Inverse Iteration Technique Research Articles

Quantum inverse iteration algorithm for programmable quantum simulators

We propose a quantum inverse iteration algorithm, which can be used to estimate ground state properties of a programmable quantum device. The method relies on the inverse power iteration technique, where the sequential application of the Hamiltonian inverse to an initial state prepares the approximate ground state. To apply the inverse Hamiltonian operation, we write it as a sum of unitary evolution operators using the Fourier approximation approach. This allows to reformulate the protocol as separate measurements for the overlap of initial and propagated wavefunction. The algorithm thus crucially depends on the ability to run Hamiltonian dynamics with an available quantum device, and can be used for analog quantum simulators. We benchmark the performance using paradigmatic examples of quantum chemistry, corresponding to molecular hydrogen and beryllium hydride. Finally, we show its use for studying the ground state properties of relevant material science models, which can be simulated with existing devices, considering an example of the Bose-Hubbard atomic simulator.

Mathematical and Numerical Analysis of the Generalized Complex-Frequency Eigenvalue Problem for Two-Dimensional Optical Microcavities

The current paper proposes a parametric eigenvalue problem for the Helmholtz equation on the plane based on two well-known physical models of emission from two-dimensional (2-D) microcavity lasers. The first model has been developed for passive, either lossy or lossless, open cavities and is usually referred to as the Complex-Frequency Eigenvalue Problem. The second model, named the Lasing Eigenvalue Problem (LEP), has been tailored to characterize emission from open cavities, filled in with gain material, on the threshold of nonattenuating in time radiation. Our generalized model contains both of them as special cases. We reduce the original problem to a nonlinear eigenvalue problem for a set of boundary integral equations with weakly singular kernels and formulate it as a parametric eigenvalue problem for a holomorphic Fredholm operator-valued function, which is convenient for mathematical and numerical analysis. Using this formulation and fundamental results of the theory of holomorphic operator-valued functions, we study the properties of the spectrum. For numerical solution of the problem, we propose a Nyström method, prove its convergence, and derive error estimates in the eigenvalue approximation. Combining this discretization with the residual inverse iteration technique, we compute approximate solutions of LEP and compare them with the exact ones and with the results obtained using other numerical methods.

Vibration Analysis of Radial Drilling Machine Structure Using Finite Element Method

Manufacturing industries now a days have stringent expectation from the machine tools in terms of productivity as well as quality of products.Vibration plays an important role in determining the quality of product.If the pattern of vibration prevailing in the machine tool during cutting is known,then machine tool structure can be designed in such a way so that natural frequency of machine tool structure can be isolated from the forced frequency.So, this study is focused on finding the natural frequency and mode shapes of radial drilling machine structure.Finite element analysis has been done to find out the natural frequencies and mode shapes of radial drilling machine structure.Assembled mass and stiffness matrices are obtained for each element and solved by using inverse iteration technique.

Hopping conductivity in nucleotide base stacks*

AbstractThe frequency‐dependent complex hopping conductivities of single nucleotide base stacks and base pair stacks were calculated on the following basis: (1) The determination of the density of states of these disordered systems with the help of the matrix block negative counting method. (2) Using the inverse iteration technique, the Anderson localization of the one‐electron functions belonging to the physically interesting levels were computed. (3) With the help of the latter quantities the hopping frequencies corresponding to the relative motion of the base pairs with respect to each other (acoustic phonons) were determined. (4) Finally, using a random‐walk theory the frequency‐dependent complex conductivities [σ(ω)] were computed. The |σ(ω)|−ω curves show a saturation at ω≈1010 s−1. The |σ(ω)|'s have the values in the base pairs case (taking into account the role of the sugar–phosphate backbone, basis set, and correlation effects) are a few times 1 Ω−1 cm−1 for ω>1010 s−1. This room temperature result is in excellent agreement with the energy loss (in a resonant cavity) experiment on lambda phage deoxyribonucleic acid (DNA) of Gruner and co‐workers. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001

The calculation of critical buckling loads for externally constrained structures

A method of identifying the critical load of structures, the buckling shape of which is constrained, is presented based on two novel techniques. The first identifies this critical or limit point buckling value using a linear, complementary algorithm to solve what is shown to be a quadratic programming problem, while the second adopts a generalized inverse iteration technique. The critical load of a circular ring confined by a rigid circular boundary and subject to thermal expansion is then obtained using this technique, and it is found that it is lower than that previously obtained from a traditional approach based on energy considerations. Copyright © 1999 John Wiley & Sons, Ltd.

