Second-order Perturbation Approximation Research Articles

Stochastic perturbation method for optimal control problem governed by parabolic PDEs with small uncertainties

In this paper, we investigate the first-order and second-order perturbation approximation schemes for an optimal control problem governed by parabolic PDEs with small uncertainties. First, we propose the finite dimensional noise assumption for the random coefficient, and expand the state and co-state variables up to a certain order with respect to a small parameter by using the perturbation technique. Then, we insert all the expansions into the known deterministic parametric optimality system, and obtain the first-order and second-order optimality systems. These two systems are discretized by the finite element method in space and the backward Euler method in time to establish the first-order and second-order approximate schemes. Further, a priori error estimates are derived for the state, co-state and control variables of the first-order and second-order approximation, respectively. Finally, some numerical experiments are presented to verify the theoretical results.

Gedanken experiments at high-order approximations: Kerr-Newman black hole cannot be overcharged and overspun

The weak cosmic censorship conjecture (WCCC) of black holes is a fundamental hypothesis in general relativity. The proof of this hypothesis has become a frontier issue in the study of gravity. Recently, Sorce and Wald proposed a new version of the gedanken experiment and examined that the WCCC of Kerr-Newman (KN) black holes is valid under the second-order approximation of the matter fields perturbation. But it is not enough to study the WCCC under the second-order approximation because there is an additional condition that makes the validity of the hypothesis impossible to judge in this case. Therefore, the higher-order approximation of the perturbation process should be further considered to examine the conjecture strictly. Based on the new version of the gedanken experiment, from the two assumptions that the spacetime geometry satisfies the stability condition and that the matter fields should obey the null energy condition, we deduce the $n$th-order perturbation inequality and demonstrate that the event horizon of KN black holes always exists under the $n$th-order approximation of the matter fields perturbation. It indicates that the hypothesis is always satisfied under the arbitrary high-order approximation and also shows that the WCCC of KN black holes has been examined rigorously at the perturbation level.

Testing the weak cosmic censorship conjecture in the massive gravitational theories

In this paper, based on the new version of the gedanken experiments proposed by Sorce and Wald, we examine the weak cosmic censorship in the process of accreting matter fields for nearly extremal static charged black holes in the massive gravitational theories coupled to a Maxwell field. In the investigation, we assume that the black hole is perturbed by some matter fields satisfied null energy condition and ultimately settles down to a static state in the asymptotic future. Moreover, we also treat the cosmological constant, the graviton mass, and the coupling constants as variables, and they can be affected by the physical process. Then, after applying the Noether charge method, we derive the first-order and second-order perturbation inequalities of the perturbation matter fields. As a result, we find that the nearly extremal black hole cannot be destroyed under the second-order approximation of perturbation. This result implies that the weak cosmic censorship conjecture is satisfied in the gravitational theories with massive graviton.

Space-Time Modulation in Lossy Dispersive Media and the Implications for Amplification and Shielding

We investigate the dispersion, attenuation, amplification, and the area of solutions of electromagnetic (EM) waves in a lossy progressively disturbed medium. Approximate, rigorous, and numerical methods are fully developed for a lossy environment, and the differences, merits, and drawbacks are discussed. A new representation of the Floquet theorem accounts for the loss exposed to both the pumped wave and the signal wave. The second-order small perturbation approximation is employed to yield closed-form solutions for transverse EM waves’ dispersion relation. It is proven to be valid in small-perturbed lossy space–time-modulated (STM) media with nonsuperluminal modulation. Analyzing the dispersion relation, an elaborated sufficiency condition is proposed. Moreover, the nonunique solutions and abnormal effects created by the loss factor are analyzed thoroughly. The special harmonic amplification and shielding properties experienced in a lossy STM media are brought to attention throughout the process. The developed approximate and rigorous analytical results are finally compared to finite-difference time-domain (FDTD) simulations in a realistic test, where a signal is going through upconversion/downconversion in an STM medium. Finally, a set of useful conclusions, implications, and applications has been raised to give more insight into how amplification and shielding might be affected in a lossy STM environment.

