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Journal articles on the topic 'Fast diagonalization method'
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1
Si, Wei, and Qiang Xu. "Virtual Boundary Element Collocation Method with RBF Interpolation on Virtual Boundary and Diagonalization Feature in Fast Multipole Method." Advanced Materials Research 378-379 (October 2011): 166–70. http://dx.doi.org/10.4028/www.scientific.net/amr.378-379.166.
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The algorithm idea of virtual boundary element collocation method with RBF interpolation on virtual boundary and diagonalization feature in fast multipole method is presented to study 2-D elasticity problems in this paper. In other words, the new fast multipole method (FMM) adopting diagonalization and the generalized minimal residual (GMRES) algorithm are jointly employed to solve the equations related to virtual boundary element collocation method (VBEM) with RBF interpolation on virtual boundary. In this paper, the numerical scheme suitable for original FMM with respect to two-dimensional problem of elasticity is optimized, through the introduction of concept of diagonalization, in terms of the radial basis function to express the unknown virtual load functions, in order to further improve the efficiency of the problem to be solved. Then large-scale numerical simulations of elastostatics might be achieved by the method. Numerical examples in the paper have proved the feasibility, efficiency and calculating precision of the method.
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Zhang Na, 张娜, 冯金超 Feng Jinchao, 李哲 Li Zhe, and 贾克斌 Jia Kebin. "Fast Photoacoustic Imaging Reconstruction Method Based on Lanczos Double Diagonalization." Chinese Journal of Lasers 45, no. 3 (2018): 0307018. http://dx.doi.org/10.3788/cjl201845.0307018.
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Nyka, Krzysztof. "Diagonalized Macromodels in Finite Element Method for Fast Electromagnetic Analysis of Waveguide Components." Electronics 8, no. 3 (February 27, 2019): 260. http://dx.doi.org/10.3390/electronics8030260.
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A new technique of local model-order reduction (MOR) in 3-D finite element method (FEM) for frequency-domain electromagnetic analysis of waveguide components is proposed in this paper. It resolves the problem of increasing solution time of the reduced-order system assembled from macromodels created in the subdomains, into which an analyzed structure is partitioned. This problem becomes particularly relevant for growing size and count of the macromodels, and when they are cloned in multiple locations of the structures or are used repeatedly in a tuning and optimization process. To significantly reduce the solution time, the diagonalized macromodels are created by means of the simultaneous diagonalization and subsequently assembled in the global system. For the resulting partially diagonal matrix, an efficient dedicated solver based on the Schur complement technique is proposed. The employed MOR method preserves frequency independence of the macromodels, which is essential for efficient diagonalization, as it can be performed once for the whole analysis bandwidth. The numerical validation of the proposed procedures with respect to accuracy and speed was carried out for varying size and count of macromodels. An exemplary finite periodical waveguide structure was chosen to investigate the influence of macromodel cloning on the resultant efficiency. The results show that the use of the diagonalized macromodels provided a significant solution speedup without any loss of accuracy.
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Montardini, M., G. Sangalli, and M. Tani. "Robust isogeometric preconditioners for the Stokes system based on the Fast Diagonalization method." Computer Methods in Applied Mechanics and Engineering 338 (August 2018): 162–85. http://dx.doi.org/10.1016/j.cma.2018.04.017.
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Liu, Fang, Yongbin Liu, Fenglin Chen, and Bing He. "Residual life prediction for ball bearings based on joint approximate diagonalization of eigen matrices and extreme learning machine." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 231, no. 9 (December 10, 2015): 1699–711. http://dx.doi.org/10.1177/0954406215621585.
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Data-driven approaches have been proved effective for remaining useful life estimation of key components (bearings for example) in rotating machinery. In such approaches, it is important to determine an appropriate degradation indicator from the collected run-to-failure life cycle data. In this paper, a new degradation indicator is introduced based on the joint approximate diagonalization of eigen matrices algorithm. First, a matrix consisting of time domain, frequency domain, and time–frequency domain features extracted from the collected data instances is created. Then a two-layer joint approximate diagonalization of eigen matrices is introduced to transform the matrix to the advanced features (a vector) that represents the behavior of the bearing’s degradation. As an independent component analysis method, the designed two-layer joint approximate diagonalization of eigen matrices is able to eliminate the redundancy of the directly extracted features. Further, the obtained vector is input into an extreme learning machine to train a remaining useful life prediction model. Finally, a set of experimental cases are utilized to verify the presented method. Results show that the two-layer joint approximate diagonalization of eigen matrices is capable of exploring features that reflects the trend of bearing’s degradation state much better. And due to the easy parameter configuration and fast learning speed, the extreme learning machine is capable of training a model that can effectively predict the remaining useful life of the bearings.