A practical algorithm for the efficient computation of eigenvector sensitivities

Derivatives of eigenvalues and eigenvectors have become increasingly important in the development of modern numerical methods for areas such as structural design optimization, dynamic system identification and dynamic control, and the development of effective and efficient methods for the calculation of such derivatives has remained to be an active research area for several decades. In this paper, a practical algorithm has been developed for efficiently computing eigenvector derivatives of generalized symmetric eigenvalue problems. For eigenvector derivative of a separate mode, the computation only requires the knowledge of eigenvalue and eigenvector of the mode itself and an inverse of system matrix accounts for most computation cost involved. In the case of two close modes, the modal information of both modes is required and the eigenvector derivatives can be accurately determined simultaneously at minor additional computational cost. Further, the proposed method has been extended to the case of practical structural design where structural modifications are made locally and the eigenderivatives of the modes concerned before are still of interest. By combining the proposed algorithm together with the proposed inverse iteration technique and singular value decomposition theory, eigenproperties and their derivatives can be very efficiently computed. Numerical results from a practical finite element model have demonstrated the practicality of the proposed method. The proposed method can be easily incorporated into commercial finite element packages to improve the computational efficiency of eigenderivatives needed for practical applications.

Electronic structure of polyparaphenylene vinylene copolymers: The relationship to light-emitting characteristics

We investigate theoretically the electronic structure of regular and statistical copolymer chains containing segments of poly(p-phenylene vinylene) (PPV) and its dimethoxy-substituted derivative, poly(2,5-dimethoxy-1,4-phenylene vinylene) (DMeOPPV). Such copolymers have recently been shown to present remarkable electroluminescence properties. The calculations on the copolymer chains are performed on the basis of the negative factor counting technique coupled to the valence effective Hamiltonian method. The eigenfunctions of the upper occupied and lower unoccupied levels are calculated explicitly using the inverse iteration technique, and the degree of localization is determined via the inverse participation number approach. It is shown that the electronic structure of PPV/DMeOPPV copolymers is strongly affected by both the concentrations and relative locations of the parent polymer moieties along the chains. The appearance of clusters of dimethoxy-substituted units leads to energy gaps that are smaller than those of the parent polymers and to a spatial localization of the upper occupied levels within the DMeOPPV clusters, while the lower unoccupied levels remain smoothly delocalized over the unsubstituted regions. Our results thus indicate that electron–hole recombination processes should take place preferentially at the interfaces between PPV and DMeOPPV segments. The evolution of the PPV/DMeOPPV copolymer band gaps as a function of the concentration in dimethoxy substituted rings is compared to experimental photoluminescence and absorption data.

Sensitivity based method for structural dynamic model improvement

Sensitivity analysis, the study of how a structure's dynamic characteristics change with design variables, has been used to predict structural modification effects in design for many decades. In this paper, methods for calculating the eigensensitivity, frequency response function sensitivity and its modified new formulation are presented. The implementation of these sensitivity analyses to the practice of finite element model improvement using vibration test data, which is one of the major applications of experimental modal testing, is discussed. Since it is very difficult in practice to measure all the coordinates which are specified in the finite element model, sensitivity based methods become essential and are, in fact, the only appropriate methods of tackling the problem of finite element model improvement. Comparisons of these methods are made in terms of the amount of measured data required, the speed of convergence and the magnitudes of modelling errors. Also, it is identified that the inverse iteration technique can be effectively used to minimize the computational costs involved. The finite element model of a plane truss structure is used in numerical case studies to demonstrate the effectiveness of the applications of these sensitivity based methods to practical engineering structures.

Influence of polymerization irregularities on the electronic properties of polythiophene

The results of quantum chemical calculations on the electronic structure of chains of poly(2,5-thienylene), i.e., polythiophene, along which some of the α–α′(2–5′) linkages are replaced randomly by α–β′(2–4′) linkages are presented. The randomness in the α–β′ linkage positions is taken into account via the negative factor counting technique, on the basis of Fock matrices calculated by the valence effective Hamiltonian method. The eigenfunctions of the upper occupied levels are explicitly calculated using the inverse iteration technique and the degree of localization is determined via the inverse participation number approach. It is shown that a random distribution in the α–β′ linkages leads to a localization of the upper occupied levels within segments containing exclusively α–α′ linkages; more extended levels start to appear some 0.3 eV below the HOMO level for α–β′ linkage concentrations of about 6%. On the contrary, when the α–β′ linkages occur regularly, delocalization remains effective even for the upper occupied levels.

On the electronic structure and conduction properties of aperiodic proteins: Study of six-component polypeptide chains

The results of the electronic density of states (DOS) of six-component periodic- and aperiodic peptide chains calculated on the basis of ab initio matrix block negative factor counting technique are presented. The six amino acid residues are glycine, serine, cysteine, asparagine, histidine and aspartic acid. In contrast to the periodic chains, aperiodic chains are found to have broader regions of allowed energy states. Large values of the fundamental band gap obtained for these systems rule out the possibility of intrinsic conduction in them. The study of the localization properties of the lowest unoccupied energy levels in the conduction band region of aperiodic chains on the basis of inverse iteration technique indicates these wavefunctions to be highly localized. In the light of these results, the possibility of charge transport through hopping conduction in proteins is discussed under the assumption of charge transfer from DNA to the polypeptide chains.

