Quantum Implementation of Powell’s Conjugate Direction Method

A quantum circuit implementation of Powell’s conjugate direction method (“Powell’s method”) is proposed based on quantum basic transformations in this study. Powell’s method intends to find the minimum of a function, including a sequence of parameters, by changing one parameter at a time. The quantum circuits that implement Powell’s method are logically built by combining quantum computing units and basic quantum gates. The main contributions of this study are the quantum realization of a quadratic equation, the proposal of a quantum one-dimensional search algorithm, the quantum implementation of updating the searching direction array (SDA), and the quantum judgment of stopping the Powell’s iteration. A simulation demonstrates the execution of Powell’s method, and future applications, such as data fitting and image registration, are discussed.

A conjugate direction method for linear systems in Banach spaces

In this article, the well-known conjugate gradient (CG) method for linear systems in Hilbert spaces is extended to a reflexive Banach space setting. In this setting, the Riesz isomorphism has to be replaced by the duality mapping. Due to the nonlinearity of the duality mapping, the short term recursion and conjugacy of search directions cannot be maintained simultaneously. The well-posedness of the proposed iteration and its global convergence are shown under appropriate conditions. Error bounds and stopping criteria are presented as well. The results extend to a limited-memory variant of the algorithm. The behavior of the method is demonstrated by numerical examples.

An orthogonal power method of solving the partial eigenproblem for a symmetric nonnegative definite matrix

Предложена и обоснована экономичная версия метода сопряженных направлений для построения нетривиального решения однородной системы линейных алгебраических уравнений с вырожденной симметричной неотрицательно определенной квадратной матрицей. Предложено однопараметрическое семейство одношаговых нелинейных итерационных процессов вычисления собственного вектора, отвечающего наибольшему собственному значению симметричной неотрицательно определенной квадратной матрицы. Это семейство включает в себя степенной метод как частный случай. Доказана сходимость возникающих последовательностей векторов к собственному вектору, ассоциированному с наибольшим характеристическим числом матрицы. Предложена двухшаговая процедура ускорения сходимости итераций этих процессов, в основе которой лежит ортогонализация в подпространстве Крылова. Приведены результаты численных экспериментов. An efficient version of the conjugate direction method to find a nontrivial solution of a homogeneous system of linear algebraic equations with a singular symmetric nonnegative definite square matrix is proposed and substantiated. A one-parameter family of one-step nonlinear iterative processes to determine the eigenvector corresponding to the largest eigenvalue of a symmetric nonnegative definite square matrix is also proposed. This family includes the power method as a special case. The convergence of corresponding vector sequences to the eigenvector associated with the largest eigenvalue of the matrix is proved. A two-step procedure is formulated to accelerate the convergence of iterations for these processes. This procedure is based on the orthogonalization in Krylov subspaces. A number of numerial results are discussed.

Research on Energy Saving for Multi-Back after Blending Water from One Station System’s Parameters Optimization

The process of multi-back after blending water from one station is used in a block of Daqing oilfield, for saving energy, reducing consumption and easy to manage. For this block’s water blending system, research on its blending parameters optimization is carried on. According to the block’s characteristic of the process, the characteristic of energy cost minimum as objective function, setting up the multi-back after blending water from one station system’s parameter optimization mathematical model[1]. The model belongs to nonlinear programming problems with constraints, the mixed penalty function method and the improved conjugate direction method are used to solve the model, the optimum water blending temperature is determined by optimization calculating as 62.1°C and quantity of blending water a year is 1.13×105 t . By doing so, running costs saved 220,000 yuan per year to achieve the goal of saving energy and reducing consumption.