Newton Conjugate Gradient Method Research Articles

Gap solitons of spin–orbit-coupled Bose–Einstein condensates in a Jacobian elliptic sine potential

Gap solitons of quasi-one-dimensional Bose–Einstein condensate loaded in a Jacobian elliptic sine potential with spin–orbit-coupled are investigated theoretically. Under the mean-field approximation, the dynamical behaviors of such a system are described using the Gross–Pitaevskii equation (GPE). Firstly, we linearize the GPE to obtain the band-gap structures. Secondly, the Newton-Conjugate-Gradient (NCG) method is used to search for gap solitons. Thirdly, the dynamic stability of the gap solitons obtained by the NCG method was investigated using linear stability analysis and nonlinear dynamic evolution methods. It is found that in this nonlinear system, there exist both stable gap solitons and structurally rich unstable gap solitons. The external potential modulus and the strength of the spin–orbit coupling are important parameters that influence the stability of the gap solitons.

On Degenerate Doubly Nonnegative Projection Problems

The doubly nonnegative (DNN) cone, being the set of all positive semidefinite matrices whose elements are nonnegative, is a popular approximation of the computationally intractable completely positive cone. The major difficulty for implementing a Newton-type method to compute the projection of a given large-scale matrix onto the DNN cone lies in the possible failure of the constraint nondegeneracy, a generalization of the linear independence constraint qualification for nonlinear programming. Such a failure results in the singularity of the Jacobian of the nonsmooth equation representing the Karush–Kuhn–Tucker optimality condition that prevents the semismooth Newton–conjugate gradient method from solving it with a desirable convergence rate. In this paper, we overcome the aforementioned difficulty by solving a sequence of better conditioned nonsmooth equations generated by the augmented Lagrangian method (ALM) instead of solving one aforementioned singular equation. By leveraging the metric subregularity of the normal cone associated with the positive semidefinite cone, we derive sufficient conditions to ensure the dual quadratic growth condition of the underlying problem, which further leads to the asymptotically superlinear convergence of the proposed ALM. Numerical results on difficult randomly generated instances and from the semidefinite programming library are presented to demonstrate the efficiency of the algorithm for computing the DNN projection to a very high accuracy.

Stability analysis on dark solitons in quasi-1D Bose\u2013Einstein condensate with three-body interactions

The stability properties of dark solitons in quasi-one-dimensional Bose–Einstein condensate (BEC) loaded in a Jacobian elliptic sine potential with three-body interactions are investigated theoretically. The solitons are obtained by the Newton-Conjugate Gradient method. A stationary cubic-quintic nonlinear Schrödinger equation is derived to describe the profiles of solitons via the multi-scale technique. It is found that the three-body interaction has distinct effect on the stability properties of solitons. Especially, such a nonlinear system supports the so-called dark solitons (kink or bubble), which can be excited not only in the gap, but also in the band. The bubbles are always linearly and dynamically unstable, and they cannot be excited if the three-body interaction is absent. Both stable and unstable kinks, depending on the physical parameters, can be excited in the BEC system.

Inferring the basal sliding coefficient field for the Stokes ice sheet model under rheological uncertainty

Abstract. We consider the problem of inferring the basal sliding coefficient field for an uncertain Stokes ice sheet forward model from synthetic surface velocity measurements. The uncertainty in the forward model stems from unknown (or uncertain) auxiliary parameters (e.g., rheology parameters). This inverse problem is posed within the Bayesian framework, which provides a systematic means of quantifying uncertainty in the solution. To account for the associated model uncertainty (error), we employ the Bayesian approximation error (BAE) approach to approximately premarginalize simultaneously over both the noise in measurements and uncertainty in the forward model. We also carry out approximative posterior uncertainty quantification based on a linearization of the parameter-to-observable map centered at the maximum a posteriori (MAP) basal sliding coefficient estimate, i.e., by taking the Laplace approximation. The MAP estimate is found by minimizing the negative log posterior using an inexact Newton conjugate gradient method. The gradient and Hessian actions to vectors are efficiently computed using adjoints. Sampling from the approximate covariance is made tractable by invoking a low-rank approximation of the data misfit component of the Hessian. We study the performance of the BAE approach in the context of three numerical examples in two and three dimensions. For each example, the basal sliding coefficient field is the parameter of primary interest which we seek to infer, and the rheology parameters (e.g., the flow rate factor or the Glen's flow law exponent coefficient field) represent so-called nuisance (secondary uncertain) parameters. Our results indicate that accounting for model uncertainty stemming from the presence of nuisance parameters is crucial. Namely our findings suggest that using nominal values for these parameters, as is often done in practice, without taking into account the resulting modeling error, can lead to overconfident and heavily biased results. We also show that the BAE approach can be used to account for the additional model uncertainty at no additional cost at the online stage.

