Hessian matrix

We show non-asymptotic exponential convergence of Sinkhorn iterates to the Schrödinger potentials, solutions of the quadratic Entropic Optimal Transport problem on R d . Our results hold under mild assumptions on the marginal inputs: in particular, we only assume that they admit an asymptotically positive log-concavity profile, covering as special cases log-concave distributions and bounded smooth perturbations of quadratic potentials. Up to the authors' knowledge, these are the first results which establish exponential convergence of Sinkhorn's algorithm in a general setting without assuming bounded cost functions or compactly supported marginals.

In order to maximize the signal-to-interference-plusnoise ratio (SINR) under a constant-envelope (CE) constraint, a fast and efficient joint design of the transmit waveform and the receive filter for colocated multiple-input multiple-output (MIMO) radars is essential. Conventional joint optimization is performed using nonlinear optimization techniques such as the semidefinite relaxation (SDR) algorithm. In this letter, we propose a novel manifold-based alternating optimization (MAO) method, which reformulates the waveform optimization subproblem as an unconstrained optimization problem on a Riemannian manifold. We present the geometrical structure of the feasible region and derive the explicit expressions for the Riemannian gradient and the Riemannian Hessian, thus the reformulated optimization could be solved by using the Riemannian trust-region (RTR) algorithm. Numerical experiments demonstrate that the proposed method has faster convergence with reduced computational cost compared with conventional SDR-based algorithm in Euclidean space. Index Terms-Multiple-input multiple-output (MIMO) radar, manifold optimization, waveform design, receive filter design. I. INTRODUCTION M ULTIPLE-INPUT multiple-output (MIMO) radar has been widely studied over the past decades [1]. For both colocated MIMO radars [2] and distributed MIMO radars [3], one of the most concerning problems is how to design probing signals appropriately. Existing approaches can be largely classified into five categories: optimizing an estimation-oriented lower bound [4], optimizing the radar ambiguity function [5], achieving a desired beampattern [6], optimizing the detection performance based on information theory [7], and joint design of transmitter and receiver to maximize the signal-to-interference-plus-noise Manuscript

The development of variational density functional theory approaches to excited electronic states is impeded by limitations of the commonly used self-consistent field (SCF) procedure. A method based on a direct optimization approach as well as the maximum overlap method is presented and the performance compared with previously proposed SCF strategies. Excited-state solutions correspond to saddle points of the energy as a function of the electronic degrees of freedom. The approach presented here makes use of a preconditioner determined with the help of the maximum overlap method to guide the convergence on a target nth-order saddle point. The method is found to be more robust and to converge faster than previously proposed SCF approaches for a set of 89 excited states of molecules. A limited-memory formulation of the symmetric rank-one method for updating the inverse Hessian is found to give the best performance. A conical intersection for the carbon monoxide molecule is calculated without resorting to fractional occupation numbers. Calculations on excited states of the hydrogen atom and a doubly excited state of the dihydrogen molecule using a self-interaction corrected functional are presented. For these systems, the self-interaction correction is found to

Assume that f ∈ C 2 (R n ) is a strict convex function with a unique minimum. We divide the vector of n variables to d ≥ 2 groups of vector subvariables. We assume that we can find the partial minimum of f with respect to each vector subvariable while other variables are fixed. We then describe an algorithm that partially minimizes each time on a specifically chosen vector subvariable. This algorithm converges geometrically to the unique minimum. The rate of convergence depends on the uniform bounds on the eigenvalues of the Hessian of f in the compact sublevel set f (x) ≤ f (x0), where x0 is the starting point of the algorithm. In the case where f (x) = x ⊤ Ax + b ⊤ x and d = n our method can be considered as a generalization of the classical conjugate gradient method. The main result of this paper is the observation that the celebrated Sinkhorn diagonal scaling algorithm for matrices, and the corresponding diagonal scaling of tensors, can be viewed as partial minimization of certain logconvex functions.

