Newton System Research Articles

Efficiently preconditioned inexact Newton methods for large symmetric eigenvalue problems

In this paper, we propose an efficiently preconditioned Newton method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices. A sequence of preconditioners based on the BFGS update formula is proposed, for the preconditioned conjugate gradient solution of the linearized Newton system to solve Au=q(u) u, q(u) being the Rayleigh quotient. We give theoretical evidence that the sequence of preconditioned Jacobians remains close to the identity matrix if the initial preconditioned Jacobian is so. Numerical results onto matrices arising from various realistic problems with size up to one million unknowns account for the efficiency of the proposed algorithm which reveals competitive with the Jacobi–Davidson method on all the test problems.

SOLITON DYNAMICS FOR THE SCHRÖDINGER–NEWTON SYSTEM

We investigate the soliton dynamics for the Schrödinger–Newton system by proving suitable modulational stability estimates in the spirit of those obtained by Weinstein for local equations.

Multiple $\mathbb{S}^{1}$-orbits for the Schrödinger-Newton system

We prove existence and multiplicity of symmetric solutions for the Schrodinger--Newton system in three-dimensional space using equivariant Morse theory.

Polynomial convergence of Mehrotra-type predictor–corrector algorithm for the Cartesian P∗(κ)-LCP over symmetric cones

We propose a new infeasible variant of Mehrotra-type predictor–corrector interior-point algorithms which can be regarded as an extension of Salahi et al. (SIAM J. Optim. 18: 1377-1397, 2007) for linear programming to the Cartesian linear complementarity problems over symmetric cone. That algorithm incorporates a safeguard in Mehrotra’s original algorithm, which allows us to prove polynomial iteration complexity. In our algorithm, the safeguard strategy is implemented by bounding the central parameter, which is different from the algorithm of Salahi et al. We modify the maximum step size in the affine scaling step and the Newton system in the corrector step, and extend the algorithm to symmetric cones using the machinery of Euclidean Jordan algebras. We show that the iteration-complexity bound of the proposed algorithm is , where is the rank of the associated Euclidean Jordan algebras, is the handicap of the problem, is the condition number of matrix and is the required precision. Some numerical tests are reported to illustrate our theoretical results.

Parallel RFSAI-BFGS Preconditioners for Large Symmetric Eigenproblems

We propose a parallel preconditioner for the Newton method in the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices. A sequence of preconditioners starting from an enhanced approximate inverse RFSAI (Bergamaschi and Martínez, 2012) and enriched by a BFGS-like update formula is proposed to accelerate the preconditioned conjugate gradient solution of the linearized Newton system to solveAu=q(u)u,q(u)being the Rayleigh quotient. In a previous work (Bergamaschi and Martínez, 2013) the sequence of preconditioned Jacobians is proven to remain close to the identity matrix if the initial preconditioned Jacobian is so. Numerical results onto matrices arising from various realistic problems with size up to 1.5 million unknowns account for the efficiency and the scalability of the proposed low rank update of the RFSAI preconditioner. The overall RFSAI-BFGS preconditioned Newton algorithm has shown comparable efficiencies with a well-established eigenvalue solver on all the test problems.

Study on the Equivalent Circuit of Quartz Crystal Microbalance (QCM) with Material Properties

The quartz crystal micro balances (QCM) principle is simply introduces. LEM model can accurately describe TSM sensors with surface the load, instead of complicated TLM model. The configuration of sensor has pure quality load, liquid load in fluid Newton system, the sticky elastic thin film, and viscoelastic membrane plus half infinite liquid. Two kinds of material properties model prediction deviate with no more than 3% error. The rule of Judging TLM to be the LEM is Zl/Zq<0.1

Low-rank update of preconditioners for the nonlinear Richards equation

Preconditioners for the Conjugate Gradient method are studied to solve the Newton system with symmetric positive definite (SPD) Jacobian. Following the theoretical work in Bergamaschi et al. (2011) [4] we start from a given approximation of the inverse of the initial Jacobian, and we construct a sequence of preconditioners by means of a low rank update, for the linearized systems arising in the Picard–Newton solution of the nonlinear discretized Richards equation. Numerical results onto a very large and realistic test case show that the proposed approach is more efficient, in terms of iteration number and CPU time, as compared to computing the preconditioner of choice at every nonlinear iteration.

