Local–global multiscale model reduction for flows in high-contrast heterogeneous media

In this paper, we study model reduction for multiscale problems in heterogeneous high-contrast media. Our objective is to combine local model reduction techniques that are based on recently introduced spectral multiscale finite element methods (see [19]) with global model reduction methods such as balanced truncation approaches implemented on a coarse grid. Local multiscale methods considered in this paper use special eigenvalue problems in a local domain to systematically identify important features of the solution. In particular, our local approaches are capable of homogenizing localized features and representing them with one basis function per coarse node that are used in constructing a weight function for the local eigenvalue problem. Global model reduction based on balanced truncation methods is used to identify important global coarse-scale modes. This provides a substantial CPU savings as Lyapunov equations are solved for the coarse system. Typical local multiscale methods are designed to find an approximation of the solution for any given coarse-level inputs. In many practical applications, a goal is to find a reduced basis when the input space belongs to a smaller dimensional subspace of coarse-level inputs. The proposed approaches provide efficient model reduction tools in this direction. Our numerical results show that, only with a careful choice of the number of degrees of freedom for local multiscale spaces and global modes, one can achieve a balanced and optimal result.     

Optimization of spectral functions of Dirichlet–Laplacian eigenvalues

We consider the shape optimization of spectral functions of Dirichlet–Laplacian eigenvalues over the set of star-shaped, symmetric, bounded planar regions with smooth boundary. The regions are represented using Fourier-cosine coefficients and the optimization problem is solved numerically using a quasi-Newton method. The method is applied to maximizing two particular nonsmooth spectral functions: the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue, both of which are generalizations of the Payne–Pólya–Weinberger ratio. The optimal values and attaining regions for n ? 13 are presented and interpreted as a study of the range of the Dirichlet–Laplacian eigenvalues. For both spectral functions and each n, the optimal attaining region has multiplicity two nth eigenvalue.     

An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces

We present a rigorous scheme that makes it possible to compute eigenvalues of the Laplace operator on hyperbolic surfaces within a given precision. The method is based on an adaptation of the method of particular solutions to the case of locally symmetric spaces and on explicit estimates for the approximation of eigenfunctions on hyperbolic surfaces by certain basis functions. It can be applied to check whether or not there is an eigenvalue in an ε-neighborhood of a given number λ > 0. This makes it possible to find all the eigenvalues in a specified interval, up to a given precision with rigorous error estimates. The method converges exponentially fast with the number of basis functions used. Combining the knowledge of the eigenvalues with the Selberg trace formula we are able to compute values and derivatives of the spectral zeta function again with error bounds. As an example we calculate the spectral determinant and the Casimir energy of the Bolza surface and other surfaces.     

Extension of Rayleigh–Ritz method for eigenvalue problems with discontinuous boundary conditions applied to vibration of rectangular plates

The Rayleigh–Ritz (R–R) method is extended to eigenvalue problems of rectangular plates with discontinuous boundary conditions (DBC). Coordinate functions are defined as sums of products of orthogonal polynomials and consist of two parts, each satisfying the BC in its respective region. These parts are matched by minimizing the mean square error of functions and their x-derivatives at the interface between regions. Matching defines a positive definite 2N²×2N² matrix Q whose eigenvectors form the orthogonal coordinate functions. The corresponding eigenvalues measure the matching error of the two parts at the interface. When applying the R–R method, the total error is the sum of the matching error and that arising from the finite number of coordinate functions. Although most of the coordinate functions correspond to the zero eigenvalue, these do not suffice and additional functions corresponding to small but finite eigenvalues must be included. In three examples with discontinuous BC of the clamped, simply supported and free kind, the calculated frequencies match closely those from a finely discretized finite element method.     

Limiting Behavior of Eigenvectors of Large Wigner Matrices

A new form of empirical spectral distribution of a Wigner matrix W _n with weights specified by the eigenvectors is defined and it is then shown to converge with probability one to the semicircular law. Moreover, central limit theorem for linear spectral statistics defined by the eigenvectors and eigenvalues is also established under some moment conditions, which suggests that the eigenvector matrix of W _n is close to being Haar distributed.     

On Eigenvalues of the Sum of Two Random Projections

We study the behavior of eigenvalues of matrix P _N +Q _N where P _N and Q _N are two N-by-N random orthogonal projections. We relate the joint eigenvalue distribution of this matrix to the Jacobi matrix ensemble and establish the universal behavior of eigenvalues for large N. The limiting local behavior of eigenvalues is governed by the sine kernel in the bulk and by either the Bessel or the Airy kernel at the edge depending on parameters. We also study an exceptional case when the local behavior of eigenvalues of P _N +Q _N is not universal in the usual sense.     

