344 research outputs found
On solving trust-region and other regularised subproblems in optimization
The solution of trust-region and regularisation subproblems which arise in unconstrained optimization is considered. Building on the pioneering work of Gay, Mor´e and Sorensen, methods which obtain the solution of a sequence of parametrized linear systems by factorization are used. Enhancements using high-order polynomial approximation and inverse iteration ensure that the resulting method is both globally and asymptotically at least superlinearly convergent in all cases, including in the notorious hard case. Numerical experiments validate the effectiveness of our approach. The resulting software is available as packages TRS and RQS as part of the GALAHAD optimization library, and is especially designed for large-scale problems
Sensitivity analysis and approximation methods for general eigenvalue problems
Optimization of dynamic systems involving complex non-hermitian matrices is often computationally expensive. Major contributors to the computational expense are the sensitivity analysis and reanalysis of a modified design. The present work seeks to alleviate this computational burden by identifying efficient sensitivity analysis and approximate reanalysis methods. For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and approximate reanalysis are classified, compared and evaluated for efficiency and accuracy. Proper eigenvector normalization is discussed. An improved method for calculating derivatives of eigenvectors is proposed based on a more rational normalization condition and taking advantage of matrix sparsity. Important numerical aspects of this method are also discussed. To alleviate the problem of reanalysis, various approximation methods for eigenvalues are proposed and evaluated. Linear and quadratic approximations are based directly on the Taylor series. Several approximation methods are developed based on the generalized Rayleigh quotient for the eigenvalue problem. Approximation methods based on trace theorem give high accuracy without needing any derivatives. Operation counts for the computation of the approximations are given. General recommendations are made for the selection of appropriate approximation technique as a function of the matrix size, number of design variables, number of eigenvalues of interest and the number of design points at which approximation is sought
Solution of an eigenvalue problem for the Laplace operator on a spherical surface
Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (also referred to as the membrane eigenvalue problem for the vibration equation) on the unit spherical surface are developed. Two specific types of spherical surface domains are considered: (1) the interior of a spherical triangle, i.e., the region bounded by arcs of three great circles, and (2) the exterior of a great circle arc extending for less than pi radians on the sphere (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is derived. The symmetric (generally five-point) finite difference equations that develop are written in matrix form and then solved by the iterative method of point successive overrelaxation. Upon convergence of this iterative method, the fundamental eigenvalue is approximated by iteration utilizing the power method as applied to the finite Rayleigh quotient
Hardware Impairments Aware Transceiver Design for Full-Duplex Amplify-and-Forward MIMO Relaying
In this work we study the behavior of a full-duplex (FD) and
amplify-and-forward (AF) relay with multiple antennas, where hardware
impairments of the FD relay transceiver is taken into account. Due to the
inter-dependency of the transmit relay power on each antenna and the residual
self-interference in an FD-AF relay, we observe a distortion loop that degrades
the system performance when the relay dynamic range is not high. In this
regard, we analyze the relay function in presence of the hardware inaccuracies
and an optimization problem is formulated to maximize the signal to
distortion-plus-noise ratio (SDNR), under relay and source transmit power
constraints. Due to the problem complexity, we propose a
gradient-projection-based (GP) algorithm to obtain an optimal solution.
Moreover, a nonalternating sub-optimal solution is proposed by assuming a
rank-1 relay amplification matrix, and separating the design of the relay
process into multiple stages (MuStR1). The proposed MuStR1 method is then
enhanced by introducing an alternating update over the optimization variables,
denoted as AltMuStR1 algorithm. It is observed that compared to GP, (Alt)MuStR1
algorithms significantly reduce the required computational complexity at the
expense of a slight performance degradation. Finally, the proposed methods are
evaluated under various system conditions, and compared with the methods
available in the current literature. In particular, it is observed that as the
hardware impairments increase, or for a system with a high transmit power, the
impact of applying a distortion-aware design is significant.Comment: Submitted to IEEE Transactions on Wireless Communication
Residual Arnoldi method, theory, package and experiments
This thesis is concerned with the solution of large-scale eigenvalue problems.
Although there are good algorithms for solving small dense eigenvalue problems,
the large-scale eigenproblem has many open issues. The major difficulty faced by
existing algorithms is the tradeoff of precision and time, especially when one
is looking for interior or clustered eigenvalues.
In this thesis, we present a new method called the residual Arnoldi method.
This method has the desirable property that certain intermediate results can be
computed in low precision without effecting the final accuracy of the solution.
Thus we can reduce the computational cost without sacrificing accuracy. This thesis
is divided into three parts. In the first, we develop the theoretical background of
the residual Arnoldi method. In the second part, we describe RAPACK,
a numerical package implementing the residual Arnoldi method. In the last part,
numerical experiments illustrate the use of the package and show the practicality of
the method
Computation methods for the eigenvalue analysis of large structures by component synthesis
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