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Eibert, T. F. "A multilevel fast spectral domain algorithm for electromagnetic analysis of infinite periodic arrays with large unit cells." Advances in Radio Science 4 (September 4, 2006): 41–47. http://dx.doi.org/10.5194/ars-4-41-2006.
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Abstract. A multilevel fast spectral domain algorithm (MLFSDA) is introduced for the efficient evaluation of the matrix vector products due to the boundary integral (BI) operator within a hybrid finite element - BI (FEBI) method for the analysis of infinite periodic arrays. The MLFSDA utilizes the diagonalization property of the spectral domain (SD) BI representation and handles the large numbers of Floquet modes required for large (with respect to wavelength) periodic unit cells by similar hierarchical techniques as applied in the multilevel fast multipole method/algorithm (MLFMM/MLFMA). With the capability of the MLFSDA to handle very large periodic unit cells, it becomes possible to model finite antennas and scatterers with the infinite periodic array model. For a cavity-backed antenna element and for a semi-finite array of 4 cavity-backed antenna elements in the finite direction, the dependence of the input impedances on the unit cell sizes is investigated and it is found that array resonances disappear for reasonably large unit cell dimensions. Finally, a semi-finite array of antipodal Vivaldi antenna elements is considered and simulation results for infinite periodic, finite, and semi-finite array configurations are compared to measured data.
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Lay-Ekuakille, A., G. Griffo, D. Pellicanò, P. Maris, and M. Cacciola. "A Hardware for Processing Magnetic Pressure Sensor Signals from Leak Detection in Waterworks." International Journal of Measurement Technologies and Instrumentation Engineering 3, no. 3 (July 2013): 35–45. http://dx.doi.org/10.4018/ijmtie.2013070103.
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Leaks in pipelines and waterworks are detected using different methods and among them spectral analysis is one of the most interesting ones. Sources of signals to be processed are different, for example: reflected signals from ground penetrating radar and acoustic sources, signals from dedicated sensors mounted on pipelines, etc… In the latter case, magnetic pressure sensors located on the pipeline acquire vibrations and oscillations of liquids (e.g. water) in the pipeline, following a leak in the pipeline. These vibrations and oscillations are transformed in electrical signal and processed using different methods and techniques like FFT (Fast Fourier Transform), ANN (Artificial Neural Network), STFT (Short-Term Fourier Transform), and Impedance Method (IM). But there are other advanced methodical approaches that can improve the quality of the signal related to the leak; one of them is FDM (Filter Diagonalization Method). Even in presence of an advanced method, recovered signal displays undesired attenuation and noisy behavior due to different reasons, namely, hardware, background noise, materials used for pipeline construction, sensors, etc.. This paper presents a complementary hardware for processing the above signals. The hardware is based on innovating approach that minimizes additional noisy components.
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Cornejo Fuentes, Joaquín, Thomas Elguedj, David Dureisseix, and Arnaud Duval. "A cheap preconditioner based on fast diagonalization method for matrix-free weighted-quadrature isogeometric analysis applied to nonlinear transient heat transfer problems." Computer Methods in Applied Mechanics and Engineering 414 (September 2023): 116157. http://dx.doi.org/10.1016/j.cma.2023.116157.
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Cervellino, Antonio, Cinzia Giannini, Antonietta Guagliardi, and Massimo Ladisa. "Unfolding a two-dimensional powder diffraction image: conformal mapping." Journal of Applied Crystallography 41, no. 4 (July 1, 2008): 701–4. http://dx.doi.org/10.1107/s0021889808019092.
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A new procedure aimed at unfolding a two-dimensional powder diffraction image into both a one-dimensional azimuthal and a radial scan is presented. In this approach, the sample-to-detector distance is the only parameter that must be adjusted in a separate step by using a standard sample. The technique consists of three steps: tracking the beam centre as the local maximum of the self-convolution of the original two-dimensional map, detector tilt and rotation determination by an intensity-tensor diagonalization, and azimuthal/radial intensity integration by a conformal mapping of the original two-dimensional powder diffraction image. The X-ray powder diffraction (XRPD) intensity profile of the NIST Si 640c standard sample is used to test the performance. The results show the robustness of the method and its capability of efficiently tagging the pixels in a two-dimensional readout system by matching the ideal geometry of the detector to the real beam–sample–detector frame. The technique is a fast, versatile and user-friendly tool for the simultaneous analysis of both azimuthal and radial spectra of two-dimensional XRPD images.
