Letteratura scientifica selezionata sul tema "Eigenvector Continuation"

Abstract Simulating quantum systems in a finite volume is a powerful theoretical tool to extract information about them. Real-world properties of the system are encoded in how its discrete energy levels change with the size of the volume. This approach is relevant not only for nuclear physics, where lattice methods for few- and many-nucleon states complement phenomenological shell-model descriptions and ab initio calculations of atomic nuclei based on harmonic oscillator expansions, but also for other fields such as simulations of cold atomic systems. This contribution presents recent progress concerning finite-volume simulations of few-body systems. In particular, it discusses details regarding the efficient numerical implementation of separable interactions and it presents eigenvector continuation as a method for performing robust and efficient volume extrapolations.

We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a notion of degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds.

This paper investigates a homotopy-based method for embedding with hundreds of thousands of data items that yields a parallel algorithm suitable for running on a distributed system. Current eigenvalue-based embedding algorithms attempt to use a sparsification of the distance matrix to approximate a low-dimensional representation when handling large-scale data sets. The main reason of taking approximation is that it is still hindered by the eigendecomposition bottleneck for high-dimensional matrices in the embedding process. In this study, a homotopy continuation algorithm is applied for improving this embedding model by parallelizing the corresponding eigendecomposition. The eigenvalue solution is converted to the operation of ordinary differential equations with initialized values, and all isolated positive eigenvalues and corresponding eigenvectors can be obtained in parallel according to predicting eigenpaths. Experiments on the real data sets show that the homotopy-based approach is potential to be implemented for millions of data sets.

Distributed energy resource (DER) has become an effective attempt in promoting use of renewable energy resources for electricity generation. The core intention of this study is to expand an approach for optimally placing several DER units to attain the most stable performance of the system and the greatest power losses decrease. The recommended technique is established on two analytical methods for analyzing voltage stability: the new modified modal analysis (MMA) and the continuation power flow (CPF) or MMA–CPF methods. The MMA evaluates voltage stability by considering incremental connection relating voltage and active power, which includes the eigenvalue and the related eigenvectors computed from the reduced modified Jacobian matrix. Furthermore, an active participation factor (APF) is computed from the eigenvectors of the reduced modified Jacobian matrix. The CPF method uses a predictor–corrector stepping pattern to reach the solution track and compute the tangent vector sensitivity (TVS). Both APF and TVS indicate each load bus sensitivity in the network. In addition, an objective function regarding losses decrease and eigenvalue is expressed to calculate the best bus position for DER allocation. The proposed MMA–CPF technique has been assessed on a 34-bus RDN and the outcomes demonstrate the effectiveness of the proposed scheme.

A new depth migration method derived in the space‐frequency domain is based on a generalized phase‐shift method for the downward continuation of surface data. For a laterally variable velocity structure, the Fourier spatial components are no longer eigenvectors of the wave equation, and therefore a rigorous application of the phase‐shift method would seem to require finding the eigenvectors by a matrix diagonalization at every depth step. However, a recently derived expansion technique enables phase‐shift accuracy to be obtained without resorting to a costly matrix diagonalization. The new technique is applied to the migration of zero‐offset time sections. As with the laterally uniform velocity case, the evanescent components of the solution need to be isolated and eliminated, in this case by the application of a spatially variant high‐cut filter. Tests performed on the new method show that it is more accurate and efficient than standard integration techniques such as the Runge‐Kutta method or the Taylor method.

The main result of this work is the estimation of the entropy mode accompanying a wave disturbance, observed at the atmosphere heights range of 90–120 km. The study is the direct continuation and development of recent results on diagnosis of the acoustic wave with the separation on direction of propagation. The estimation of the entropy mode contribution relies upon the measurements of the three dynamic variables (the temperature, density, and vertical velocity perturbations) of the neutral atmosphere measured by the method of the resonant scattering of radio waves on the artificial periodic irregularities of the ionospheric plasma. The measurement of the atmosphere dynamic parameters was carried out on the SURA heating facility. The mathematical foundation of the mode separation algorithm is based on the dynamic projection operators technique. The operators are constructed via the eigenvectors of the coordinate evolution operator of the transformed system of balance equations of the hydro-thermodynamics.

In this paper we derive explicit expressions for the Green’s functions in the case of an anisotropic elastic half-space and bimaterial subjected to a line force and a line dislocation. In contrast to previous studies in this area, our analysis includes the contributions of both anisotropic surface elasticity and surface van der Waals interaction forces. By means of the Stroh sextic formalism, analytical continuation and the state-space approach, the corresponding boundary value problem is reduced to a system of six (for a half-space) or 12 (for a bimaterial) coupled first-order differential equations. By employing the orthogonality relations among the corresponding eigenvectors, the coupled system of differential equations is further decoupled to six (for a half-space) or 12 (for a bimaterial) independent first-order differential equations. The latter is solved analytically using exponential integrals. In addition, we identify four and seven non-zero intrinsic material lengths for a half-space and a bimaterial, respectively, due entirely to the incorporation of the surface elasticity and surface van der Waal forces. We prove that these material lengths can be only either real and positive or complex conjugates with positive real parts.