A posteriori error estimates are presented for the Laplace equation and meshes with large aspect ratio. Error estimates are presented in the natural H 1 seminorm or in the framework of goal oriented error control. The proposed estimator relies on anisotropic interpolation estimates derived by Formaggia and Perotto [19, and on Zienckiewicz-Zhu post-processing techniques, thus avoiding the use of numerical techniques in order to compute approximations of the Hessian of the solution. All the constant involved in the error estimates are independent of the mesh size and aspect ratio, which enables the use of anisotropic, adaptive finite elements.

Using the embedded gradient vector field method we explicitly compute the list of critical points of the free energy for a Cosserat body model. We also formulate necessary and sufficient conditions for critical points in the abstract case of the special orthogonal group SO(n). Each critical point is then characterized using an explicit formula for the Hessian operator of a cost function defined on the orthogonal group. We also give a positive answer to an open question posed in L. Borisov, A. Fischle, P. Neff, Optimality of the relaxed polar factors by a characterization of the set of real square roots of real symmetric matrices, ZAMM (2019), namely if all local minima of the optimization problem are global minima. We point out a few examples with physical relevance, in contrast to some theoretical (mathematical) situations that do not hold such a relevance.

We give an explicit construction of the Newton algorithm on orthogonal Stiefel manifolds. In order to do this we introduce a local frame appropriate for the computation of the Hessian matrix for a cost function defined on Stiefel manifolds. For a Brockett cost function defined on St 4 2 we give a classification of the critical points and we present numerical simulations of the Newton algorithm.

Using the embedded gradient vector field method we explicitly compute the list of critical points of the free energy for a Cosserat body model. We also formulate necessary and sufficient conditions for critical points in the abstract case of the special orthogonal group SO(n). Each critical point is then characterized using an explicit formula for the Hessian operator of a cost function defined on the orthogonal group. We also give a positive answer to an open question posed in L. Borisov, A. Fischle, P. Neff, Optimality of the relaxed polar factors by a characterization of the set of real square roots of real symmetric matrices, ZAMM (2019), namely if all local minima of the optimization problem are global minima. We point out a few examples with physical relevance, in contrast to some theoretical (mathematical) situations that do not hold such a relevance.

On a constraint manifold we give an explicit formula for the Hessian matrix of a cost function that involves the Hessian matrix of a prolonged function and the Hessian matrices of the constraint functions. We give an explicit formula for the case of the orthogonal group O(n) by using only Euclidean coordinates on R n 2 . An optimization problem on SO(3) is completely carried out. Its applications to nonlinear stability problems are also analyzed.

An idea to analyze the fluctuational mode or/and spectrum of water molecules in an inhomogeneous environment around a biomolecule such as protein and DNA is proposed based on the generalized Langevin equation (GLE) for water interacting with a biomolecule. The idea is to solve an eigen-value problem of the collective frequency matrix of water, which has a physical meaning similar to the Hessian matrix in the case of a connected harmonic-oscillator. The diagonalization of the matrix gives the mode and frequency of the density fluctuation of water around the biomolecule. The treatment is a natural extension of that made for water in a homogeneous environment by Chong and Hirata in 2008, in which the authors have found the four modes for density fluctuation; one for an acoustic mode, and the other three for optical modes. It is the point of the new approach how the mode and frequency of the density fluctuation are modulated by the field from the biomolecule. The method provides a way to distinguish the mode of fluctuation with color. A few possible applications of the theoretical approach to problems of the life science including medicine are also proposed.

This paper explores the profound relationship between information entropy and quantum dynamics through the lens of differential geometry, demonstrating that quantum behaviour naturally emerges from the geometry of probability distributions when evolution is constrained by maximum entropy principles. By constructing a statistical manifold equipped with the Fisher-Rao metric, the Schrödinger equation is derived as a geodesic flow on this information geometry. This discussion provides novel geometric interpretations of quantum phenomena including uncertainty relations, entanglement, and measurement, all unified through the mathematical language of information theory. Further rigorous connections are established between quantum Fisher information and spacetime geometry, suggesting deeper links between quantum mechanics, general relativity, and thermodynamics.