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Liu, Wang, Huang, and Yang. "An Improved Second-Order Blind Identification (SOBI) Signal De-Noising Method for Dynamic Deflection Measurements of Bridges Using Ground-Based Synthetic Aperture Radar (GBSAR)." Applied Sciences 9, no. 17 (August 30, 2019): 3561. http://dx.doi.org/10.3390/app9173561.
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Ground-based synthetic aperture radar (GBSAR) technology has been widely used for bridge dynamic deflection measurements due to its advantages of non-contact measurements, high frequency, and high accuracy. To reduce the influence of noise in dynamic deflection measurements of bridges using GBSAR—especially for noise of the instantaneous vibrations of the instrument itself caused by passing vehicles—an improved second-order blind identification (SOBI) signal de-noising method is proposed to obtain the de-noised time-series displacement of bridges. First, the obtained time-series displacements of three adjacent monitoring points in the same time domain are selected as observation signals, and the second-order correlations among the three time-series displacements are removed using a whitening process. Second, a mixing matrix is calculated using the joint approximation diagonalization technique for covariance matrices and to further obtain three separate signal components. Finally, the three separate signal components are converted in the frequency domain using the fast Fourier transform (FFT) algorithm, and the noise signal components are identified using a spectrum analysis. A new, independent, separated signal component matrix is generated using a zeroing process for the noise signal components. This process is inversely reconstructed using a mixing matrix to recover the original amplitude of the de-noised time-series displacement of the middle monitoring point among three adjacent monitoring points. The results of both simulated and on-site experiments show that the improved SOBI method has a powerful signal de-noising ability.
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Shen, Yizhi, Katherine Klymko, James Sud, David B. Williams-Young, Wibe A. de Jong, and Norm M. Tubman. "Real-Time Krylov Theory for Quantum Computing Algorithms." Quantum 7 (July 25, 2023): 1066. http://dx.doi.org/10.22331/q-2023-07-25-1066.
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Quantum computers provide new avenues to access ground and excited state properties of systems otherwise difficult to simulate on classical hardware. New approaches using subspaces generated by real-time evolution have shown efficiency in extracting eigenstate information, but the full capabilities of such approaches are still not understood. In recent work, we developed the variational quantum phase estimation (VQPE) method, a compact and efficient real-time algorithm to extract eigenvalues on quantum hardware. Here we build on that work by theoretically and numerically exploring a generalized Krylov scheme where the Krylov subspace is constructed through a parametrized real-time evolution, which applies to the VQPE algorithm as well as others. We establish an error bound that justifies the fast convergence of our spectral approximation. We also derive how the overlap with high energy eigenstates becomes suppressed from real-time subspace diagonalization and we visualize the process that shows the signature phase cancellations at specific eigenenergies. We investigate various algorithm implementations and consider performance when stochasticity is added to the target Hamiltonian in the form of spectral statistics. To demonstrate the practicality of such real-time evolution, we discuss its application to fundamental problems in quantum computation such as electronic structure predictions for strongly correlated systems.
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Greenwood, Eric. "Time-dependent particle production and particle number in cosmological de Sitter space." International Journal of Modern Physics D 24, no. 05 (March 18, 2015): 1550031. http://dx.doi.org/10.1142/s0218271815500315.
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In this paper, we consider the occupation number of induced quasi-particles which are produced during a time-dependent process using three different methods: Instantaneous diagonalization, the usual Bogolyubov transformation between two different vacua (more precisely the instantaneous vacuum and the so-called adiabatic vacuum), and the Unruh–DeWitt detector methods. Here we consider the Hamiltonian for a time-dependent Harmonic oscillator, where both the mass and frequency are taken to be time-dependent. From the Hamiltonian we derive the occupation number of the induced quasi-particles using the invariant operator method. In deriving the occupation number we also point out and make the connection between the Functional Schrödinger formalism, quantum kinetic equation, and Bogolyubov transformation between two different Fock space basis at equal times and explain the role in which the invariant operator method plays. As a concrete example, we consider particle production in the flat FRW chart of de Sitter spacetime. Here we show that the different methods lead to different results: The instantaneous diagonalization method leads to a power law distribution, while the usual Bogolyubov transformation and Unruh–DeWitt detector methods both lead to thermal distributions (however the dimensionality of the results are not consistent with the dimensionality of the problem; the usual Bogolyubov transformation method implies that the dimensionality is 3D while the Unruh–DeWitt detector method implies that the dimensionality is 7D/2). It is shown that the source of the descrepency between the instantaneous diagonalization and usual Bogolyubov methods is the fact that there is no notion of well-defined particles in the out vacuum due to a divergent term. In the usual Bogolyubov method, this divergent term cancels leading to the thermal distribution, while in the instantaneous diagonalization method there is no such cancelation leading to the power law distribution. However, to obtain the thermal distribution in the usual Bogolyubov method, one must use the large mass limit. On physical grounds, one should expect that only the modes which have been allowed to sample the horizon would be thermal, thus in the large mass limit these modes are well within the horizon and, even though they do grow, they remain well within the horizon due to the mass. Thus, one should not expect a thermal distribution since the modes will not have a chance to thermalize.