The topological RN-AdS black holes cannot be overcharged by the new version of gedanken experiment

In this paper, we test the weak cosmic censorship conjecture (WCCC) for n≥4 dimensional nearly extremal RN-AdS black holes with non-trivial topologies, namely plane and hyperbola, using the new version of gedanken experiment proposed by Sorce and Wald. Provided that the non-electromagnetic part of the stress tensor of matter fields satisfies the null energy condition and the linear stability condition holds, we find that the black holes cannot be overcharged under the second-order perturbation approximation, which includes the self-force and finite-size effects. As a result, we conclude that the violation of Hubeny type never occurs and the WCCC holds for the topological RN-AdS black hole.

New Gedanken experiment on RN-AdS black holes surrounded by quintessence

In this paper, we use the new version of Gedanken experiment to investigate the weak cosmic censorship conjecture(WCCC) for RN-AdS black holes surrounded by quintessence. The process of matter fields falling into the black hole can be regarded as a dynamic process. Since the perturbation of matter fields doesn’t affect the spacetime geometry, we propose the stability condition and assume the process of matter fields falling into the black hole satisfies the null energy condition. Based on the stability condition and the null energy condition, the first-order and second-order perturbation inequalities are derived. As a result, we show that the WCCC for RN-AdS black holes surrounded by quintessence cannot be violated under the second-order approximation of matter fields perturbation.

Gedanken experiments at high-order approximation: Kerr black hole cannot be overspun

Sorce and Wald proposed a new version of gedanken experiments to examine the weak cosmic censorship conjecture (WCCC) in Kerr-Newmann black holes. However, their discussion only includes the second-order approximation of perturbation and there exists an optimal condition such that the validity of the WCCC is determined by the higher-order approximations. Therefore, in this paper, we extended their discussions into the high-order approximations to study the WCCC in a nearly extremal Kerr black hole. After assuming that the spacetime satisfies the stability condition and the perturbation matter fields satisfy the null energy condition, based on the Noether charge method by Iyer and Wald, we completely calculate the first four order perturbation inequalities and discuss the corresponding gedanken experiment to overspin the Kerr black hole. As a result, we find that the nearly extremal Kerr black holes cannot be destroyed under the fourth-order approximation of perturbation. Then, by using the mathematical induction, we strictly prove the nth order perturbation inequality when the first (n − 1) order perturbation inequalities are saturated. Using these results, we discuss the first 100 order approximation of the gedanken experiments and find that the WCCC in Kerr black hole is valid under the higher-order approximation of perturbation. Our investigation implies that the WCCC might be strictly satisfied in Kerr black holes under the perturbation level.

Constructing shortcuts to the adiabatic passage for implementation of nongeomeric phase gates in a two-atom system

Through combining ‘Lewis–Riesenfeld invariants’ and ‘transitionless quantum driving’ with ‘quantum zeno subspace’ and ‘second-order perturbation approximation’, respectively, we put forward two different methods to construct shortcuts via nonadiabatic processes to present robust quantum phase gates in a two-atom system. The influence of various decoherence processes such as spontaneous emission and photon loss on the fidelity is discussed. The result shows that this scheme is insensitive to both the two error sources. Moreover, the interaction time of the scheme is very short and precise control is not required. Additionally, this scheme is not only implemented without requiring extra complex conditions, but is also insensitive to variations of the parameters. Therefore, this scheme may provide a new perspective for quantum computation.

New version of the gedanken experiments to test the weak cosmic censorship in charged dilaton-Lifshitz black holes

In this paper, based on the new version of the gedanken experiments proposed by Sorce and Wald, we examine the weak cosmic censorship in the perturbation process of accreting matter fields for the charged dilaton-Lifshitz black holes. In the investigation, we assume that the black hole is perturbed by some extra matter source satisfied the null energy condition and ultimately settle down to a static charged dilaton-Lifshitz black hole in the asymptotic future. Then, after applying the Noether charge method, we derive the first-order and second-order perturbation inequalities of the perturbation matter fields. As a result, we find that the nearly extremal charged dilaton-Lifshitz black hole cannot be destroyed under the second-order approximation of perturbation. This result implies that the weak cosmic censorship conjecture might be a general feature of the Einstein gravity, and it is independent of the asymptotic behaviors of the black holes.

