Restarted Hessenberg method for solving shifted nonsymmetric linear systems

It is known that the restarted full orthogonalization method (FOM) outperforms the restarted generalized minimum residual (GMRES) method in several circumstances for solving shifted linear systems when the shifts are handled simultaneously. Many variants of them have been proposed to enhance their performance. We show that another restarted method, the restarted Hessenberg method [M. Heyouni, M\'ethode de Hessenberg G\'en\'eralis\'ee et Applications, Ph.D. Thesis, Universit\'e des Sciences et Technologies de Lille, France, 1996] based on Hessenberg procedure, can effectively be employed, which can provide accelerating convergence rate with respect to the number of restarts. Theoretical analysis shows that the new residual of shifted restarted Hessenberg method is still collinear with each other. In these cases where the proposed algorithm needs less enough CPU time elapsed to converge than the earlier established restarted shifted FOM, weighted restarted shifted FOM, and some other popular shifted iterative solvers based on the short-term vector recurrence, as shown via extensive numerical experiments involving the recent popular applications of handling the time fractional differential equations.Comment: 19 pages, 7 tables. Some corrections for updating the reference

Efficient variants of the CMRH method for solving a sequence of multi-shifted non-Hermitian linear systems simultaneously

Multi-shifted linear systems with non-Hermitian coefficient matrices arise in numerical solutions of time-dependent partial/fractional differential equations (PDEs/FDEs), in control theory, PageRank problems, and other research fields. We derive efficient variants of the restarted Changing Minimal Residual method based on the cost-effective Hessenberg procedure (CMRH) for this problem class. Then, we introduce a flexible variant of the algorithm that allows to use variable preconditioning at each iteration to further accelerate the convergence of shifted CMRH. We analyse the performance of the new class of methods in the numerical solution of PDEs and FDEs, also against other multi-shifted Krylov subspace methods.Comment: Techn. Rep., Univ. of Groningen, 34 pages. 11 Tables, 2 Figs. This manuscript was submitted to a journal at 20 Jun. 2016. Updated version-1: 31 pages, 10 tables, 2 figs. The manuscript was resubmitted to the journal at 9 Jun. 2018. Updated version-2: 29 pages, 10 tables, 2 figs. Make it concise. Updated version-3: 27 pages, 10 tables, 2 figs. Updated version-4: 28 pages, 10 tables, 2 fig

A flexible and adaptive Simpler GMRES with deflated restarting for shifted linear systems

In this paper, two efficient iterative algorithms based on the simpler GMRES method are proposed for solving shifted linear systems. To make full use of the shifted structure, the proposed algorithms utilizing the deflated restarting strategy and flexible preconditioning can significantly reduce the number of matrix-vector products and the elapsed CPU time. Numerical experiments are reported to illustrate the performance and effectiveness of the proposed algorithms.Comment: 17 pages. 9 Tables, 1 figure; Newly update: add some new numerical results and correct some typos and syntax error

BiCGCR2: A new extension of conjugate residual method for solving non-Hermitian linear systems

In the present paper, we introduce a new extension of the conjugate residual (CR) for solving non-Hermitian linear systems with the aim of developing an alternative basic solver to the established biconjugate gradient (BiCG), biconjugate residual (BiCR) and biconjugate A-orthogonal residual (BiCOR) methods. The proposed Krylov subspace method, referred to as the BiCGCR2 method, is based on short-term vector recurrences and is mathematically equivalent to both BiCR and BiCOR. We demonstrate by extensive numerical experiments that the proposed iterative solver has often better convergence performance than BiCG, BiCR and BiCOR. Hence, it may be exploited for the development of new variants of non-optimal Krylov subspace methods

Mechanism of Collapse, Sensitivity to Ground Motion Features, and Rapid Estimation of the Response of Tall Steel Moment Frame Buildings to Earthquake Excitation