An improved Newton projection method for nonnegative deblurring of Poisson-corrupted images with Tikhonov regularization

In this paper a quasi-Newton projection method for image deblurring is presented. The image restoration problem is mathematically formulated as a nonnegatively constrained minimization problem where the objective function is the sum of the Kullback---Leibler divergence, used to express fidelity to the data in the presence of Poisson noise, and of a Tikhonov regularization term. The Hessian of the objective function is approximated so that the Newton system can be efficiently solved by using Fast Fourier Transforms. The numerical results show the potential of the proposed method both in terms of relative error reduction and computational efficiency.

Quasi-Newton projection methods and the discrepancy principle in image restoration

In this work, the problem of the restoration of images corrupted by space invariant blur and noise is considered. This problem is ill-posed and regularization is required. The image restoration problem is formulated as a nonnegatively constrained minimization problem whose objective function depends on the statistical properties of the noise corrupting the observed image. The cases of Gaussian and Poisson noise are both considered. A Newton-like projection method with early stopping of the iterates is proposed as an iterative regularization method in order to determine a nonnegative approximation to the original image. A suitable approximation of the Hessian of the objective function is proposed for a fast solution of the Newton system. The results of the numerical experiments show the effectiveness of the method in computing a good solution in few iterations, when compared with some methods recently proposed as best performing.

Local path-following property of inexact interior methods in nonlinear programming

We study the local behavior of a primal-dual inexact interior point methods for solving nonlinear systems arising from the solution of nonlinear optimization problems or more generally from nonlinear complementarity problems. The algorithm is based on the Newton method applied to a sequence of perturbed systems that follows by perturbation of the complementarity equations of the original system. In case of an exact solution of the Newton system, it has been shown that the sequence of iterates is asymptotically tangent to the central path (Armand and Benoist in Math. Program. 115:199---222, 2008). The purpose of the present paper is to extend this result to an inexact solution of the Newton system. We give quite general conditions on the different parameters of the algorithm, so that this asymptotic property is satisfied. Some numerical tests are reported to illustrate our theoretical results.

The Q Method for Symmetric Cone Programming

The Q method of semidefinite programming, developed by Alizadeh, Haeberly and Overton, is extended to optimization problems over symmetric cones. At each iteration of the Q method, eigenvalues and Jordan frames of decision variables are updated using Newton’s method. We give an interior point and a pure Newton’s method based on the Q method. In another paper, the authors have shown that the Q method for second-order cone programming is accurate. The Q method has also been used to develop a “warm-starting” approach for second-order cone programming. The machinery of Euclidean Jordan algebra, certain subgroups of the automorphism group of symmetric cones, and the exponential map is used in the development of the Newton method. Finally we prove that in the presence of certain non-degeneracies the Jacobian of the Newton system is nonsingular at the optimum. Hence the Q method for symmetric cone programming is accurate and can be used to “warm-start” a slightly perturbed symmetric cone program.

Low-rank update of preconditioners for the inexact Newton method with SPD Jacobian

In this note preconditioners for the Conjugate Gradient method are studied to solve the Newton system with a symmetric positive definite Jacobian. In particular, we define a sequence of preconditioners built by means of BFGS rank-two updates. Reasonable conditions are derived which guarantee that the preconditioned matrices are not far from the identity in a matrix norm. Some notes on the implementation of the corresponding inexact Newton method are given and some numerical results on a number of model problems illustrate the efficiency of the proposed preconditioners.

Decomposition–coordination interior point method and its application to multi-area optimal reactive power flow

A decomposition–coordination interior point method (DIPM) is presented and applied to the multi-area optimal reactive power flow (ORPF) problem in this paper. In the method, the area distributed ORPF problem is first formed by introducing duplicated border variables. Then the nonlinear primal dual interior point method (IPM) is directly applied to the distributed ORPF problem in which a Newton system with border-matrix-blocks is formulated. Finally the overall ORPF problem is solved in decomposition iterations with the Newton system being decoupled. The proposed DIPM inherits the good performance of the traditional IPM with a feature appropriate for distributed calculations among multiple areas. It can be easily extended to other distributed optimization problems of power systems. Numeric results of five IEEE Test Systems are demonstrated and comparisons are made with those obtained using the traditional auxiliary problem principle (APP) method. The results show that the DIPM for the multi-area OPRF problem requires less iterations and CPU time, has better stability in convergence, and reaches better optimality compared to the traditional auxiliary problem principle method.