Exact forward recursions for the score vector and observed information matrix of the Markov-modulated Poisson process (MMPP) are developed. The recursions are motivated by similar recursions developed for hidden Markov models by Lystig and Hughes who extended earlier work by LeGland and Mèvel. Explicit expressions for the first derivative and Hessian of the MMPP transition density matrix are developed and coupled with the recursions. The recursions are implemented and applied to confidence interval estimation in a simulation study.

We find bilateral global bounds for the fundamental solutions associated with some quasilinear and fully nonlinear operators perturbed by a nonnegative zero order term with natural growth under minimal assumptions. Important model problems involve the equationsp u = σ |u| p-2 u + δ x 0 , for p > 1, and F k (-u) = σ |u| k-1 u + δ x 0 , for k ≥ 1. Here p and F k are the p-Laplace and k-Hessian operators respectively, and σ is an arbitrary positive measurable function (or measure). We will in addition consider the Sobolev regularity of the fundamental solution away from its pole.

Diabetic Retinopathy (DR) harm retinal blood vessels in the eye causing visual deficiency. The appearance and structure of blood vessels in retinal images play an essential part in the diagnoses of an eye sicknesses. We proposed a less computational unsupervised automated technique with promising results for detection of retinal vasculature by using morphological hessian based approach and region based Otsu thresholding. Contrast Limited Adaptive Histogram Equalization (CLAHE) and morphological filters have been used for enhancement and to remove low frequency noise or geometrical objects, respectively. The hessian matrix and eigenvalues approach used has been in a modified form at two different scales to extract wide and thin vessel enhanced images separately. Otsu thresholding has been further applied in a novel way to classify vessel and non-vessel pixels from both enhanced images. Finally, postprocessing steps has been used to eliminate the unwanted region/segment, non-vessel pi...

Genetic algorithms are widely used for solving a multitude of complex problems. Methods for increasing the genetic algorithms accuracy are of great importance. One such method consist of using the fixed point theory combined with an improved genetic algorithm. This paper aims to provide such an algorithm with an increased convergence accuracy. In order to increase the algorithm operates on a simplicial triangulation over the searching space. We use the crossover, mutation and increase dimension genetic operators to obtain a convergent population that contains only fully labelled simplexes. In order to obtain better results, we use a custom increase dimension operator that significantly boosts the overall fitness. The increase dimension operator converts non-labelled individuals into nearly labelled/fully labelled ones. The solution, the global optimum point, is obtained after applying the Hessian matrix onto the final population. Our obtained results clearly show an improvement over existing work due to implementation particularities of the algorithm.

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Deep learning has become the method of choice for many machine learning tasks in recent years, and especially for multi-class classification. The most common loss function used in this context is the cross-entropy loss. While this function is insensitive to the identity of the assigned class in the case of misclassification, in practice it very common to have imbalanced sensitivity to error, meaning some wrong assignments are much worse than others. Here we present the bilinear-loss (and related log-bilinear-loss) which differentially penalizes the different wrong assignments of the model. We thoroughly test the proposed method using standard models and benchmark image datasets.

This paper presents the way to obtain the Newton gradient by using a traction given by the perturbation for the Lagrange multiplier. Conventionally, the second order adjoint model using the Hessian/Vector Products expressed by the product of the Hessian matrix and the perturbation of the design variables has been researched. (6 However, in case that the boundary value would like to be obtained, this model can not be directly applied. Therefore, the conventional second order adjoint technique is extended to the boundary value determination problem and the second order adjoint technique is applied to the conduit flow problem in this paper. As the minimization technique, the Newton based method is employed. The BFGS method is applied to calculate the Hessian matrix which is used in the Newton based method and the a traction given by the perturbation for the Lagrange multiplier is used in the BFGS method.