Time-dependent optimized coupled-cluster method for multielectron dynamics. III. A second-order many-body perturbation approximation.

We report successful implementation of the time-dependent second-order many-body perturbation theory using optimized orthonormal orbital functions called time-dependent optimized second-order many-body perturbation theory to reach out to relatively larger chemical systems for the study of intense-laser-driven multielectron dynamics. We apply this method to strong-field ionization and high-order harmonic generation of Ar. The calculation results are benchmarked against ab initio time-dependent complete-active-space self-consistent field, time-dependent optimized coupled-cluster double, and time-dependent Hartree-Fock methods, as well as a single active electron model to explore the role of electron correlation.

Examining the weak cosmic censorship conjecture of RN-AdS black holes via the new version of the gedanken experiment

Based on the new version of the gedanken experiment proposed by Sorce and Wald, we investigate the weak cosmic censorship conjecture (WCCC) for Reissner-Nordström-AdS (RN-AdS) black holes under the second-order approximation of the perturbation that comes from the matter fields. First of all, we propose that the cosmological constant is a portion of the matter fields. It means that the cosmological constant can be seen as a variable, and its value will vary with the process of the perturbation. Moreover, we assume that the perturbation satisfies the stability condition, which states that after the perturbation, the spacetime geometry also belongs to the class of RN-AdS solutions. Based on the stability condition and the null energy condition, while using the method of the off-shell variation, the first two order perturbation inequalities are derived. In the two inequalities, the term which contains the thermodynamic pressure and volume is involved firstly. Furthermore, based on the first-order optimal option and the second-order inequality, the WCCC for nearly extremal RN-AdS black holes is examined. It is shown that considering the cosmological constant varying with the perturbation, the WCCC for RN-AdS black holes cannot be violated under the second-order approximation of the matter fields perturbation.

Gedanken experiments at high-order approximation: nearly extremal Reissner-Nordstr\xf6m black holes cannot be overcharged

The new version of the gedanken experiment proposed by Sorce and Wald has been used to examine the weak cosmic censorship conjecture (WCCC) for black holes at the second-order approximation of the matter fields perturbation. However, only considering the perturbation until the second-order approximation is incomplete because there is an optimal option such that the existing condition of the event horizon vanishes at second- order. For this circumstance, we cannot judge whether the WCCC is satisfied at this order. In our investigation, the kth-order perturbation inequality is generally derived. Using the inequalities, we examine the WCCC for nearly extremal Reissner-Nordstöm black holes at higher-order approximation. It is shown that the WCCC cannot be violated yet after the perturbation. From this result, it can be indicated that the WCCC is strictly satisfied at the perturbation level for nearly extremal RN black holes.

Response of second-harmonic generation of Lamb wave propagation to microdamage thickness in a solid plate

The response features of second-harmonic generation (SHG) of primary Lamb wave propagation to the thickness of microdamage layer (MDL) in a solid plate have been theoretically and numerically investigated in this paper. Here the solid plate with a MDL is regarded as a double-layered plate in analysis of nonlinear Lamb wave propagation. On the basis of a second-order perturbation approximation and modal expansion analysis, the physical process of cumulative SHG by primary Lamb wave propagation in a solid plate with a MDL has been investigated. The influence of variation in the thickness of MDL on the effect of SHG of primary S0 mode, which satisfies an approximate phase velocity matching in the low frequency region, has been theoretically analyzed, and then the finite element (FE) simulation has been carried out to validate the results of the theoretical predictions. A close agreement between the theoretical analyses and FE simulations validates the effectiveness of using the effect of SHG of primary S0 mode for characterizing changes in the thickness of MDL. Moreover, change mechanism of nonlinear acoustic parameter with the thickness of MDL is revealed It is expected that the results obtained can provide a convenient means for accurately characterizing nonhomogeneous microdamage (MDL thickness) in layered plates.

Correlation effects beyond coupled cluster singles and doubles approximation through Fock matrix dressing.