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Sabri, K., M. El Badaoui, F. Guillet, A. Adib, and D. Aboutajdine. "On Blind MIMO System Identification Based on Second-Order Cyclic Statistics." Research Letters in Signal Processing 2008 (2008): 1–5. http://dx.doi.org/10.1155/2008/539139.
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This letter introduces a new frequency domain approach for either MIMO System Identification or Source Separation of convolutive mixtures in cyclostationary context. We apply the joint diagonalization algorithm to a set of cyclic spectral density matrices of the measurements to identify the mixing system at each frequency up to permutation and phase ambiguity matrices. An efficient algorithm to overcome the frequency dependent permutations and to recover the phase, even for non-minimum-phase channels, based on cyclostationarity is also presented. The new approach exploits the fact that each input has a different and specific cyclic frequency. A comparison with an existing MIMO method is proposed.
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Leung, P. W., and K. K. Ng. "Asymmetry in the Hole-Doped and Electron-Doped t-J Model." International Journal of Modern Physics B 17, no. 18n20 (August 10, 2003): 3367–69. http://dx.doi.org/10.1142/s0217979203021009.
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We model electron-doped high Tc materials using the t-t′-t′′-J model. This model is solved on a cluster with 32 sites using the method of exact diagonalization. Our purpose is to study the symmetry and asymmetry between hole- and electron-doped high Tc materials by comparing the results with those of the t-J model. In the electron-doped model with one charge carrier, we find a strong quasiparticle peak at (π,0). Compared to the t-J model, the bandwidth is larger, reflecting the fact that the charge carrier moves more freely in the electron-doped model. In the two-carrier model the ground state has robust dx2-y2 symmetry. This is in contrary to the two-hole t-J model whose ground state is a competition between low-energy states with dx2-y2 and p symmetries. The spatial distribution function of the carriers shows that they move almost freely on different sublattices.
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Herman, Przemyslaw. "Controller for an Asymmetric Underactuated Hovercraft in Terms of Quasi-Velocities." Applied Sciences 13, no. 8 (April 14, 2023): 4965. http://dx.doi.org/10.3390/app13084965.
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In this paper, a nonlinear controller for tracking a desired trajectory for an underactuated hovercraft is considered. It is a modification of a method known from the literature. However, the control algorithm considered here has two important features that differ from the mentioned control strategy. First, it concerns the case when the center of mass does not coincide with the geometric center, which results in additional forces and moments of force. The lack of symmetry causes the original trajectory tracking method not to take this fact into account, while the proposed approach is a generalization of the known concept. Here, a diagonalization of the inertia matrix has been applied, by means of a velocity transformation, which made it possible to reduce the symmetric matrix to a diagonal form. Secondly, the transformed quasi-velocity equations of motion allow some insight into the dynamics of the vehicle as it moves, which was not shown in the source work. The offered approach was verified by numerical tests for a hovercraft model with three DOF and for two desired trajectories. The method can be useful in preliminary simulation studies at the controller selection stage without experimental validation.
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Zhao, J.-S., H.-X. Zhou, Z.-J. Feng, and J. S. Dai. "An algebraic methodology to identify the principal screws and pitches of screw systems." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 223, no. 8 (April 16, 2009): 1931–41. http://dx.doi.org/10.1243/09544062jmes1394.
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This article investigates the principal screws and their pitches of screw systems from the viewpoint of line geometry. It starts from the fact that any n-independent screws pertaining to a specified n-screw system can be utilized to determine the system, and the pitch of a unit screw is the half of the reciprocal product with itself. The reciprocal product matrix of a screw matrix is therefore defined to determine the principal pitches and the directions of the principal screw axes of the system. With matrix diagonalization procedure, one can immediately obtain the principal pitches and the principal screws of a screw system. The whole procedure is straightforward and easily understood by an engineer with primary knowledge of linear algebra. Besides, the computational effort is greatly reduced and the efficiency is better compared with other methods. As a matter of fact, the pitch of any screw of a screw system can be represented by a homogeneous quadratic form of the linear combination coefficients. The homogeneous quadratic equation of the pitch of a screw system represents a conic curve in a two-system and an ellipsoid or a hyperboloid in a three-system. Therefore, this article also provides such visualizations of the principal axes in two- and three-dimensional spaces at last.