An algorithm for solving smooth nonconvex optimization problems is proposed that, in the worst-case, takes ${\mathscr O}(\varepsilon ^{-3/2})$ iterations to drive the norm of the gradient of the objective function below a prescribed positive real number $\varepsilon $ and can take ${\mathscr O}(\varepsilon ^{-3})$ iterations to drive the leftmost eigenvalue of the Hessian of the objective above $-\varepsilon $. The proposed algorithm is a general framework that covers a wide range of techniques including quadratically and cubically regularized Newton methods, such as the Adaptive Regularization using Cubics (arc) method and the recently proposed Trust-Region Algorithm with Contractions and Expansions (trace). The generality of our method is achieved through the introduction of generic conditions that each trial step is required to satisfy, which in particular allows for inexact regularized Newton steps to be used. These conditions center around a new subproblem that can be approxima...

The main objective for this study is to examine the efficiency of block iterative method namely Four-Point Explicit Group Successive Over Relaxation (4EGSOR) iterative method. The nonlinear Burger's equation is then solved through the application of nonlocal arithmetic mean discretization (AMD) scheme to form a linear system. Next, to scrutinize the efficiency of 4EGSOR with Gauss-Seidel (GS) and Successive Over Relaxation (SOR) iterative methods, the numerical experiments for four proposed problems are being considered. By referring to the numerical results obtained, we concluded that 4EGSOR is more superior than GS and SOR iterative methods in aspects of number of iterations and execution time.

Let R be a ring with unity and let M be an R-module. Let R(+)M be the idealization of the ring R by the R-module M. In this article, we study the Eurelian property of zero-divisor graphs. We investigate when some special idealization rings are Eulerian graphs.

In thermal glasses at temperatures sufficiently lower than the glass transition, the constituent particles are trapped in their cages for sufficiently long time such that their time-averaged positions can be determined before diffusion and structural relaxation takes place. The effective forces are those that hold these average positions in place. In numerical simulations the effective forces Fij between any pair of particles can be measured as a time average of the bare forces fij(rij (t)). In general even if the bare forces come from two-body interactions, thermal dynamics dress the effective forces to contain many-body interactions. Here we develop the effective theory for systems with generic interactions, where the effective forces are derivable from an effective potential and in turn they give rise to an effective Hessian whose eigenvalues are all positive when the system is stable. In this Letter we offer analytic expressions for the effective theory, and demonstrate the usefulness and the predictive power of the approach.

Coins square measure integral a part of our day to day life. We tend to use coins everyplace like grocery market, banks, buses, trains etc. Therefore it is a basic want that coin is recognized and counted. The target of this paper is to classify the Indian coins of different denomination discharged recently. The objective is to notice the Indian coins and count its total worth. The system is projected to design coin recognition by applying Advanced HarrisHessian Algorithm, supported the parameters of Indian coins such as size, shape, weight, surface and so on . This paper presents a coin recognition methodology with rotation invariance. For circle detection use Hough Transform. Keywords-smoothing, Interest point detection, Hough Transform technique

Diabetes mellitus is common disease nowadays which could cause blindness. Earlier detection of diabetes signs from retina fundus images could help predicting and preventing the damages. Image processing methods could process the matrix data of pictures as blood vessel segmentation and exudate detection. In this research, the CLAHE algorithm with morphological transformations are used to blood vessel segmentation and determination of the Hessian matrix of images are utilized to detect the exudate blobs.

Convexity plays a prominent role in a number of problems, but practical considerations frequently give rise to non-convex functions. We suggest a method for determining convex regions, and also for assessing the lack of convexity in the other regions. The method relies on a specially constructed decomposition of symmetric matrices, such as the Hessian. We illustrate theoretical results using several examples, one of which analyses a problem arising in risk measurement and management in insurance and finance.

We introduce a modification of the index of increase that works in both deterministic and random environments, and thus allows us to assess monotonicity of functions that are prone to random measurement errors. We prove consistency of the empirical index and show how its rate of convergence is influenced by deterministic and random parts of the data. In particular, the obtained results allow us to determine the frequency at which observations should be taken in order to reach any pre-specified level of estimation precision. We illustrate the performance of the suggested estimator using simulated data arising from purely deterministic and error-contaminated monotonic and non-monotonic functions.