We present an accurate single reference coupled cluster theory in which the conventional Fock operator matrix is suitably dressed to simulate the effect of triple and higher excitations within a singles and doubles framework. The dressing thus invoked originates from a second-order perturbative approximation of a similarity transformed Hamiltonian and induces higher rank excitations through local renormalization of individual occupied and unoccupied orbital lines. Such a dressing is able to recover a significant amount of correlation effects beyond singles and doubles approximation, but only with an economic n5 additional cost. Due to the inclusion of higher rank excitations via the Fock matrix dressing, this method is a natural improvement over conventional coupled cluster theory with singles and doubles approximation, and this method would be demonstrated via applications on some challenging systems. This highly promising scheme has a conceptually simple structure which is also easily generalizable to a multi-reference coupled cluster scheme for treating strong degeneracy. We shall demonstrate that this method is a natural lowest order perturbative approximation to the recently developed iterative n-body excitation inclusive coupled cluster singles and doubles scheme [R. Maitra et al., J. Chem. Phys. 147, 074103 (2017)].

Eigenvalue sensitivity and veering in gyroscopic systems with application to high-speed planetary gears

Abstract This study investigates eigenvalue sensitivity to model parameters and veering in gyroscopic systems. The eigenvalue perturbation approach is formulated such that the results apply to discrete, continuous, and hybrid discrete-continuous gyroscopic systems. Third-order perturbation approximations for distinct eigenvalues are determined. Perturbations through second-order are derived for degenerate eigenvalues. The perturbation results are applied to a high-speed planetary gear model, where the results are shown to be accurate over a wide range of rotation speeds. The sensitivity of the eigenvalues to model parameters are written in terms of modal kinetic and potential energies. From the second-order perturbation approximation an eigenvalue veering parameter is defined and used to analyze veering in high-speed planetary gears, which is prominent in planetary gears that have disrupted cyclic symmetry.

Influence of Change in Inner Layer Thickness of Composite Circular Tube on Second-Harmonic Generation by Primary Circumferential Ultrasonic Guided Wave Propagation**Supported by the National Natural Science Foundation of China under Grant Nos 11474361, 11474093 and 11274388.

The influence of change in inner layer thickness of a composite circular tube is investigated on second-harmonic generation (SHG) by primary circumferential ultrasonic guided wave (CUGW) propagation. Within a second-order perturbation approximation, the nonlinear effect of primary CUGW propagation is treated as a second-order perturbation to its linear response. It is found that change in inner layer thickness of the composite circular tube will influence the efficiency of SHG by primary CUGW propagation in several aspects. In particular, with change in inner layer thickness, the phase velocity matching condition that is originally satisfied for the primary and double-frequency CUGW mode pair selected may no longer be satisfied. This will remarkably influence the efficiency of SHG by primary CUGW propagation. Theoretical analyses and numerical results show that the effect of SHG by primary CUGW propagation is very sensitive to change in inner layer thickness, and it can be used to accurately monitor a minor change in inner layer thickness of the composite circular tube.

Improved virtual orbitals in state specific multireference perturbation theory for prototypes of quasidegenerate electronic structure.

The state-specific multireference perturbation theory (SSMRPT) with an improved virtual orbital complete active space configuration interaction (IVO-CASCI) reference function [called as IVO-SSMRPT] is used to investigate the energy surface, geometrical parameters, molecular properties of spectroscopic interest for the systems/situations [such as BeH2, BeCH2, MgCH2, Si2H4, unimolecular dissociation of H2CO, and intramolecular reaction pathways of 1,3-butadiene] where the effect of quasidegeneracy cannot be neglected. The merit of using the IVO-CASCI rather than complete active space self-consistent field (CASSCF) is that it is free from iterations beyond those in the initial SCF calculation and the convergence difficulties that plague CASSCF calculations with increasing size of the CAS. While IVO-CASCI describes the non-dynamical correlation, the SSMRPT scheme is a good second-order perturbative approximation to account for the rest of the correlation energy. Our IVO-SSMRPT method is instrumental in avoiding intruder states in an size-extensive manner and allows the revision of the content of wave function in the model space. It can treat model as well as real systems with predictive accuracy, as is evident from the fairly nice accordance between our estimates, and high-level theoretical results. Our estimates also corroborate well with some experimental findings.

