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Nonlinear RF and Microwave Circuit Analysis
Published in Mike Golio, Commercial Wireless Circuits and Components Handbook, 2018
Michael B. Steer, John F. Sevic
Harmonic Balance analysis of circuits achieves significant computation savings by assuming that the signals in a circuit are steady state, described by a sum of sinusoids. The coefficients and phases of these sinusoids are solved for and not the transient response. Harmonic balance has a significant computation time advantage over SPICE for small to medium RF and microwave circuits. However, the time increases rapidly as circuit size increases. HB lends itself well to optimization and to analysis of multifunction circuits including amplifiers, oscillators, mixers, frequency converters, and numerous types of control circuits such as limiters and switches, if transient effects are not of concern. Another major advantage of the harmonic balance method is that linear circuits can be of practically any size, with no significant decrease in speed if additional internal nodes are added, or if elements of widely varying time constants are used (such is not the case with time domain simulators). Two extensions, separately implemented, also increase the usefulness of Harmonic Balance. The method of time-variant phasors enables digitally modulated signals to be handled. The second extension using matrix-free methods enables Harmonic Balance to handle very rich spectra and thus also approximately treat digitally modulated signals.
Solving the Neutron Transport Equation for Microreactor Modeling Using Unstructured Meshes and Exascale Computing Architectures
Published in Nuclear Science and Engineering, 2023
William C. Dawn, Scott Palmtag
Generalized eigensolver algorithms are sometimes referred to as “all-at-once” methods as they solve for the angular flux in all discrete angles and energy groups simultaneously. However, the most significant drawback of the generalized eigensolver methodology is the memory required. Although matrix-free methods are technically possible, most implementations require the construction and storage of the and matrices in memory. Consider the matrix, which contains all energy groups and associated scattering data, as well as all angles and associated quadrature weights. As such, the number of nonzero entries in the matrix scales quadratically with the number of angles in a quadrature set. Recall from Eq. (7) that the number of angles in a level-symmetric quadrature scales quadratically with the quadrature order ( for SN). Therefore, the number of nonzero entries in the matrix scales quarticly with the quadrature order [i.e., for SN]. Especially in exascale computing architectures, with a strong dependence on GPUs with limited memory, storing can be prohibitive to implementing a generalized eigensolver.
The HyTeG finite-element software framework for scalable multigrid solvers
Published in International Journal of Parallel, Emergent and Distributed Systems, 2019
Nils Kohl, Dominik Thönnes, Daniel Drzisga, Dominik Bartuschat, Ulrich Rüde
When solving very large systems of equations obtained from the discretisation of PDEs, the memory consumption of matrix-based implementations may be impracticably high, even when using appropriate sparse matrix formats. HyTeG therefore employs matrix-free methods based on stencils (cf. Figure 9). Stencil operations additionally allow for efficient parallel kernels to carry out matrix-vector multiplications or point-wise smoothers.
On the Barzilai–Borwein gradient methods with structured secant equation for nonlinear least squares problems
Published in Optimization Methods and Software, 2022
Aliyu Muhammed Awwal, Poom Kumam, Lin Wang, Mahmoud Muhammad Yahaya, Hassan Mohammad
As calculation of the exact Hessian matrix requires the computation of the second derivatives of the objective function which may sometimes be cumbersome and very expensive, one of the alternative method that avoids the computation of the exact Hessian is the quasi-Newton method. Quasi-Newton method also uses (2) to generate its sequence of iterates, however, the Hessian matrix is being approximated with an matrix, denoted as that is updated in every iteration and satisfies the classical secant equation where and . If is nonsingular, then setting in (3) gives Different matrix-free methods for solving problem (1) have been presented in the literature. One of such methods is the two-point step size gradient methods by Raydan [30] originated from Barzilai and Borwein [4], popularly known as BB method. Starting from a suitable initial point, BB method computes its next iterate via (2) where the search direction, is defined as . The BB approach gave birth to two stepsizes in such a way that the diagonal matrices and approximately satisfy (3) and (4) respectively, where I is an identity matrix. The two stepsizes were obtained by solving the following least squares problems: where their solutions are respectively given as Assuming then by Cauchy Schwarz inequality, we have and which mean .