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Ren, Hualing, and Zhiheng Dong. "Modeling traffic flow on a comprehensive network with competition and cooperation between urban logistic alliances." Modern Physics Letters B 32, no. 33 (November 30, 2018): 1850402. http://dx.doi.org/10.1142/s021798491850402x.
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The traffic during rush hour in metropolises is always in heavy congestion, so the wagon trucks of logistic transportation are mostly forbidden during rush hour. In fact, quite a few logistic transportations are fulfilled by converted passenger vehicles. This paper aims to model the logistic companies’ behaviors and see through the effect of their cooperation modes on the companies and on the whole traffic conditions. The logistic companies decide their transportation pricing, warehouse using and route choosing separately or as a group (cooperation) to compete with others. It is also assumed that the heavy freight vehicles, light freight vehicles and the auto vehicles share the same network. Three models based on variational inequalities are proposed to describe the companies’ behaviors when there is no cooperation, simple cooperation and deep cooperation, respectively, and the modified diagonalization method is used to solve these models. The total transportation demand is elastic, which depends on the users’ choice of logistics company and further depends on all the companies’ prices, warehouse and route choices. The goodness of different coalitions to the companies and the whole traffic system is well investigated in the numerical examples.
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Wu, Chen-Huan. "Time evolution and thermodynamics for a nonequilibrium system in phase-space." Canadian Journal of Physics 97, no. 6 (June 2019): 609–36. http://dx.doi.org/10.1139/cjp-2017-0913.
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The integrable system is constrained strictly by the conservation law during the time evolution, and the prethermal state from the nearly integrable system is also constrained by the conserved parameters (the constants of motion) with the corresponding generalized Gibbs ensemble (GGE), which is indubitably a powerful tool in the prediction of the relaxation dynamics. For stochastic evolution dynamics with considerable noise, the two-point correlation of local operators (like the density of kinks or transverse magnetization correlators), which do not exhibit the thermal features, display the behaviors of nonthermalization and an asymptotic GGE. In fact it is an asymptotic quasi-steady state with an infinite temperature, therefore the required distance to the nonthermal steady state is in an infinite time average. In this paper, we unambiguously investigate the relaxation of a nonequilibrium system in a canonical ensemble for integrable and nonintegrable systems. Temporal behavior of the many-body quantum system and the corresponding linear-coupling between the harmonic oscillators are discussed. The matrix-method in entropy ensemble is utilized to discuss the boundary and the diagonalization algebraically. The approximation results for nonintegrable system under the considerable perturbations are also presented.
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Bermúdez-Montaña, Marisol, Marisol Rodríguez-Arcos, Renato Lemus, José M. Arias, Joaquín Gómez-Camacho, and Emilio Orgaz. "Algebraic DVR Approaches Applied to Describe the Stark Effect." Symmetry 12, no. 10 (October 19, 2020): 1719. http://dx.doi.org/10.3390/sym12101719.
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Two algebraic approaches based on a discrete variable representation are introduced and applied to describe the Stark effect in the non-relativistic Hydrogen atom. One approach consists of considering an algebraic representation of a cutoff 3D harmonic oscillator where the matrix representation of the operators r2 and p2 are diagonalized to define useful bases to obtain the matrix representation of the Hamiltonian in a simple form in terms of diagonal matrices. The second approach is based on the U(4) dynamical algebra which consists of the addition of a scalar boson to the 3D harmonic oscillator space keeping constant the total number of bosons. This allows the kets associated with the different subgroup chains to be linked to energy, coordinate and momentum representations, whose involved branching rules define the discrete variable representation. Both methods, although originating from the harmonic oscillator basis, provide different convergence tests due to the fact that the associated discrete bases turn out to be different. These approaches provide powerful tools to obtain the matrix representation of 3D general Hamiltonians in a simple form. In particular, the Hydrogen atom interacting with a static electric field is described. To accomplish this task, the diagonalization of the exact matrix representation of the Hamiltonian is carried out. Particular attention is paid to the subspaces associated with the quantum numbers n=2,3 with m=0.
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Tchoffo, Martin, William Degaulle Waladi Gueagni, Georges Collince Fouokeng, Lionel Tenemeza Kenfack, and Lukong Cornelius Fai. "Thermodynamic properties of a multiferroic antiferromagnetic spin system: The influence of Dzyaloshinskii Moriya interaction." International Journal of Modern Physics B 33, no. 08 (March 30, 2019): 1950051. http://dx.doi.org/10.1142/s0217979219500516.