A simulation comparison of quantile approximation techniques for compound distributions popular in operational risk

Many banks currently use the loss distribution approach (LDA) for estimating economic and regulatory capital for operational risk under Basel's advanced measurement approach. The LDA requires the modeling of the aggregate loss distribution in each operational risk category (ORC), among others. The aggregate loss distribution is a compound distribution resulting from a random sum of losses, where the losses are distributed according to some severity distribution, and the number (of losses) is distributed according to some frequency distribution. In order to estimate the economic or regulatory capital in a particular ORC, an extreme quantile of the aggregate loss distribution has to be estimated from the fitted severity and frequency distributions. Since a closed-form expression for the quantiles of the resulting estimated compound distribution does not exist, the quantile is usually approximated using a brute force Monte Carlo simulation, which is computationally intensive. However, a number of numerical approximation techniques have been proposed to lessen the computational burden. Such techniques include Panjer recursion, the fast Fourier transform and different orders of both the single-loss approximation and perturbative approximation. The objective of this paper is to compare these methods in terms of their practical usefulness and potential applicability in an operational risk context. We find that the second-order perturbative approximation, a closed-form approximation, performs very well at the extreme quantiles and over a wide range of distributions, and it is very easy to implement. This approximation can then be used as an input to the recursive fast Fourier algorithm to gain further improvements at the less extreme quantiles.

Modal expansion analysis of nonlinear circumferential guided wave propagation in a circular tube

Within the second-order perturbation approximation, the nonlinear effect of primary circumferential guided wave propagation in a circular tube is investigated using modal expansion analysis for waveguide excitation. The nonlinearity of the wave equation governing the wave propagation ensures the second-harmonic generation accompanying primary circumferential guided wave propagation. This nonlinearity may be treated as a second-order perturbation of the linear elastic response. The fields of the second harmonics of primary circumferential guided wave propagation are considered as superpositions of the fields of a series of double frequency circumferential guided wave (DFCGW) modes. Based on the momentum theorem and mathematical formulae of nonlinear stress tensor and its divergence under the cylindrical coordinate system, the mathematical expressions of the corresponding double frequency traction stress tensors and bulk driving forces are deduced for a certain primary circumferential guided wave mode. Subsequently, the equation governing the DFCGW mode expansion coefficient is established. Finally, the mathematical expression of second-harmonic field of the primary circumferential guided wave mode in a tube is derived. The results of the theoretical analyses and numerical calculations indicate that the degree of cumulative growth of the DFCGW mode with circumferential angle is obviously influenced by that of phase velocity matching between the primary and double frequency wave modes. It is found that the amplitude of the DFCGW mode can grow with circumferential angle when its phase velocity matches with that of the primary circumferential guided wave, and that the amplitude of the DFCGW mode will show a beat effect with circumferential angle when its phase velocity is not equal to that of the primary wave mode. The DFCGW mode, whose phase velocity matches with that of the primary wave mode, plays a dominant role in the field of second harmonic generated by the primary wave mode propagation, and the contribution of the other DFCGW modes to the said second-harmonic field is negligible after the primary wave mode has propagated some circumferential angle.

Influences of Electrical Boundary Conditions on Second-Harmonic Generation of Ultrasonic Guided Wave Propagation in a Piezoelectric Plate

The influences of electrical boundary conditions on second-harmonic generation (SHG) of ultrasonic guided wave propagation in a piezoelectric plate are analyzed. Based on the modal expansion analysis for waveguide excitation, an accurate description for the SHG effect of primary ultrasonic guided wave propagation in a piezoelectric plate has been presented within a second-order perturbation approximation. The formal solution of the double frequency guided waves, constituting the field of second harmonic, has been developed. The analytical results clearly reveal that the SHG effect of primary guided wave propagation is closely related to the electric boundary conditions of the piezoelectric plate. It is found that under different electrical boundary conditions there is an evident difference in the SHG effect of ultrasonic guided waves, and that the SHG effect is highly sensitive to the electrical boundary conditions.