A note on Schrödinger–Newton systems with decaying electric potential

We prove the existence of solutions for the singularly perturbed Schrödinger–Newton system { ħ 2 Δ ψ − V ( x ) ψ + U ψ = 0 ħ 2 Δ U + 4 π γ | ψ | 2 = 0 in R 3 with an electric potential V that decays polynomially fast at infinity. The solution ψ concentrates, as ħ → 0 , around (structurally stable) critical points of the electric potential. As a particular case, isolated strict extrema of V are allowed.

Conjectures and reputations: The composition and reception of James Bradley's paper on the aberration of light with some reference to a third unpublished version.

In January 1729 a paper written by James Bradley was read at two meetings of the Royal Society. On a newly discovered motion of the fixed stars, later described as the theory of the aberration of light, it was to transform the science of astrometry. The paper appeared as a narrative of a programme of observation first begun at Kew and finalized at Wanstead, but it was, in reality, a careful reconstruction devised to enhance his reputation in response to a recognition that the programme was initially conducted in terms that were inimical to what he conceived to be his interest. The planned attempt to repeat Robert Hooke's celebrated experiment by James Pound, Samuel Molyneux and George Graham was set up at Molyneux's residence in Kew with James Bradley replacing Pound after his untimely and sudden demise. The unexpected and counterintuitive behaviour of the object star γ Draconis and the eradication of any suspicion of instrumental or systemic error led to the abandonment of the attempt to measure annual parallax and the initiation of new conjectures. An annual nutation was proposed but after the observation of a control star, 35 Camelopardalis, this conjecture was abandoned. Unknown to Bradley and Graham a premature approach was made by Molyneux to Newton claiming that the 'nutation' negated the whole of Newton's system. In the abandonment of the nutation yet another conjecture opposed to Newtonian theory was proposed and abandoned. Bradley determined to use his own instrument designed on different principles by Graham to observe the phenomenon in Wanstead. At Wanstead Bradley observed many stars to determine the parameters of the phenomenon. With the law of the motion described, Bradley proposed a hypothesis to explain it. Drawn from his earlier work on the ephemerides of Jupiter's satellites his hypothesis of the 'new-discovered motion' was quickly presented to the Royal Society as Bradley was working on a later and more definitive version of his paper. It is this later, third, unpublished version that is commonly referred to throughout this essay. It issued a challenge to 'anti-Copernicans' to offer an explanation of the observed phenomenon in geostatic terms. One such astronomer, Eustachio Manfredi, had examined the phenomenon of 'aberrations' in detail, the term being his. It was Bradley who first applied the term to the 'new-discovered motion' and within a short time 'aberration' was being applied by astronomers in the reduction of their observations. Annual aberration was widely accepted as evidence of the motion of the Earth. The paper enhanced Bradley's reputation and projected him into the forefront of European astronomers.

An interior-point piecewise linear penalty method for nonlinear programming

We present an interior-point penalty method for nonlinear programming (NLP), where the merit function consists of a piecewise linear penalty function and an l 2-penalty function. The piecewise linear penalty function is defined by a set of break points that correspond to pairs of values of the barrier function and the infeasibility measure at a subset of previous iterates and this set is updated at every iteration. The l 2-penalty function is a traditional penalty function defined by a single penalty parameter. At every iteration the step direction is computed from a regularized Newton system of the first-order equations of the barrier problem proposed in Chen and Goldfarb (Math Program 108:1–36, 2006). Iterates are updated using a line search. In particular, a trial point is accepted if it provides a sufficient reduction in either of the penalty functions. We show that the proposed method has the same strong global convergence properties as those established in Chen and Goldfarb (Math Program 108:1–36, 2006). Moreover, our method enjoys fast local convergence. Specifically, for each fixed small barrier parameter μ, iterates in a small neighborhood (roughly within o(μ)) of the minimizer of the barrier problem converge Q-quadratically to the minimizer. The overall convergence rate of the iterates to the solution of the nonlinear program is Q-superlinear.

