speed, and good robustness. Our proposed algorithm can further increase the practicality of BLT in biomedical applications.

A damped Newton algorithm for generated Jacobian equations

Generated Jacobian Equations have been introduced by Trudinger (Discrete Contin Dyn Syst A 34(4):1663–1681, 2014) as a generalization of Monge–Ampère equations arising in optimal transport. In this paper, we introduce and study a damped Newton algorithm for solving these equations in the semi-discrete setting, meaning that one of the two measures involved in the problem is finitely supported and the other one is absolutely continuous. We also present a numerical application of this algorithm to the near-field parallel reflector problem arising in non-imaging problems.

Computational semi-discrete optimal transport with general storage fees

We propose and analyze a modified damped Newton algorithm to solve the semi-discrete optimal transport with storage fees. We prove global linear convergence for a wide range of storage fee functions, the main assumption being that each warehouse's storage costs are independent. We show that if F is an arbitrary storage fee function that satisfies this independence condition then F can be perturbed into a new storage fee function so that our algorithm converges. We also show that the optimizers are stable under these perturbations. Furthermore, our results come with quantitative rates.

Critical point calculations by numerical inversion of functions

In this work, we propose a new approach to the problem of critical point calculation, based on the formulation of Heidemann and Khalil. This leads to a 2 × 2 system of nonlinear algebraic equations in temperature and molar volume, which makes possible the prediction of critical points of the mixture through an adaptation of the technique of inversion of functions from the plane to the plane, proposed by Malta, Saldanha, and Tomei. The results are compared to those obtained by three methodologies: (i) the classical method of Heidemann and Khalil, which uses a double-loop structure, also in terms of temperature and molar volume; (ii) the algorithm of Dimitrakopoulos, Jia, and Li, which employs a damped Newton algorithm and (iii) the methodology proposed by Nichita and Gomez, based on a stochastic algorithm. The proposed methodology proves to be robust and accurate in the prediction of critical points, as well as provides a global view of the nonlinear problem.

Power penalty method for solving HJB equations arising from finance

Nonlinear Hamilton–Jacobi–Bellman (HJB) equation commonly occurs in financial modeling. Implicit numerical scheme is usually applied to the discretization of the continuous HJB so as to find its numerical solution, since it is generally difficult to obtain its analytic viscosity solution. This type of discretization results in a nonlinear discrete HJB equation. We propose a power penalty method to approximate this discrete equation by a nonlinear algebraic equation containing a power penalty term. Under some mild conditions, we give the unique solvability of the penalized equation and show its convergence to the original discrete HJB equation. Moreover, we establish a sharp convergence rate of the power penalty method, which is of an exponential order with respect to the power of the penalty term. We further develop a damped Newton algorithm to iteratively solve the lower order penalized equation. Finally, we present a numerical experiment solving an incomplete market optimal investment problem to demonstrate the rates of convergence and effectiveness of the new method. We also numerically verify the efficiency of the power penalty method by comparing it with the widely used policy iteration method.

Convergence of a Newton algorithm for semi-discrete optimal transport

A popular way to solve optimal transport problems numerically is to assume that the source probability measure is absolutely continuous while the target measure is finitely supported. We introduce a damped Newton algorithm in this setting, which is experimentally efficient, and we establish its global linear convergence for cost functions satisfying an assumption that appears in the regularity theory for optimal transport.

A damped Newton algorithm for computing viscoplastic fluid flows

For the first time, a Newton method is proposed for the unregularized viscoplastic fluid flow problem. It leads to a superlinear convergence for Herschel–Bulkley fluids when 0 < n < 1, where n is the power law index. Performances are enhanced by using the inexact variant of the Newton method and, for solving the Jacobian system, by using an efficient preconditioner based on the regularized problem. A demonstration is provided by computing a viscoplastic flow in a pipe with a square cross section. Comparisons with the augmented Lagrangian algorithm show a dramatic reduction of the required computing time while this new algorithm provides an equivalent accuracy for the prediction of the yield surfaces.

Incorporating the FAVAD Leakage Equation into Water Distribution System Analysis

The standard formulation of the hydraulic network equations incorporates a power function for modeling pressure-dependent consumption such as leakage. However, recent research has shown that the FAVAD leakage model more accurately accounts for the behavior of leaks in practice. The objective of this paper is to propose a method for solving the hydraulic network equations incorporating the FAVAD model. An energy minimization problem is used to prove the existence and the uniqueness of the solution to this problem, and to provide the basis for a robust solver. A damped Newton algorithm is proposed for solving the system equations.

An efficient numerical algorithm for the L2 optimal transport problem with periodic densities

We present an extension of the numerical method of Loeper and Rapetti (2005, Numerical solution of the Monge–Ampère equation by a Newton's algorithm. C.R. Acad. Sci. Paris, I, 319–324) for the Monge–Ampère equation to non-uniform target densities and adopt it to solve the optimal transport problem with quadratic cost. The method employs a damped Newton algorithm to solve the Monge–Ampère equation. We show that the algorithm converges for sufficiently large damping coefficients, for the case where the source and target densities are sufficiently smooth, periodic and bounded away from zero. At each Newton iteration, we solve a non-constant coefficient linear partial differential equation. To improve the efficiency of the procedure, we use an analytically preconditioned fast Fourier transform method coupled with GMRES (Strain, J. (1994) Fast spectrally-accurate solution of variable-coefficients elliptic problems. Proc. Amer. Math. Sci., 122, 843–850) to solve this equation, as opposed to a more straightforward approach based on a second-order finite-difference discretization combined with biconjugate gradient used in the original LOEPER and RAPETTI paper. Finally, we present some numerical experiments in image processing to demonstrate the efficiency of the method.