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In this paper, we address the influence of Dzyaloshinskii Moriya (DM) interaction on the thermodynamic properties of a multiferroic antiferromagnetic spin system using the spin wave theory (SWT) as a diagonalization method which, associated to the statistical physics, helps to evaluate the statistical sum. The basic factors in thermodynamics, such as the Boltzmann entropy and the specific heat capacity at thermal equilibrium, are obtained. Analyzing the numerical results, it follows that the DM interaction is the best candidate that can help to maintain in long time the system in its coherent state. Furthermore, we find that the influence of the DM interaction is more accentuated at low-temperature and that it enables the system to release heat energy. On the other hand, the fact that the specific heat capacity tends to the constant clearly shows that the system obeys the Dulong–Petit law. In addition, our results also show that for high magnetic field (greater than B[Formula: see text]=[Formula: see text]30 Tesla), the magnetic field dependence of both entropy and specific heat shows a peak-like behavior and that the DM interaction raises the corresponding critical magnetic field but lowers the peak amplitude. It follows that the spin-flop transition occurs in the system and that the strong magnetic field withdraw the antiferromagnetic phase. Overall, we find that the DM interaction is a promising candidate for the control of our system.
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Finkelstein, Joshua, Christian F. A. Negre, and Jean-Luc Fattebert. "A fast, dense Chebyshev solver for electronic structure on GPUs." Journal of Chemical Physics 159, no. 10 (September 11, 2023). http://dx.doi.org/10.1063/5.0164255.
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Matrix diagonalization is almost always involved in computing the density matrix needed in quantum chemistry calculations. In the case of modest matrix sizes (≲4000), performance of traditional dense diagonalization algorithms on modern GPUs is underwhelming compared to the peak performance of these devices. This motivates the exploration of alternative algorithms better suited to these types of architectures. We newly derive, and present in detail, an existing Chebyshev expansion algorithm [Liang et al., J. Chem. Phys. 119, 4117–4125 (2003)] whose number of required matrix multiplications scales with the square root of the number of terms in the expansion. Focusing on dense matrices of modest size, our implementation on GPUs results in large speed ups when compared to diagonalization. Additionally, we improve upon this existing method by capitalizing on the inherent task parallelism and concurrency in the algorithm. This improvement is implemented on GPUs by using CUDA and HIP streams via the MAGMA library and leads to a significant speed up over the serial-only approach for smaller (≲1000) matrix sizes. Finally, we apply our technique to a model system with a high density of states around the Fermi level, which typically presents significant challenges.
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Faulkner, J. S. "Experiences with the Quadratic Korringa-Kohn-Rostoker Band Theory Method." MRS Proceedings 253 (1991). http://dx.doi.org/10.1557/proc-253-27.
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ABSTRACTThe Quadratic Korringa-Kohn-Rostoker method is a fast band theory method in the sense that all eigenvalues for a given k are obtained from one matrix diagonalization, but it differs from other fast band theory methods in that it is derived entirely from multiple-scattering theory, without the introduction of a Rayleigh-Ritz variational step. In this theory, the atomic potentials are shifted by Δασ(r) with Δ equal to E-E0 and σ(r) equal to one when r is inside the Wigner-Seitz cell and zero otherwise, and it turns out that the matrix of coefficients is an entire function of Δ. This matrix can be terminated to give a linear KKR, quadratic KKR, cubic KKR, …, or not terminated at all to give the pivoted multiple-scattering equations. Full potentials are no harder to deal with than potentials with a shape approximation.
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Foltyn, Ladislav, Dalibor Lukáš, and Marco Zank. "Robust PRESB Preconditioning of a 3-Dimensional Space-Time Finite Element Method for Parabolic Problems." Computational Methods in Applied Mathematics, January 25, 2024. http://dx.doi.org/10.1515/cmam-2023-0085.
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Abstract We present a recently developed preconditioning of square block matrices (PRESB) to be used within a parallel method to solve linear systems of equations arising from tensor-product discretizations of initial boundary-value problems for parabolic partial differential equations. We consider weak formulations in Bochner–Sobolev spaces and tensor-product finite element approximations for the heat and eddy current equations. The fast diagonalization method is employed to decouple the arising linear system of equations into auxiliary spatial complex-valued linear systems that can be solved concurrently. We prove that the real part of the system matrix is positive definite, which allows us to accelerate the flexible generalized minimal residual method (FGMRES) by the PRESB preconditioner. The action of PRESB on a vector includes two solutions with positive definite matrices. The spectrum of the preconditioned system lies between 1/2 and 1. Finally, we combine the PRESB-FGMRES method with multigrid-CG iterations and illustrate the numerical efficiency and the robustness for spatial discretizations up to 12 millions degrees of freedom.