Primal-dual interior-point method for thermodynamic gas-particle partitioning

A mathematical model for the computation of the phase equilibrium and gas-particle partitioning in atmospheric organic aerosols is presented. The thermodynamic equilibrium is determined by the global minimum of the Gibbs free energy under equality and inequality constraints for a system that involves one gas phase and many liquid phases. A primal-dual interior-point algorithm is presented for the efficient solution of the phase equilibrium problem and the determination of the active constraints. The first order optimality conditions are solved with a Newton iteration. Sequential quadratic programming techniques are incorporated to decouple the different scales of the problem. Decomposition methods that control the inertia of the matrices arising in the resolution of the Newton system are proposed. A least-squares initialization of the algorithm is proposed to favor the convergence to a global minimum of the Gibbs free energy. Numerical results show the efficiency of the approach for the prediction of gas-liquid-liquid equilibrium for atmospheric organic aerosol particles.

An iterative solver-based long-step infeasible primal-dual path-following algorithm for convex QP based on a class of preconditioners

In this paper, we present a long-step infeasible primal-dual path-following algorithm for convex quadratic programming (CQP) whose search directions are computed by means of a preconditioned iterative linear solver. In contrast to the authors’ previous paper [Z. Lu, R.D.C. Monteiro, and J.W. O'Neal. An iterative solver-based infeasible primal-dual path-following algorithm for convex quadratic programming, SIAM J. Optim. 17(1) (2006), pp. 287–310], we propose a new linear system, which we refer to as the hybrid augmented normal equation (HANE), to determine the primal-dual search directions. Since the iterative linear solver can only generate an approximate solution to the HANE, this solution does not yield a primal-dual search direction satisfying all equations of the primal-dual Newton system. We propose a recipe to compute an inexact primal-dual search direction, based on a suitable approximate solution to the HANE. The second difference between this paper and [Z. Lu, R.D.C. Monteiro, and J.W. O'Neal. An iterative solver-based infeasible primal-dual path-following algorithm for convex quadratic programming, SIAM J. Optim. 17(1)(2006), pp. 287–310] is that, instead of using the maximum weight basis (MWB) preconditioner in the aforesaid recipe for constructing the inexact search direction, this paper proposes the use of any member of a whole class of preconditioners, of which the MWB preconditioner is just a special case. The proposed recipe allows us to: (i) establish a polynomial bound on the number of iterations performed by our path-following algorithm and (ii) establish a uniform bound, depending on the quality of the preconditioner, on the number of iterations performed by the iterative solver.

Atoms, Pneuma and Tranquillity: Epicurean and Stoic Themes in European Thought

Introduction Margaret J. Osler and Letizia A. Panizza 1. Ethics and logic in stoicism Gerard Verbeke 2. Medieval connectives, Hellenistic connections: the strange case of propositional logic Calvin G. Normore 3. Stoic psychotherapy in the Middle Ages and Renaissance: Petrarch's De Remediis Letizia A. Panizza 4. Alonso de Cartagena and John Calvin as interpreters of Seneca's De Clementia Nicholas G. Round 5. The Epicurean in Lorenzo Valla's On Pleasure Maristella de P. Lorch 6. Seneca's role in popularizing Epicurus in the Renaissance Louise Fothergill-Payne 7. Stoic contributions to early modern science Peter Barker 8. Fortune, fate, and divination: Gassendi's voluntarist theology and the baptism of Epicureanism Margaret J. Osler 9. Epicureanism and the creation of a privatist ethic in early seventeenth-century France Lisa Tunick Sarasohn 10. Robert Boyle on Epicurean atheism and atomism J. J. Macintosh 11. Stoic and Epicurean doctrines in Newton's system of the world B. J. T. Dobbs 12. Locke, Willis, and the seventeenth-century Epicurean soul John P. Wright 13. The Epicurean way of ideas: Gassendi, Locke, and Berkeley Thomas M. Lennon 14. The stoic legacy in the early Scottish enlightenment M. A. Stewart.

A Clifford–Finslerian physical unification and fractal dynamics

A Clifford–Finslerian physical unification is proposed based on Clifford–Finslerian mathematical structures and three physical principles. In the Clifford–Finslerian mathematical structure, spontaneous symmetry breaking is automatically embedded in fractal branches. With the action principle, connection principle and computation principle, physics can be unified, in which the Riemman–Einstein system and Euclid–Newton system are naturally included when quaternion are reduced to complex and real phases.