Le noyau atomique, système quantique de nucléons en interaction, constitue un problème difficle à résoudre exactement. Pour contourner cette difficulté, des méthodes de résolution approchées on été introduites, comme la Projected Generator Coordinate Method (PGCM). La force de la PGCM est de construire un espace de faible dimension, motivé par des considérations physiques, dans lequel trouver un solution approchée est facile. Cependant, le coût numérique du calcul d'un espace PGCM rend cette méthode mal adaptée pour une étude statistique de sensibilité des observables nucléaires vis-à-vis des paramètres du modèle d'interaction, laquelle nécessite un grand nombre de calculs PGCM. Afin de rendre ce type d'études possibles, cette thèse explore la notion d'émulateur PGCM. Dans ce travail, une combinaison de PGCM avec la méthode Eigenvector Continuation (EC) est construite et étudiée. Cette combinaison (l'émulateur PGCM-EC) tire parti des ressemblances formelles entre PGCM et EC, et surtout de la possibilité de décomposer l'hamiltonien comme combinaison linéaire de termes indépendants des paramètres du modèle d'interaction. Cette dernière propriété permet de concentrer la plus grande partie du coût numérique sur le calcul de quantités indépendantes des paramètres de l'interaction (les kernels élémentaires), et ainsi rend possible l'émulation massive de calculs PGCM, au prix d'avoir en amont effectué le calcul très lourd des kernels élémentaires. Les limites de cet émulateur sont aussi étudiées, en introduisant notamment la notion de sur-entraînement, qui provient précisément du fait que la PGCM est une méthode non-exacte de résolution du problème à N-corps nucléaire. Cette thèse démontre au final qu'il est possible d'émuler des millions de calculs PGCM avec une erreur ne dépassant pas 3% sur la spectroscopie collective des noyaux, et avec un faible coût numérique représentant une fraction de 1% du coût des millions de calculs PGCM
An atomic nucleus is a quantum system of interacting nucleons and constitutes a problem difficult to solve exactly. For this reason, a diversity of approximate resolution methods has been designed, and Projected Generator Coordinate Method (PGCM) is one of them. The strong point of PGCM is to construct a physically inspired small dimensional space, in which an approximate solution of the nuclear many-body problem is easily found. However the numerical cost of PGCM space computation make this method inadapted for sensibility analysis of nuclear observables with restect to parametrisation of the interaction model, this analysis requiring an huge number of PGCM computations. In order to make this type of study possible, this thesis explore the concept of PGCM emulator. In this work, a combination of PGCM with Eigenvector Continuation (EC) is constructed and studied. This combination (the PGCM-EC emulator) takes advantage of mathematical similarities between PGCM and EC, and above all of the decomposition of the hamiltonian as a linear combination of parameter-independent terms. The latter property is used to concentrate the heavier numerical cost in the computation of parameter-independent quantities (the elementary kernels), and open the feasability of massive PGCM emulations, the price being having first-handedly computed the costly elementary kernels. Limits of the emulator are also explored, by introducing the concept of over-training, which is exactly a consequence of the aproximativeness of a PGCM computation. Eventually this thesis demonstrates the possibility to emulate millions of PGCM computations with an error on collective spectroscopy less than 3%, and with a low numerical cost fraction of 1% of the million PGCM calculations cost

For symmetric (hermitian) (dense or banded) matrices the computation of eigenvalues and eigenvectors Ax = λBx is an important task, e.g. in electronic structure calculations. If a larger number of eigenvectors are needed, often direct solvers are applied. On parallel architectures the ELPA implementation has proven to be very efficient, also compared to other parallel solvers like EigenExa or MAGMA. The main improvement that allows better parallel efficiency in ELPA is the two-step transformation of dense to band to tridiagonal form. This was the achievement of the ELPA project. The continuation of this project has been targeting at additional improvements like allowing monitoring and autotuning of the ELPA code, optimizing the code for different architectures, developing curtailed algorithms for banded A and B, and applying the improved code to solve typical examples in electronic structure calculations. In this paper we will present the outcome of this project.

Abstract This paper is a continuation of previous papers by the authors that focuses on predicting the mistuned embedded rotor blade forced response. The compressor under consideration is a part of a 3.5stage rig located at Purdue University. Previously, the current authors have discussed the impact of sideband travelling wave forcing functions on the mistuned response and pinned down the reason for a constant underprediction in the amplification factor. This prompted further research to determine the sensitivity of the response to a known change in the system mode. In the first section of the current paper, the authors perturb the system modes frequencies in a probabilistic manner and compute the influence of the system mode on the mistuning amplification factor. The second part of this study involves determining the impact of a perturbation in the structural damping on the mistuned response. The study is also extended to an aerodynamic dynamic analysis wherein the impact of a perturbation in system mode frequency and structural damping is determined both individually and as a combined influence. Finally, a brief investigation of system eigenvalues and eigenvectors is conducted to understand the impact of mistuning on aerodynamic damping suppression. This analysis is joined with a broad examination of the relationship between damping and the extent of mistuning. The key conclusions from this paper are: 1) The mistuned forced response was highly sensitive to the system mode input, i.e., although the input was probabilistic, the output was deterministic. 2) Since the aerodynamic damping dominates in the case study, a change in the structural damping parameter has minimal effect on the mistuned response, particularly the mistuning amplification factor. 3) The results of the flutter analysis show that a perturbation in the system mode frequency stabilizes the system much faster than a perturbation in the structural damping, and the latter has minimal influence on the eigenvalues of the solution. 4) Under the application of mistuning, compression of the damping components of the complex eigenvalues is dominant over the expansion of the frequency components and is amplified when the structural coupling is not considered. 5) In cases of both random and near alternate blade mistuning, the relationship between mistuning and damping is positive up to a point, after which further increasing mistuning begins to reduce system damping.