Optimization Algorithms on Subspaces: Revisiting Missing Data Problem in Low-Rank Matrix

Low-rank matrix approximation has applications in many fields, such as 3D reconstruction from an image sequence and 2D filter design. In this paper, one issue with low-rank matrix approximation is re-investigated: the missing data problem. Much effort was devoted to this problem, and the Wiberg algorithm or the damped Newton algorithm were recommended in previous studies. However, the Wiberg or damped Newton algorithms do not suit for large (especially "long") matrices, because one needs to solve a large linear system in every iteration. In this paper, we revitalize the usage of the Levenberg-Marquardt algorithm for solving the missing data problem, by utilizing the property that low-rank approximation is a minimization problem on subspaces. In two proposed implementations of the Levenberg-Marquardt algorithm, one only needs to solve a much smaller linear system in every iteration, especially for "long" matrices. Simulations and experiments on real data show the superiority of the proposed algorithms. Though the proposed algorithms achieve a high success rate in estimating the optimal solution by random initialization, as illustrated by real examples; it still remains an open issue how to properly do the initialization in a severe situation (that is, a large amount of data is missing and with high-level noise).

Self-concordant functions for optimization on smooth manifolds

This paper discusses self-concordant functions on smooth manifolds. In Euclidean space, such functions are utilized extensively as barrier functions in interior-point methods for polynomial time optimization algorithms. Here, the self-concordant function is carefully defined on a differential manifold in such a way that the properties of self-concordant functions in Euclidean space are preserved. A Newton decrement is defined and analyzed for this class of functions. Based on this, a damped Newton algorithm is proposed for the optimization of self-concordant functions. Under reasonable technical assumptions such as geodesic completeness of the manifold, this algorithm is guaranteed to fall in any given small neighborhood of the optimal solution in a finite number of steps. The existence and uniqueness of the optimal solution is also proved in this paper. Hence, the optimal solution is a global one. Furthermore, it ensures a quadratic convergence within a neighborhood of the minimal point. This neighborhood can be specified in terms of the Newton decrement. The computational complexity bound of the proposed approach is also given explicitly. This complexity bound is shown to be of the order $$O(-{\rm ln}(\epsilon)),$$ where $$\epsilon$$ is the desired precision. Some interesting optimization problems are given to illustrate the proposed concept and algorithm.

An advanced equation assembly module

We present an advanced equation assembly module which has been developed for the simulation of semiconductor devices based on the Finite Boxes discretization scheme and is currently used in the general purpose device and circuit simulator Minimos-NT. Such simulations require the solution of a specific set of nonlinear partial differential equations which are discretized on a grid. The resulting nonlinear problem is solved by a damped Newton algorithm that demands the solution of a linear equation system at each step. The presented module is responsible for assembling these systems and takes into account several requirements of the simulation process. The underlying concepts, namely the representation of boundary conditions, physically motivated variable transformation, preelimination and numerical conditioning, are presented together with some examples.

Newton’s algorithm in Euclidean Jordan algebras, with applications to robotics

We consider a convex optimization problem on linearly constrained cones in Euclidean Jordan algebras. The problem is solved using a damped Newton algorithm. Quadratic convergence to the global minimum is shown using an explicit step-size selection. Moreover, we prove that the algorithm is a smooth discretization of a Newton flow with Lipschitz continuous derivative. 1. Introduction. It is a great pleasure to contribute this paper to the special issue, honoring John Moore on the occasion of his 60th birthday. I (U.H.) first met John in 1989 during the end of a short visit of mine at the ANU. Communication between us went along easily right from the start, despite quite dierent backgrounds in mathematics and control. Initial ideas for future research were immediately flying high at fast speed and at the end of the day we had a new Ph.D. student starting to work on balanced realization flows as well as an emerging plan for longer term research collaborations. The excitement and fun in working with John has been steady ever since and added a lot to my own enjoyment of research and life. The motivation for this paper comes from a problem in robotics that John started to work on a few years ago, i.e. that of force optimization in dextrous hand grasp- ing. Here the main goal is to achieve the minimization of contact forces without violating nonlinear friction cone constraints. In earlier work (2), (8), linear program- ming techniques have been used, based on piecewise linear approximations of the cone constraints, but these approaches led to ill conditioning problems. Therefore, an alter- native approach had to be found that avoids approximations of the constraints. In the pioneering joint paper (1) by John Moore with M. Buss and H. Hashimoto, the task was reformulated as a convex optimization problem on linearly constrained cones of positive definite matrices, thus making contact with the well established area of con- vex (semi)definite optimization theory. Linearly convergent gradient flow algorithms were applied to find the global minimum and this led to the first real time solution approach. A challenge remained in finding faster computational schemes. This prob- lem has been resolved in subsequent work (6), where a damped Newton algorithm is

A nonsmooth Newton method for variational inequalities, II: Numerical results

This paper presents the results of extensive computational testing of the modified damped Newton algorithm for solving variational inequality problems presented in Part I [8].

A nonsmooth Newton method for variational inequalities, I: Theory

This paper presents a modified damped Newton algorithm for solving variational inequality problems based on formulating this problem as a system of equations using the Minty map. The proposed modified damped-Newton method insures convergence and locally quadratic convergence under the assumption of regularity. Under the assumption ofweak regularity and some mild conditions, the modified algorithm is shown to always create a descent direction and converge to the solution. Hence, this new algorithm is often suitable for many applications where regularity does not hold. Part II of this paper presents the results of extensive computational testing of this new method.