This paper deals with variational inclusions of the form 0 ∈ Kf (x) where f : R n → R m is a semismooth function and K is a nonempty closed convex cone in R m . We show that the previous problem can be solved by a Newton-type method using the Clarke generalized Jacobian of f . The results obtained in this paper extend those obtained by Robinson in the famous paper (Robinson in Numer. Math. 19:341-347, 1972). We provide a semilocal method with a superlinear convergence that is new in the context of semismooth functions. Finally, numerical results are also given to illustrate the convergence.

In this paper, we propose a unified framework, the Hessian discretisation method (HDM), which is based on four discrete elements (called altogether a Hessian discretisation) and a few intrinsic indicators of accuracy, independent of the considered model. An error estimate is obtained, using only these intrinsic indicators, when the HDM framework is applied to linear fourth order problems. It is shown that HDM encompasses a large number of numerical methods for fourth order elliptic problems: finite element methods (conforming and non-conforming) as well as finite volume methods. We also use the HDM to design a novel method, based on conforming P 1 finite element space and gradient recovery operators. Results of numerical experiments are presented for this novel scheme and for a finite volume scheme.

In this paper, we propose a unified framework, the Hessian discretisation method (HDM), which is based on four discrete elements (called altogether a Hessian discretisation) and a few intrinsic indicators of accuracy, independent of the considered model. An error estimate is obtained, using only these intrinsic indicators, when the HDM framework is applied to linear fourth order problems. It is shown that HDM encompasses a large number of numerical methods for fourth order elliptic problems: finite element methods (conforming and non-conforming) as well as finite volume methods. We also use the HDM to design a novel method, based on conforming P 1 finite element space and gradient recovery operators. Results of numerical experiments are presented for this novel scheme and for a finite volume scheme.

     The focus of this article is to add a new class of rank one of  modified Quasi-Newton techniques to solve the problem of unconstrained optimization by updating the inverse Hessian matrix with an update of rank 1, where a diagonal matrix is the first component of the next inverse Hessian approximation, The inverse Hessian matrix is  generated by the method proposed which is symmetric and it satisfies the condition of modified quasi-Newton, so the global convergence is retained. In addition, it is positive definite that  guarantees the existence of the minimizer at every iteration of the objective function. We use  the program MATLAB to solve an algorithm function to introduce the feasibility of the proposed procedure. Various numerical examples are given`.

Several attempts have been made to modify the quasi-Newton condition in order to obtain rapid convergence with complete properties (symmetric and positive definite) of the inverse of  Hessian matrix (second derivative of the objective function). There are many unconstrained optimization methods that do not generate positive definiteness of the inverse of Hessian matrix. One of those methods is the symmetric rank 1( H-version) update (SR1 update), where this update satisfies the quasi-Newton condition and the symmetric property of inverse of Hessian matrix, but does not preserve the positive definite property of the inverse of Hessian matrix where the initial inverse of Hessian matrix is positive definiteness. The positive definite property for the inverse of Hessian matrix is very important to guarantee the existence of the minimum point of the objective function and determine the minimum value of the objective function.

In this paper, we propose a new modify of DFP update with a new extended quasi-Newton condition for unconstrained optimization problem so called update. This update is based on a new Zhang Xu condition we show that update preserves the value of determinant of the next Hessian matrix equal to the value of determinant of current Hessian matrix theoretically and practically. Global convergence of the modify is established. Local and super linearly convergence are obtained for the proposed method. Numerical results are given to compare a performance of the modify method with the standard DFP method on same function is selected.

The study presents the modification of the Broyden-Flecher-Goldfarb-Shanno (BFGS) update (H-Version) based on the determinant property of inverse of Hessian matrix (second derivative of the objective function), via updating of the vector s ( the difference between the next solution and the current solution), such that the determinant of the next inverse of Hessian matrix is equal to the determinant of the current inverse of Hessian matrix at every iteration. Moreover, the sequence of inverse of Hessian matrix generated by the method would never  approach a near-singular matrix, such that the program would never break before the minimum value of the objective function is obtained. Moreover, the new modification of BFGS update (H-version) preserves the symmetric property and the positive definite property without any condition.