Restarted primal–dual Newton conjugate gradient method for enhanced spatial resolution of reconstructed cone-beam x-ray luminescence computed tomography images

Cone-beam x-ray luminescence computed tomography (CB-XLCT) has been proposed as a promising imaging tool, which enables three-dimensional imaging of the distribution of nanophosphors (NPs) in small animals. However, the reconstruction performance is usually unsatisfactory in terms of spatial resolution due to the ill-posedness of the CB-XLCT inverse problem. To alleviate this problem and to achieve high spatial resolution, a reconstruction method consisting of inner and outer iterations based on a restarted strategy is proposed. In this method, the primal–dual Newton conjugate gradient method (pdNCG) is adopted in the inner iterations to get fast reconstruction, which is used for resetting the permission region and increasing the convergence speed of the outer iteration. To assess the performance of the method, both numerical simulation and physical phantom experiments were conducted with a CB-XLCT system. The results demonstrate that compared with conventional reconstruction methods, the proposed re-pdNCG method can accurately and efficiently resolve the adjacent NPs with the least relative error.

Stokes waves with constant vorticity: folds, gaps and fluid bubbles

The Stokes wave problem in a constant vorticity flow is formulated, by virtue of conformal mapping techniques, as a nonlinear pseudodifferential equation, involving the periodic Hilbert transform, which becomes the Babenko equation in the irrotational flow setting. The associated linearized operator is self-adjoint, whereby the modified Babenko equation is efficiently solved by the Newton-conjugate gradient method. For strong positive vorticity, a ‘fold’ appears in the wave speed versus amplitude plane, and a ‘gap’ as the vorticity strength increases, bounded by two touching waves, whose profile contacts with itself at the trough line, enclosing a bubble of air. More folds and gaps follow for stronger vorticity. Touching waves at the beginnings of the lowest gaps tend to the limiting Crapper wave as the vorticity strength increases indefinitely or, equivalently, gravitational acceleration vanishes, while the profile encloses a circular bubble of fluid in rigid body rotation at the ends of the gaps. Touching waves at the beginnings of the second gaps tend to the circular vortex wave on top of the limiting Crapper wave in the infinite vorticity limit, or the zero gravity limit, and the circular vortex wave on top of itself at the ends of the gaps. Touching waves for higher gaps accommodate more circular bubbles of fluid.

Dynamical stability of gap solitons in quasi-1D Bose–Einstein condensate loaded in a Jacobian elliptic sine potential

The dynamical stability properties of gap solitons in a quasi-one-dimensional Bose–Einstein Condensate (BEC) loaded in a Jacobian elliptic sine potential are investigated numerically. The Gross–Pitaevskii equation (GPE) is adopted to describe the dynamical behaviors for such a system under the mean-field approximation. Firstly, we presented the band-gap structures via linearizing the GPE. Secondly, the gap solitons are gotten by the Newton-Conjugate-Gradient (NCG) method. Thirdly, the linear stability analysis and nonlinear dynamical evolution method are used to investigate the dynamical stability properties of gap solitons given by the NCG method. It is found that there exist both stable gap solitons and unstable gap solitons with rich structures in such a nonlinear system. The modulus of external potential has distinctly influence on the instability property of the gap solitons.

Sparse Solutions in Optimal Control of PDEs with Uncertain Parameters: The Linear Case

We study sparse solutions of optimal control problems governed by PDEs with uncertain coefficients. We propose two formulations, one where the solution is a deterministic control optimizing the mean objective, and a formulation aiming at stochastic controls that share the same sparsity structure. In both formulations, regions where the controls do not vanish can be interpreted as optimal locations for placing control devices. In this paper, we focus on linear PDEs with linearly entering uncertain parameters. Under these assumptions, the deterministic formulation reduces to a problem with known structure, and thus we mainly focus on the stochastic control formulation. Here, shared sparsity is achieved by incorporating the $L^1$-norm of the mean of the pointwise squared controls in the objective. We reformulate the problem using a norm reweighting function that is defined over physical space only and thus helps to avoid approximation of the random space using samples or quadrature. We show that a fixed point algorithm applied to the norm reweighting formulation leads to a variant of the well-studied iterative reweighted least squares (IRLS) algorithm, and we propose a novel preconditioned Newton-conjugate gradient method to speed up the IRLS algorithm. We combine our algorithms with low-rank operator approximations, for which we provide estimates of the truncation error. We carefully examine the computational complexity of the resulting algorithms. The sparsity structure of the optimal controls and the performance of the solution algorithms are studied numerically using control problems governed by the Laplace and Helmholtz equations. In these experiments the Newton variant clearly outperforms the IRLS method.