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Gibbs, Joe, Kaitlin Gili, Zoë Holmes, Benjamin Commeau, Andrew Arrasmith, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger. "Long-time simulations for fixed input states on quantum hardware." npj Quantum Information 8, no. 1 (November 19, 2022). http://dx.doi.org/10.1038/s41534-022-00625-0.
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AbstractPublicly accessible quantum computers open the exciting possibility of experimental dynamical quantum simulations. While rapidly improving, current devices have short coherence times, restricting the viable circuit depth. Despite these limitations, we demonstrate long-time, high fidelity simulations on current hardware. Specifically, we simulate an XY-model spin chain on Rigetti and IBM quantum computers, maintaining a fidelity over 0.9 for 150 times longer than is possible using the iterated Trotter method. Our simulations use an algorithm we call fixed state Variational Fast Forwarding (fsVFF). Recent work has shown an approximate diagonalization of a short time evolution unitary allows a fixed-depth simulation. fsVFF substantially reduces the required resources by only diagonalizing the energy subspace spanned by the initial state, rather than over the total Hilbert space. We further demonstrate the viability of fsVFF through large numerical simulations, and provide an analysis of the noise resilience and scaling of simulation errors.
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Lam, Jordy Homing, Aiichiro Nakano, and Vsevolod Katritch. "Scalable computation of anisotropic vibrations for large macromolecular assemblies." Nature Communications 15, no. 1 (April 24, 2024). http://dx.doi.org/10.1038/s41467-024-47685-8.
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AbstractThe Normal Mode Analysis (NMA) is a standard approach to elucidate the anisotropic vibrations of macromolecules at their folded states, where low-frequency collective motions can reveal rearrangements of domains and changes in the exposed surface of macromolecules. Recent advances in structural biology have enabled the resolution of megascale macromolecules with millions of atoms. However, the calculation of their vibrational modes remains elusive due to the prohibitive cost associated with constructing and diagonalizing the underlying eigenproblem and the current approaches to NMA are not readily adaptable for efficient parallel computing on graphic processing unit (GPU). Here, we present eigenproblem construction and diagonalization approach that implements level-structure bandwidth-reducing algorithms to transform the sparse computation in NMA to a globally-sparse-yet-locally-dense computation, allowing batched tensor products to be most efficiently executed on GPU. We map, optimize, and compare several low-complexity Krylov-subspace eigensolvers, supplemented by techniques such as Chebyshev filtering, sum decomposition, external explicit deflation and shift-and-inverse, to allow fast GPU-resident calculations. The method allows accurate calculation of the first 1000 vibrational modes of some largest structures in PDB ( > 2.4 million atoms) at least 250 times faster than existing methods.
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Ugandi, Mihkel, and Michael Roemelt. "A configuration‐based heatbath‐CI for spin‐adapted multireference electronic structure calculations with large active spaces." Journal of Computational Chemistry, August 17, 2023. http://dx.doi.org/10.1002/jcc.27203.
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AbstractThis work reports on a spin‐pure configuration‐based implementation of the heatbath configuration interaction (HCI) algorithm for selective configuration interaction. Besides the obvious advantage of being spin‐pure, the presented method combines the compactness of the configurational ansatz with the known efficiency of the HCI algorithm and a variety of algorithmic and conceptual ideas to achieve a high level of performance. In particular, through pruning of the selected configurational space after HCI selection by means of a more strict criterion, a more compact wavefunction representation is obtained. Moreover, the underlying logic of the method allows us to minimize the number of redundant matrix‐matrix multiplications while making use of just‐in‐time compilation to achieve fast diagonalization of the Hamiltonian. The critical search for 2‐electron connections within the configurational space is facilitated by a tree‐based representation thereof as suggested previously by Gopal et al. Usage of a prefix‐based parallelization and batching during the calculation of the PT2‐correction leads to a good load balancing and significantly reduced memory requirements for these critical steps of the calculation. In this way, the need for a semistochastic approach to the PT2 correction is avoided even for large configurational spaces. Finally, several test‐cases are discussed to demonstrate the strengths and weaknesses of the presented method.
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Munch, Peter, and Martin Kronbichler. "Cache-optimized and low-overhead implementations of additive Schwarz methods for high-order FEM multigrid computations." International Journal of High Performance Computing Applications, December 6, 2023. http://dx.doi.org/10.1177/10943420231217221.