On the electronic structure and conduction properties of aperiodic DNA and proteins. I. Strategy and methods of investigations

The possibilites for applying the ab initio matrix block negative factor counting technique (NFC) for non-periodic nucleotide base stacks and for 4- and 5-component aperiodic model proteins are described. It is outlined how also correlation effects could be taken into account in an approximate way. After a brief summary of the basic equations of the NFC method the inverse iteration technique to study the Anderson localization of the wavefunctions belonging to the different energy levels is shortly reviewed. Also the calculation of hopping probabilities between different localization sites is sketched. Finally, the possibilites are outlined for the experiment verification of hopping conduction in proteins and (in a minor extent) in DNA, assuming the generation of free charge carriers in them through charge transfer.

Whirl Speeds and Unbalance Response of Multibearing Rotors Using Finite Elements

This paper presents dynamic modeling of rotor bearing systems with rigid disks, distributed parameter finite rotor elements and flexible, discrete multibearings. The previous works have included the effects of rotary inertia, gyroscopic moment, axial load, internal viscous and hysteretic damping and transverse shear deformations, but have not considered them all in the same model. A computer program is developed in this work to calculate the forward and backward whirl speeds, the corresponding mode shapes, the dynamic unbalance response of multibearing rotor systems and to evaluate rotor stability. It utilizes the banded property of the system matrices to reduce the computational effort for the complex eigensolution. A combined bisection and inverse iteration technique is used for the complex eigen-value problem. The combined effect of shear deformations and internal damping on the stability, forward and backward whirl speeds, as well as on the response to unbalance excitation, are investigated with the numerical examples given. This study confirms that any inherent material damping in the shaft itself does not affect the unbalance response if the bearings are isotropic, which is, however, not the case when the bearings are orthotropic causing elliptical synchronous whirl orbits.

On a numerical solution of the supersonic panel flutter eigenproblem

AbstractAn automated digital computer procedure is presented in this paper which enables efficient solution of the eigenvalue problem associated with the supersonic panel flutter phenomena. The step‐by‐step incremental solution procedure is based on an inverse iteration technique which effectively utilizes solution results from the previous step in determining such results during the current solution step. Also, the computations are limited to the determination of a few specific roots only, which are expected to contain the flutter mode, and this is achieved at each step without having to compute any other root.The structural discretization achieved by the finite element method yields highly banded stiffness, mass, and aerodynamic matrices; the aerodynamic matrix evaluated by the linearized piston theory is real but unsymmetric in nature. The solution algorithm presented in this paper fully exploits the banded form of the associated matrices, and the resulting computer program written in FORTRAN V for the JPL UNIVAC 1108 computer proves to be most efficient and economical when compared to existing procedures of such analysis. Numerical results are presented for a two‐dimensional panel flutter problem.

Author's Reply. On a combined sturm sequence and inverse iteration technique for eigenproblem solution of spinning structures

Author's Reply. On a combined sturm sequence and inverse iteration technique for eigenproblem solution of spinning structures

Eigenproblem solution of damped structural systems

AbstractThis article presents an efficient numerical algorithm a complete listing of the associated computer program, developed for the eigenproblem solution of discrete damped structures, including spinning ones. The numerically stable procedure is based on a combined Sturm sequence and inverse iteration technique, which fully exploits the banded form of the relevant matrices and proves to be most substantially economical when compared to similar existing softwares. Numerical results are presented for representative structures, solved by the present computer program written in FORTRAN V for the JPL UNIVAC 1108 computer.

On a combined sturm sequence and inverse iteration technique for eigenproblem solution of spinning structures

AbstractThis paper describes an efficient numerical algorithm for the accurate computation of specific eigenvalues and related vectors of spinning structures. The Sturm sequence technique is first applied to isolate the required roots, which are then individually located by an inverse iteration technique adopting an efficient matrix formulation specially developed for this purpose. The associated digital computer procedure is numerically stable, which also fully exploits the banded form of relevant matrices. Furthermore, the algorithm proves to be much faster than other related existing procedures.A computer program, based on the current algorithm, has been developed in FORTRAN V for the JPL UNIVAC 1108 computer. Numerical results for representative structures, computed by the program, are also presented in detail. Extensive applications of the program are envisaged in the attitude control of spacecraft and in the natural frequency analysis of spinning structures, discretized by the finite element method.

Eigenproblem solution by a combined Sturm sequence and inverse iteration technique

AbstractThis article presents an efficient and numerically stable algorithm, along with a complete listing of the associated computer program, developed for the accurate computation of specified roots and associated vectors of the eigenvalue problem Aq = λBq with band symmetric A and B, B being also positive definite. The desired roots are first isolated by the Sturm sequence procedure; then a special variant of the inverse iteration technique is applied for the individual determination of each root along with its vector. The algorithm fully exploits the banded form of relevant matrices, and the associated program written in FORTRAN V for the JPL UNIVAC 1108 computer proves to be most significantly economical in comparison to similar existing procedures.The program may be conveniently utilized for the efficient solution of practical engineering problems including free vibration and buckling analysis of structures. Results of such analyses are presented for representative structures.