It is well known that the unconstrained Optimization often arises in economies, finance, trade, law, meteorology, medicine, biology, chemistry, engineering, physics, education, history, sociology, psychology, and so on. The classical Unconstrained Optimization is based on the Updating of Hessian matrix and computed of its inverse which make the solution is very expensive. In this work we will updating the LU factors of the Hessian matrix so we don't need to compute the inverse of Hessian matrix, so called the Cholesky Update for unconstrained optimization. We introduce the convergent of the update and report our findings on several standard problems, and make a comparison on its performance with the well-accepted BFGS update.

The minimization of the loss function is of paramount importance in deep neural networks. On the other hand, many popular optimization algorithms have been shown to correspond to some evolution equation of gradient flow type. Inspired by the numerical schemes used for general evolution equations we introduce a second order stochastic Runge Kutta method and show that it yields a consistent procedure for the minimization of the loss function. In addition it can be coupled, in an adaptive framework, with a Stochastic Gradient Descent (SGD) to adjust automatically the learning rate of the SGD, without the need of any additional information on the Hessian of the loss functional. The adaptive SGD, called SGD-G2, is successfully tested on standard datasets.

In nonlinear model predictive control (NMPC), a control task is approached by repeatedly solving an optimal control problem (OCP) over a receding horizon. Popularly, the OCP is approximated with a finite-dimensional nonlinear program (NLP). Since computing the solution of an NLP can be a complex and time-consuming task, tailored optimization algorithms have emerged to (approximately) solve the NLPs. Most methods rely on repeatedly solving a quadratic approximation of the NLP. Since computing this approximation is generally computationally demanding, it can form a bottelenck in obtaining a real-time applicable control law. This paper proposes DOPUS, a novel update scheme for the quadratic approximation of the NLP. DOPUS exploits the structure of the NLP and the repeated nature at which it is solved, to reduce the number of computations at the price of a small reduction of the convergence speed. Foreseen application areas include (economic) NMPC for fast-changing control tasks and fast time-varying systems. The convergence properties of DOPUS are studied and the performance is illustrated in a numerical case study considering a control task for a planar robot arm.

In many applications spanning from sensor to social networks, transportation systems, gene regulatory networks or big data, the signals of interest are defined over the vertices of a graph. The aim of this paper is to propose a least mean square (LMS) strategy for adaptive estimation of signals defined over graphs. Assuming the graph signal to be band-limited, over a known bandwidth, the method enables reconstruction, with guaranteed performance in terms of mean-square error, and tracking from a limited number of observations over a subset of vertices. A detailed mean square analysis provides the performance of the proposed method, and leads to several insights for designing useful sampling strategies for graph signals. Numerical results validate our theoretical findings, and illustrate the performance of the proposed method. Furthermore, to cope with the case where the bandwidth is not known beforehand, we propose a method that performs a sparse online estimation of the signal supp...

In this paper, we consider the manifold of covariance matrices of order n parametrized by reflection coefficients which are derived from Levinson's recursion of autoregressive model. The explicit expression of the reparametrization and its inverse are obtained. With the Riemannian metric given by the Hessian of a Kähler potential, we show that the manifold is in fact a Cartan-Hadamard manifold with lower sectional curvature bound -4. The explicit expressions of geodesics are also obtained. After that we introduce the notion of Riemannian median of points lying on a Riemannian manifold and give a simple algorithm to compute it. Finally, some simulation examples are given to illustrate the applications of the median method to radar signal processing.

A rigorous thermodynamic framework is developed for performing free energy calculations of polymer glasses described by classical molecular forcefields. The proper free energy connected to all combinations of imposed external conditions (strain, stress) is derived from a well defined Helmholtz energy calculated from the detailed atomistic configurations. The quasi-harmonic approximation (QHA) is employed around local minima of the system on its potential energy landscape. The ingredients of the methodology, i.e. the derivatives of the free energy with respect to atomic positions (forces, elements of the Hessian) and the relevant derivatives with respect to strain (stresses and elastic constants), are calculated through tractable analytic expressions which are thoroughly presented and discussed. The derivation is general and readily applicable to any system described by classical molecular forcefields. As a proof of concept, we study the mechanical response of long-chain glassy atactic polystyrene, in the elastic and the plastic regime (compressive strain up to 100%). The macroscopic stress-strain curve for polystyrene is obtained and the evolution of the microscopic polymer configurations in the course of deformation is studied.