A note on distributionally robust optimization under moment uncertainty

AbstractWe considers a distributionally robust optimization problem when the ambiguity set specifies the support as well as the mean and the covariance matrix of the uncertain parameters. After deriving a general deterministic reformulation for the distributionally robust optimization problem, we obtain tractable optimization reformulations when the support set is the whole space and when it is a convex polyhedral set. A hybrid method of Gurobi and a smoothing Newton conjugate gradient method is suggested to solve the tractable optimization problems and numerical results of the hybrid method for solving an illustrative example are reported.

A Randomized Maximum A Posteriori Method for Posterior Sampling of High Dimensional Nonlinear Bayesian Inverse Problems

We present a randomized maximum a posteriori (rMAP) method for generating approximate samples of posteriors in high dimensional Bayesian inverse problems governed by large-scale forward problems. We derive the rMAP approach by (1) casting the problem of computing the MAP point as a stochastic optimization problem; (2) interchanging optimization and expectation; and (3) approximating the expectation with a Monte Carlo method. For a specific randomized data and prior mean, rMAP reduces to the randomized maximum likelihood (RML) approach. It can also be viewed as an iterative stochastic Newton method. An analysis of the convergence of the rMAP samples is carried out for both linear and nonlinear inverse problems. Each rMAP sample requires solution of a PDE-constrained optimization problem; to solve these problems, we employ a state-of-the-art trust region inexact Newton conjugate gradient method with sensitivity-based warm starts. An approximate Metropolization approach is presented to reduce the bias in rMA...

A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors

We present a scalable solver for approximating the maximum a posteriori (MAP) point of Bayesian inverse problems with Besov priors based on wavelet expansions with random coefficients. It is a subspace trust region interior reflective Newton conjugate gradient method for bound constrained optimization problems. The method combines the rapid locally-quadratic convergence rate properties of Newton's method, the effectiveness of trust region globalization for treating ill-conditioned problems, and the Eisenstat--Walker idea of preventing oversolving. We demonstrate the scalability of the proposed method on two inverse problems: a deconvolution problem and a coefficient inverse problem governed by elliptic partial differential equations. The numerical results show that the number of Newton iterations is independent of the number of wavelet coefficients $n$ and the computation time scales linearly in $n$. It will be numerically shown, under our implementations, that the proposed solver is two times faster than the split Bregman approach, and it is an order of magnitude less expensive than the interior path following primal-dual method. Our results also confirm the fact that the Besov $\mathbb{B}_{11}^1$ prior is sparsity promoting, discretization-invariant, and edge-preserving for both imaging and inverse problems governed by partial differential equations.

A Preconditioner for A Primal-Dual Newton Conjugate Gradient Method for Compressed Sensing Problems

In this paper we are concerned with the solution of compressed sensing (CS) problems where the signals to be recovered are sparse in coherent and redundant dictionaries. We extend the primal-dual Newton Conjugate Gradient method (pdNCG) in [T. F. Chan, G. H. Golub, and P. Mulet, SIAM J. Sci. Comput., 20 (1999), pp. 1964--1977] to CS problems. We provide an inexpensive and provably effective preconditioning technique for linear systems using pdNCG. Numerical results are presented on CS problems which demonstrate the performance of pdNCG with the proposed preconditioner compared to state-of-the-art existing solvers.

Convergence Analysis Of Gradient Based Iterative Algorithm For Solving Pde Constrained Optimization Problems

In this paper, by considering distributed optimal control over a PDE, a gradient based iterative Algorithm is proposed for solving is proposed and analyzed. Galerkin finite element method is used for solving underlying PDE, then the adjoint base technique for derivative computation to implementation of the optimal control issue in preconditioned Newton's conjugate gradient method isused. The interface and connection between quadratic programming extracted from discretizing the problem and Newton's type method, as well as the convergence rate of the algorithm in each iteration is established. Updating control values at discretization points in each iteration yields optimal control of the problem, where the corresponding state values at these points approximate the desired function. Numerical experiments are presented for illustrating the theoretical results.

Optical solitary waves in thermal media with non-symmetric boundary conditions

Optical spatial solitary waves are considered in a nonlocal thermal focusing medium with non-symmetric boundary conditions. The governing equations consist of a nonlinear Schrödinger equation for the light beam and a Poisson equation for the temperature of the medium. Three numerical methods are investigated for calculating the ground and excited solitary wave solutions of the coupled system. It is found that the Newton conjugate gradient method is the most computationally efficient and versatile numerical technique. The solutions show that by varying the ambient temperature, the solitary wave is deflected towards the warmer boundary. Solitary wave stability is also examined both theoretically and numerically, via power versus propagation constant curves and numerical simulations of the governing partial differential equations. Both the ground and excited state solitary waves are found to be stable. The Newton conjugate gradient method should also prove extremely useful for calculating solitary waves of other related optical systems, which support nonlocal spatial solitary waves, such as nematic liquid crystals.