This study explores the behavior of two tall steel moment frame buildings and their variants under strong earthquake ground shaking through parametric analysis using idealized ground motion waveforms. Both fracture-susceptible as well as perfect-connection conditions are investigated. Ground motion velocity waveforms are parameterized using triangular (sawtooth-like) wave-trains with a characteristic period (T), amplitude(peak ground velocity, PGV ), and duration (number of cycles, N). This idealized representation has the desirable feature that the response of the target buildings under the idealized waveforms closely mimics their response under the emulated true ground motion waveforms. A suite of nonlinear analyses are performed on four tall building models subjected to these idealized wave-trains, with T varying from 0.5s to 6.0s, PGV varying from 0.125 m/s to 2.5 m/s, and N taking the values of 1 to 5, and 10. This range of parameters should be adequate to characterize the ground motions that can be expected to occur during earthquakes in the 6-8 magnitude range at some distance (say, > 2km) away from the fault. Databases of peak transient and residual interstory drift ratio (IDR), and permanent roof drift are created for each model. The sensitivity of structural response to T, PGV , and N is studied. Severe dynamic response is induced only in the long-period, large-amplitude excitation regime. Through a simple examination of the energy balance during earthquake shaking, it can be shown that the input excitation energy is small for excitation with periods shorter than the structural period, whereas it is proportional to the square of the ground velocity if the excitation periods are much longer than the structural periods. Thus, collapse-level response can be induced only by long-period, moderate to large PGV ground excitation. The collapse initiation regime expands to lower ground motion periods and amplitudes with increasing number of ground motion cycles. It should be noted that the energy balance analysis is not appropriate for excitation velocities that are extreme where conservation of momentum may be more applicable. However, peak ground velocity from earthquakes seldom exceeds 2.5m/s and energy balance would generally be applicable. The close examination of one instance of collapse shows damage (yielding and/or fracture) localizing in a few stories in the form of a "quasi-shear" band (QSB) comprising of plastic hinges at the top of all columns in the uppermost story of the band, at the bottom of all columns in the lowermost story of the band, and at both ends of all beams in the intermediate stories. Such a pattern of hinging results in shear-like deformation in these stories, resembling plastic shear bands in ductile solids. Most of the lateral deformation due to seismic shaking is concentrated in this band. When the overturning 1st-order and 2nd-order (P - ) moments from the inertia of the overriding block of stories exceed the moment-carrying capacity of the quasi-shear band, it loses stability and collapses. This initiates gravity-driven progressive collapse of the overriding block of stories. Thus, the collapse mechanism initiates as a sidesway mechanism that is taken over by gravity once the quasi-shear band is destabilized. There are Ns(Ns+1) 2 possible quasi-shear bands (and an equal number of sidesway collapse mechanisms) in either principal direction of an Ns-story moment frame building. More than one quasi-shear bands could occur during the entire duration of strong earthquake shaking. The band exhibiting the greatest distress (termed the "primary" quasi-shear band) iv ultimately evolves into a sidesway collapse mechanism. The formation of the quasi-shear band under single-cycle excitations is explained through the classical uniform shear-beam analogy to moment frame buildings. Under low-intensity motions (PGV < 0.25m/s)with periods in the 0.5s-6s range excitation energy is low. As a result, structural response is predominantly elastic and is analogous to that of a uniform elastic shear-beam through which a shear wave propagates. For moderate-intensity excitations (0.25m/s PGV < 1.5m/s), the reverse phase of the incident pulse constructively interferes with the reflected forward phase causing yielding in the region of positive interference,very similar to what would occur in a uniform inelastic shear-beam. The primary quasi-shear band migrates down the building with increasing pulse period. However, this migration slows down with increasing period and gets arrested nominally between floors 3 and 9 for the existing building, and between floors 3 and 8 for the redesigned building, whereas the peak strain in the corresponding inelastic uniform shear-beam continues to migrate to the very bottom. This is a direct result of the non-uniformity of the buildings. Going from the top of the building to the bottom, there is a gradual increase in the strength and stiffness of the structure. The increased strength at the bottom does not allow yielding to permeate into those stories. Now,excitation energy imparted to the structure can be large enough only under