Data assimilation combines information from an imperfect model, sparse and noisy observations, and error statistics, to produce a best estimate of the state of a physical system. Different observational data points have different contributions to reducing the uncertainty with which the state is estimated. Quantifying the observation impact is important for analyzing the effectiveness of the assimilation system, for data pruning, and for designing future sensor networks. This paper is concerned with quantifying observation impact in the context of four dimensional variational data assimilation. The main computational challenge is posed by the solution of linear systems, where the system matrix is the Hessian of the variational cost function. This work discusses iterative strategies to efficiently solve this system and compute observation impacts.

This paper presents a practical computational approach to quantify the effect of individual observations in estimating the state of a system. Such an analysis can be used for pruning redundant measurements, and for designing future sensor networks. The mathematical approach is based on computing the sensitivity of the reanalysis (unconstrained optimization solution) with respect to the data. The computational cost is dominated by the solution of a linear system, whose matrix is the Hessian of the cost function, and is only available in operator form. The right hand side is the gradient of a scalar cost function that quantifies the forecast error of the numerical model. The use of adjoint models to obtain the necessary first and second order derivatives is discussed. We study various strategies to accelerate the computation, including matrix-free iterative solvers, preconditioners, and an in-house multigrid solver. Experiments are conducted on both a small-size shallow-water equations model, and on a large-scale numerical weather prediction model, in order to illustrate the capabilities of the new methodology.

We present a new scheme to calculate isotope effects. Only selected frequencies at the target level of theory are calculated. The frequencies are selected by an analysis of the Hessian from a lower level of theory. We obtain accurate isotope effects without calculating the full Hessian at the target level of theory. The calculated frequencies are very accurate. The scheme converges to the correct isotope effect.

This paper presents an approach that directly utilizes the Hessian matrix to investigate the existence and uniqueness of global solutions for the ECQP problem. The novel features of this proposed algorithm are its uniqueness and faster rate of convergence to the solution. The merit of this algorithm is base on cost, accuracy and number of operations.

The problem of variational data assimilation (estimation) for a nonlinear model is considered in general operator formulation. Hessian-based methods are presented to compute the estimation error covariances. The importance of dynamic formulation and the role of the Hessian and its inverse are discussed.

The problem of variational data assimilation for a nonlinear evolution model is considered to identify the initial condition. An equation for the error of the optimal solution through the statistical errors of input data is derived, based on the Hessian of the misfit functional and secondorder adjoint techniques. The covariance operator of the optimal solution error is expressed through the covariance operators of input errors. Numerical algorithms are developed for constructing the covariance operator of the optimal solution error using the covariance operators of input errors.

Sampling noisy intermediate-scale quantum devices is a fundamental step that converts coherent quantumcircuit outputs to measurement data for running variational quantum algorithms that utilize gradient and Hessian methods in cost-function optimization tasks. This step, however, introduces estimation errors in the resulting gradient or Hessian computations. To minimize these errors, we discuss tunable numerical estimators, which are the finite-difference (including their generalized versions) and scaled parameter-shift estimators [introduced in Phys. Rev. A 103, 012405 (2021)], and propose operational circuit-averaged methods to optimize them. We show that these optimized numerical estimators offer estimation errors that drop exponentially with the number of circuit qubits for a given sampling-copy number, revealing a direct compatibility with the barrenplateau phenomenon. In particular, there exists a critical sampling-copy number below which an optimized difference estimator gives a smaller average estimation error in contrast to the standard (analytical) parametershift estimator, which exactly computes gradient and Hessian components. Moreover, this critical number grows exponentially with the circuit-qubit number. Finally, by forsaking analyticity, we demonstrate that the scaled parameter-shift estimators beat the standard unscaled ones in estimation accuracy under any situation, with comparable performances to those of the difference estimators within significant copy-number ranges, and are the best ones if larger copy numbers are affordable.