A smoothing trust-region Newton-CG method for minimax problem

This paper presents a smooth approximate method with a new smoothing technique and a standard unconstrained minimization algorithm in the solution to the finite minimax problems. The new smooth approximations only replace the original problem in some neighborhoods of the kink points with a twice continuously differentiable function, its gradient and Hessian matrix are the combination of the first and the second order derivative of the original functions respectively. Compared to the other smooth functions such as the exponential penalty function, the remarkable advantage of the new smooth function is that the combination coefficients of its gradient and the Hessian matrix have sparse properties. Furthermore, the maximal possible difference value between the optimal values of the smooth approximate problem and the original one is determined by a fixed parameter selected previous. An algorithm to solve the equivalent unconstrained problem by using the trust-region Newton conjugate gradient method is proposed in the solution process. Finally, some numerical examples are reported to compare the proposed algorithm with SQP algorithm that implements in MATLAB toolbox and the algorithm in [E. Polak, J.O. Royset, R.S. Womersley, Algorithms with adaptive smoothing for finite minimax problems, Journal of Optimization Theory and Applications 119 (3) (2003) 459–484] based on the exponential penalty function, the numerical results prove that the proved algorithm is efficient.

Design Of 2-D Recursive Digital Filters Using Nonsymmetric Half-Plane Allpass Filters

A novel structure using recursive nonsymmetric half-plane (NSHP) digital allpass filters (DAFs) is presented for designing 2-D recursive digital filters. First, several important properties of 2-D recursive DAFs with NSHP support regions for filter coefficients are investigated. The stability of the 2-D recursive NSHP DAFs is guaranteed by using a spectral factorization-based algorithm. Then, the important characteristics regarding the proposed novel structure are discussed. The design problem of 2-D recursive digital filters using the novel structure is considered. We formulate the problem by forming an objective function consisting of the weighted sum of magnitude, group delay, and stability-related errors. A design technique using a trust-region Newton-conjugate gradient method in conjunction with the analytic derivatives of the objective function is presented to efficiently solve the resulting optimization problem. The novelty of the presented 2-D structure is that it possesses the advantage of better performance in designing a variety of 2-D recursive digital filters over existing 2-D filter structures. Finally, several design examples are provided for conducting illustration and comparison.

Variational ground states of two-dimensional antiferromagnets in the valence bond basis

We study a variational wave function for the ground state of the two-dimensional $S=1∕2$ Heisenberg antiferromagnet in the valence bond basis. The expansion coefficients are products of amplitudes $h(x,y)$ for valence bonds connecting spins separated by $(x,y)$ lattice spacings. In contrast to previous studies, in which a functional form for $h(x,y)$ was assumed, we here optimize all the amplitudes for lattices with up to $32\ifmmode\times\else\texttimes\fi{}32$ spins. We use two different schemes for optimizing the amplitudes; a Newton conjugate-gradient method and a stochastic method which requires only the signs of the first derivatives of the energy. The latter method performs significantly better. The energy for large systems deviates by only $\ensuremath{\approx}0.06%$ from its exact value (calculated using unbiased quantum Monte Carlo simulations). The spin correlations are also well reproduced, falling $\ensuremath{\approx}2%$ below the exact ones at long distances (corresponding to an $\ensuremath{\approx}1%$ underestimation of the sublattice magnetization). The amplitudes $h(r)$ for valence bonds of long length $r$ decay as ${r}^{\ensuremath{-}3}$. We also discuss some results for small frustrated lattices.

Image reconstruction in optical tomography with the aid of reverse differentiation in the complex domain

In Optical Tomography, an efficient image reconstruction algorithm is needed in order to deduce the tissue optical properties from information obtained from measurements on the boundary. The image reconstruction in this work is treated as a non-linear optimisation problem and uses the Truncated Newton Conjugate Gradient method with Trast region. Reverse differentiation is used in order to speed up the reconstruction. The algorithm is attractive because of the speed increase gained from the use of reverse differentiation, and the storage benefit from the use of the Truncated Newton method. Reverse differentiation is applied to the inverse problem in Optical Tomography in the frequency domain. The problem of using complex variables in reverse differentiation is discussed, and the solution for complex linear systems of equations obtained. Error functions are outlined for phase shift, modulation and amplitude. The speed of the reconstruction using reverse differentiation compared to forward differentiation is discussed.