The convergence properties of the Fletcher-Reeves method for unconstrained optimization are further studied with the technique of generalized line search. Two conditions are given which guarantee the global convergenc...
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The convergence properties of the Fletcher-Reeves method for unconstrained optimization are further studied with the technique of generalized line search. Two conditions are given which guarantee the global convergence of the Fletcher-Reeves method using generalized Wolfe line searches or generalized Arjimo line searches, whereas an example is constructed showing that the conditions cannot be relaxed in certain senses.
The Beale\|Powell restart algorithm is highly useful for large\|scale unconstrained optimization. An example is taken to show that the algorithm may fail to converge. The global convergence of a slightly modified algo...
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The Beale\|Powell restart algorithm is highly useful for large\|scale unconstrained optimization. An example is taken to show that the algorithm may fail to converge. The global convergence of a slightly modified algorithm is proved.
The two-sided rank-one (TR1. update method was introduced by Griewank and Walther (2002) for solving nonlinear equations. It generates dense approximations of the Jacobian and thus is not applicable to large-scale spa...
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The two-sided rank-one (TR1. update method was introduced by Griewank and Walther (2002) for solving nonlinear equations. It generates dense approximations of the Jacobian and thus is not applicable to large-scale sparse problems. To overcome this difficulty, we propose sparse extensions of the TR1.update and give some convergence analysis. The numerical experiments show that some of our extensions are superior to the TR1.update method. Some convergence analysis is also presented.
For large sparse non-Hermitian positive definite system of linear equations, we present several variants of the Hermitian and skew-Hermitian splitting (HSS) about the coefficient matrix and establish correspondingly s...
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For large sparse non-Hermitian positive definite system of linear equations, we present several variants of the Hermitian and skew-Hermitian splitting (HSS) about the coefficient matrix and establish correspondingly several HSS-based iterative schemes. Theoretical analyses show that these methods are convergent unconditionally to the exact solution of the referred system of linear equations, and they may show advantages on problems that the HSS method is ineffective.
The compressible Navier-Stokes equations discretized with a fourth order accurate compact finite difference scheme with group velocity control are used to simulate the Richtmyer-Meshkov (R-M) instability problem produ...
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The compressible Navier-Stokes equations discretized with a fourth order accurate compact finite difference scheme with group velocity control are used to simulate the Richtmyer-Meshkov (R-M) instability problem produced by cylindrical shock-cylindrical material interface with shock Mach number Ms=1.2 and density ratio 1.20 (interior density/outer density). Effect of shock refraction, reflection, interaction of the reflected shock with the material interface, and effect of initial perturbation modes on R-M instability are investigated numerically. It is noted that the shock refraction is a main physical mechanism of the initial phase changing of the material surface. The multiple interactions of the reflected shock from the origin with the interface and the R-M instability near the material interface are the reason for formation of the spike-bubble structures. Different viscosities lead to different spike-bubble structure characteristics. The vortex pairing phenomenon is found in the initial double mode simulation. The mode interaction is the main factor of small structures production near the interface.
The image restoration problems play an important role in remote sensing and astronomical image analysis. One common method for the recovery of a true image from corrupted or blurred image is the least squares error (L...
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The image restoration problems play an important role in remote sensing and astronomical image analysis. One common method for the recovery of a true image from corrupted or blurred image is the least squares error (LSE) method. But the LSE method is unstable in practical applications. A popular way to overcome instability is the Tikhonov regularization. However, difficulties will encounter when adjusting the so-called regularization parameter a. Moreover, how to truncate the iteration at appropriate steps is also challenging. In this paper we use the trust region method to deal with the image restoration problem, meanwhile, the trust region subproblem is solved by the truncated Lanczos method and the preconditioned truncated Lanczos method. We also develop a fast algorithm for evaluating the Kronecker matrix-vector product when the matrix is banded. The trust region method is very stable and robust, and it has the nice property of updating the trust region automatically. This releases us from tedious finding the regularization parameters and truncation levels. Some numerical tests on remotely sensed images are given to show that the trust region method is promising.
The Integrated Sensing and Communications (ISAC) paradigm is anticipated to be a cornerstone of the upcoming 6G networks. In order to optimize the use of wireless resources, 6G ISAC systems need to harness the communi...
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In this paper, we analyze the convergence of the adaptive conforming P 1.element method with the red-green refinement. Since the mesh after refining is not nested into the one before, the Galerkin-orthogonality does n...
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In this paper, we analyze the convergence of the adaptive conforming P 1.element method with the red-green refinement. Since the mesh after refining is not nested into the one before, the Galerkin-orthogonality does not hold for this case. To overcome such a difficulty, we prove some quasi-orthogonality instead under some mild condition on the initial mesh (Condition A). Consequently, we show convergence of the adaptive method by establishing the reduction of some total error. To weaken the condition on the initial mesh, we propose a modified red-green refinement and prove the convergence of the associated adaptive method under a much weaker condition on the initial mesh (Condition B).
Advances in quantum computers pose great threats on the currently used public key cryptographic algorithms such as RSA and ECC. As a promising candidate secure against attackers equipped with quantum computational pow...
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Advances in quantum computers pose great threats on the currently used public key cryptographic algorithms such as RSA and ECC. As a promising candidate secure against attackers equipped with quantum computational power, multivariate public key cryptosystems (MPKCs) have attracted increasing attention in recently years. Unfortunately, the existing MPKCs can only be used as a multivariate signature scheme, and it remains unknown how to construct an efficient MPKC enabling secure encryption. Furthermore, some multivariate signature schemes have been shown insecure in recent years, and it is also not trivial to build MPKC which can serve as a secure signature scheme. By employing the basic MQ-trapdoors, this paper proposes a novel MPKC and shows how it can be used as a multivariate signature scheme and a multivariate encryption scheme, respectively. The goal is achieved by incorporating our new hash authentication techniques and some modification methods such as the Shamir's minus method. Thorough analysis shows that our schemes are secure and efficient. Our MPKC gives a positive response to the challenges in multivariate public key cryptography.
In this paper we solve large scale ill-posed problems, particularly the image restoration problem in atmospheric imaging sciences, by a trust region-CG algorithm. Image restoration involves the removal or minimization...
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In this paper we solve large scale ill-posed problems, particularly the image restoration problem in atmospheric imaging sciences, by a trust region-CG algorithm. Image restoration involves the removal or minimization of degradation (blur, clutter, noise, etc.) in an image using a priori knowledge about the degradation phenomena. Our basic technique is the so-called trust region method, while the subproblem is solved by the truncated conjugate gradient method, which has been well developed for well-posed *** trust region method, due to its robustness in global convergence, seems to be a promising way to deal with ill-posed problems.
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