Full textAbstract:
This contribution presents data-locality optimizations of the additive Schwarz method (ASM) based on the fast-diagonalization method defined on overlapping cell-centric and vertex-star patches in the context of high-order matrix-free finite-element computations on modern CPU-based hardware. The developments are guided by detailed performance models of the ASM in the context of Chebyshev iterations when used as smoothers for p-multigrid. The proposed efficient implementation of ASM adopts concepts known from cell-loop infrastructures for efficient operator evaluation, in particular, the storage of information per geometric entity and the cache-friendly interleaving of cell loops and vector updates as a means to increase data locality. We use the latter concept for both applying the weights required by ASM and performing the vector updates required by the Chebyshev iteration, which are memory-bound operations with non-negligible costs in comparison to efficient operator evaluation. Furthermore, the solution of a scalar Poisson problem on a highly anisotropic and an unstructured mesh with p-multigrid using the developed smoothers indicates that efficient implementations of the additive Schwarz method can outperform optimized point-Jacobi preconditioners already for simple problems despite being more than twice as expensive per iteration. Even though ASM introduces additional communication steps per smoother application, the reduced number of iterations can lead to improved parallel scalability for intermediate problem sizes. At the scaling limit, the results are inconclusive due to these two opposing effects.
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Song, Qi, Baoyuan Liu, Junfeng Wu, Wenli Zou, Yubin Wang, Bingbing Suo, and Yibo Lei. "GUGA-based MRCI approach with core-valence separation approximation (CVS) for the calculation of the core-excited states of molecules." Journal of Chemical Physics 160, no. 9 (March 5, 2024). http://dx.doi.org/10.1063/5.0189443.
Full textAbstract:
We develop and demonstrate how to use the Graphical Unitary Group Approach (GUGA)-based MRCISD with Core–Valence Separation (CVS) approximation to compute the core-excited states. First, perform a normal Self-Consistent-Field (SCF) or valence MCSCF calculation to optimize the molecular orbitals. Second, rotate the optimized target core orbitals and append to the active space, form an extended CVS active space, and perform a CVS-MCSCF calculation for core-excited states. Finally, construct the CVS-MRCISD expansion space and perform a CVS-MRCISD calculation to optimize the CI coefficients based on the variational method. The CVS approximation with GUGA-based methods can be implemented by flexible truncation of the Distinct Row Table. Eliminating the valence-excited configurations from the CVS-MRCISD expansion space can prevent variational collapse in the Davidson iteration diagonalization. The accuracy of the CVS-MRCISD scheme was investigated for excitation energies and compared with that of the CVS-MCSCF and CVS-CASPT2 methods using the same active space. The results show that CVS-MRCISD is capable of reproducing well-matched vertical core excitation energies that are consistent with experiments by combining large basis sets and a rational reference space. The calculation results also highlight the fact that the dynamic correlation between electrons makes an undeniable contribution in core-excited states.
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Wang, Jie, Jacopo Surace, Irénée Frérot, Benoît Legat, Marc-Olivier Renou, Victor Magron, and Antonio Acín. "Certifying Ground-State Properties of Many-Body Systems." Physical Review X 14, no. 3 (July 11, 2024). http://dx.doi.org/10.1103/physrevx.14.031006.
Full textAbstract:
A ubiquitous problem in quantum physics is to understand the ground-state properties of many-body systems. Confronted with the fact that exact diagonalization quickly becomes impossible when increasing the system size, variational approaches are typically employed as a scalable alternative: Energy is minimized over a subset of all possible states and then different physical quantities are computed over the solution state. Despite remarkable success, rigorously speaking, all that variational methods offer are upper bounds on the ground-state energy. On the other hand, so-called relaxations of the ground-state problem based on semidefinite programming represent a complementary approach, providing lower bounds to the ground-state energy. However, in their current implementation, neither variational nor relaxation methods offer provable bound on other observables in the ground state beyond the energy. In this work, we show that the combination of the two classes of approaches can be used to derive certifiable bounds on the value of any observable in the ground state, such as correlation functions of arbitrary order, structure factors, or order parameters. We illustrate the power of this approach in paradigmatic examples of 1D and 2D spin-1/2 Heisenberg models. To improve the scalability of the method, we exploit the symmetries and sparsity of the considered systems to reach sizes of hundreds of particles at much higher precision than previous works. Our analysis therefore shows how to obtain certifiable bounds on many-body ground-state properties beyond energy in a scalable way. Published by the American Physical Society 2024