LIMITED MEMORY BFGS METHOD BY USING LINEAR INDEPENDENT SEARCH DIRECTIONS

The degree of numerical linear independence is proposed and discussed. Based on this linear independence theory, a modified limited memory BFGS method is deve loped. Similar to the standard limited memory method, this new method determines the new update by applying the updating formula m times to an initial positive diagonal matrix using the m previous pairs of the change in iteration and gradient. Besides the most recent pair of the change, which guarantees the quadratic termination, the choice of the other ( m -1) pairs of the change in the new method is dependent on the degree of numerical linear independence of previous search directions. In addition, the numerical linear independence theory is further discussed and the computation of the degree of linear independence is simplified. Theoretical and numerical results show that this new modified method improves efficiently the standard limited memory method.

Improved hybrid iterative optimization method for seismic full waveform inversion

In full waveform inversion （FWI）, Hessian information of the misfit function is of vital importance for accelerating the convergence of the inversion; however, it usually is not feasible to directly calculate the Hessian matrix and its inverse. Although the limited memory Broyden-Fletcher-Goldfarb-Shanno （L-BFGS） or Hessian-free inexact Newton （HFN） methods are able to use approximate Hessian information, the information they collect is limited. The two methods can be interlaced because they are able to provide Hessian information for each other; however, the performance of the hybrid iterative method is dependent on the effective switch between the two methods. We have designed a new scheme to realize the dynamic switch between the two methods based on the decrease ratio （DR） of the misfit function （objective function）, and we propose a modified hybrid iterative optimization method. In the new scheme, we compare the DR of the two methods for a given computational cost, and choose the method with a faster DR. Using these steps, the modified method always implements the most efficient method. The results of Marmousi and overthrust model testings indicate that the convergence with our modified method is significantly faster than that in the L-BFGS method with no loss of inversion quality. Moreover, our modified outperforms the enriched method by a little speedup of the convergence. It also exhibits better efficiency than the HFN method.

MIMO Soft-sensor Model of Nutrient Content for Compound Fertil- izer Based on Hybrid Modeling Technique

In compound fertilizer production, several quality variables need to be monitored and controlled simultaneously. It is very diifficult to measure these variables on-line by existing instruments and sensors. So, soft-sensor technique becomes an indispensable method to implement real-time quality control. In this article, a new model of multi-inputs multi-outputs （MIMO） soft-sensor, which is constructed based on hybrid modeling technique, is proposed for these interactional variables. Data-driven modeling method and simplified first principle modelingmethod are combined in this model. Data-driven modeling method based on limited memory partial least squares（LM-PLS） al.gorithm is used to build soft-senor models for some secondary variables.then, the simplified first principle model is used to compute three primary variables on line. The proposed model has been used in practicalprocess; the results indicate that the proposed model is precise and efficient, and it is possible to realize on line quality control for compound fertilizer process.

LIMITED MEMORY BFGS METHOD FOR NONLINEAR MONOTONE EQUATIONS

In this paper, we propose an algorithm for solving nonlinear monotone equations by combining the limited memory BFGS method （L-BFGS） with a projection method. We show that the method is globally convergent if the equation involves a Lipschitz continuous monotone function. We also present some preliminary numerical results.

Limited Memory BFGS Method for Least Squares Semidefinite Programming with Banded Structure

This work is intended to solve the least squares semidefinite program with a banded structure. A limited memory BFGS method is presented to solve this structured program of high dimension.In the algorithm, the inverse power iteration and orthogonal iteration are employed to calculate partial eigenvectors instead of full decomposition of n × n matrices. One key feature of the algorithm is that it is proved to be globally convergent under inexact gradient information. Preliminary numerical results indicate that the proposed algorithm is comparable with the inexact smoothing Newton method on some large instances of the structured problem.

Global Convergence of a Modified Limited Memory BFGS Method for Non-convex Minimization

In this paper, a modified limited memory BFGS method for solving large-scale unconstrained optimization problems is proposed. A remarkable feature of the proposed method is that it possesses global convergence property without convexity assumption on the objective function. Under some suitable conditions, the global convergence of the proposed method is proved. Some numerical results are reported which illustrate that the proposed method is efficient.

Multiple Schubert's Updating Matrix and Its Compact Representation

Schubert's method for solving systems of sparse equations has achieved a great deal of computational success. In this paper, Schubert's method was extended to multiple version, and the compact representation of multple Schubert's updating matrix was derived. The compact representation could be used to efficiently implement limited memory methods for large problems.

A new approach for Bayesian model averaging

Bayesian model averaging(BMA) is a recently proposed statistical method for calibrating forecast ensembles from numerical weather models.However,successful implementation of BMA requires accurate estimates of the weights and variances of the individual competing models in the ensemble.Two methods,namely the Expectation-Maximization(EM) and the Markov Chain Monte Carlo(MCMC) algorithms,are widely used for BMA model training.Both methods have their own respective strengths and weaknesses.In this paper,we first modify the BMA log-likelihood function with the aim of removing the addi-tional limitation that requires that the BMA weights add to one,and then use a limited memory quasi-Newtonian algorithm for solving the nonlinear optimization problem,thereby formulating a new approach for BMA(referred to as BMA-BFGS).Several groups of multi-model soil moisture simulation experiments from three land surface models show that the performance of BMA-BFGS is similar to the MCMC method in terms of simulation accuracy,and that both are superior to the EM algo-rithm.On the other hand,the computational cost of the BMA-BFGS algorithm is substantially less than for MCMC and is al-most equivalent to that for EM.