long-period ground motion in the context of the target buildings. Therefore, collapse-level response must be accompanied by the formation of the primary quasi-shear band in the vicinity of the stories where the downward migration of the QSB (with increasing T) is arrested. For high-intensity excitations (PGV > 1.5m/s) that are sufficiently long-period, the pulse may yield the structure on its way up the building. The strength of the building drops as the pulse travels up the building. However, inertial forces drop as well, as a result of fewer stories above contributing to the mass. The narrow band of stories with an optimal combination of low-enough strength and high-enough inertial force demand is where peak yielding occurs. This region is identical to the region where the downward migration of the primary quasi-shear band is arrested under moderate-intensity, long-period excitation. This is because the governing factor dictating the location of the band in both cases is strength non-uniformity. As the wave travels up the building, it is reflected off the roof with a change in sign. Because the period of the incident wave is sufficiently long (a necessary condition for large input excitation energy), the reverse phase of the incident pulse constructively interferes with the reflected forward phase causing greatest yielding in the same region as the pre-reflection yielding. To summarize, under both moderate-intensity and highintensity ground motions, input excitation energy large enough to collapse the building requires long-period excitation. Such long-period excitation always causes the formation of the primary quasi-shear band in an optimal set of stories governed by the mass and strength distribution of the building over its height, which are characteristics solely of the structure and not the ground motion. When T and PGV are large enough, it is this band that evolves into a collapse mechanism. This points to the existence of a "characteristic" collapse mechanism or only a few preferred collapse mechanisms (out of the Ns(Ns+1)2 possible mechanisms) in either principal direction of the building. If multiple preferred collapse mechanisms exist, they would be clustered together with significant story-overlap amongst them. The simulations of the four models under idealized ground motion waveforms where collapse occurs do not show the formation of a single (unique) collapse mechanism. However, in each model only one to five v collapse mechanisms occur out of a possible 153 mechanisms in each principal direction of the building. Furthermore, if two or more preferred mechanisms do exist, they have significant story-overlap, typically separated by just one story. For example, the strongly preferred collapse mechanisms in the existing building model (perfect connections) under X direction excitation occur between floors 3 and 9, and floors 4 and 9, while the weakly preferred mechanisms occur between floors 3 and 8, and floors 4 and 8 (four preferred mechanisms out of 153 possible mechanisms, all clustered together within a narrow story zone; two of these mechanisms are in fact a subset of the other two mechanisms). The characteristic and/or preferred collapse mechanisms can be identified by applying the Principle of Virtual Work to all possible quasi-shear bands in a building. Based on plastic analysis principles, the band that is destabilized by the smallest acceleration of the over-riding block of stories is the characteristic collapse mechanism. If one or more bands exist that have destabilizing accelerations close to that of the characteristic collapse band, say within 5%, then these bands may evolve into collapse mechanisms as well. This method identifies all the preferred collapse mechanisms in all four building models satisfactorily. One application of the structural response database built for the sensitivity study is the rapid estimation of structural response immediately following an earthquake if the ground motion records become available. The best fit of the idealized wave-trains in the database to the ground motion record can be determined using the least absolute deviation method. The corresponding key structural response metrics can be extracted from the database using a simple table look-up approach. Such a method, when applied to a suite of nearsource records, predicts peak transient IDR remarkably well. Gaussian mean estimation error on the peak transient IDR is 0.0006, with a standard deviation of 0.0069. A minor modification to this approach is needed when applying it to multi-cycle far-field records. This modified approach is used to estimate the peak transient IDR response of the buildings under synthetic waveforms from a large hypothetical San Andreas fault earthquake. The Gaussian mean error for this estimation is 0.0011, with a standard deviation of 0.0209, slightly worse than for the near-source records, nevertheless within one "performance level" - good enough for emergency response decision-making. The same approach can be used for ball-park estimation of structural response under any given earthquake record, in lieu of comprehensive nonlinear analysis