A mechanical definition of the thermodynamic pressure

A dynamical definition of pressure for grand-canonical Gibbs measures in bounded regions Λ is rigorously discussed: It measures the momentum transferred to the walls of the container by the elastically colliding particles. The local pressureP(r, δΛ) so obtained is proportional to the temperature and the local density at the boundaries of Λ. This allows us to obtain a rigorous proof of the virial theorem of Clausius. In this picture the thermodynamic pressureP ^d (Λ) is obtained as the average ofP(r, δΛ) onδΛ. Its relationship with the usual equilibrium pressureP _eq(Λ) = (βsΛ¦)^?1lnZ _Λ (Z _Λ is the grand-canonical partition function) is then discussed. In the particular case in which the regions A are spheres, it is shown that P_d(Λ) converges in average so that, if the limit of P_d(Λ) exists, it equals P_eq, the thermodynamic limit of the equilibrium pressure P_eq(Λ). Finally, convergence ofP _d(Λ) is proven to hold in the particular case of one-dimensional hard cores in the absence of phase transitions.     

Low Lying Spectrum of Weak-Disorder Quantum Waveguides

We study the low-lying spectrum of the Dirichlet Laplace operator on a randomly wiggled strip. More precisely, our results are formulated in terms of the eigenvalues of finite segment approximations of the infinite waveguide. Under appropriate weak-disorder assumptions we obtain deterministic and probabilistic bounds on the position of the lowest eigenvalue. A Combes-Thomas argument allows us to obtain a so-called ‘initial length scale decay estimate’ as they are employed in the proof of spectral localization using the multiscale analysis method.     

三维量子自旋玻璃理论(Ⅱ)——稳定性分析

利用我们提出的三维量子自旋玻璃理论,通过引入一个量子的Goldbart去耦方案,将三维量子自旋玻璃的稳定性Hessian矩阵的本征值问题转化为类伊辛自旋玻璃的相应问题,并求出了所有的本征值。通过分析最小本征值,获得了Almeida-Thouless(AT)下临界线和Gabay-Toulouse(GT)上临界线所满足的方程。最后,数值研究了不同自旋的AT不稳定性并讨论了我们的理论与热场动力学方法、集团展开方法的联系。     

Commutator eigenvalue problem for finite groups

The commutator eigenproblem for an arbitrary finite group is defined by analogy to that of the Lie group. After considering possible arrangements of the group basis of a Frobenius algebra, two convenient matrix forms of commutator eigenproblem are found. They enabled us to find an analytical formula for the eigenvalues and allowed to propose a method for finding the eigenvectors for this problem. The results are demonstrated on the point group T_d of tetrahedron symmetry.     

Complex energy spectra in reggeon quantum mechanics with cubic plus generalized quartic interactions

We investigate the energy eigenvalue spectra in reggeon quantum mechanics when the hamiltonian contains (non-hermitian) cubic, symmetric quartic and asymmetric quartic interactions. We describe two new methods for finding eigenvalues numerically. When the asymmetric quartic coupling is zero, the energy eigenvalues cross the vacuum state sequentially as predicted by Bronzan as long as r?0.7. For r?0.7 the energy eigenvalues above the ground state pinch together pairwise above the energy axis, leaving the ground state to oscillate about the vacuum. The addition of an asymmetric quartic term appears to dilute the effects produced by increasing the cubic coupling strength. Quantitative graphs of the functional dependence of the eigenvalues on the parameter are given. An alternative derivation of Bronzan's formula for vanishing eigenvalues is given in the appendix.     

Non-isospectral Hamiltonians, intertwining operators and hidden hermiticity

We have recently proposed a strategy to produce, starting from a given Hamiltonian h₁ and a certain operator x for which [h₁,xx†]=0 and x†x is invertible, a second Hamiltonian h₂ with the same eigenvalues as h₁ and whose eigenvectors are related to those of h₁ by x†. Here we extend this procedure to build up a second Hamiltonian, whose eigenvalues are different from those of h₁, and whose eigenvectors are still related as before. This new procedure is also extended to crypto-hermitian Hamiltonians.     

Scalars coupled to fermions in 1+1 dimensions

Using a method introduced in an earlier paper, we study a Bose field coupled to a Fermi field in 1+1 space-time dimensions. We employ the standard Hamiltonian formalism in which one computes the eigenvalues and eigenvectors of the Hamiltonian matrix. The matrix elements are computed using states defined on a lattice in momentum space. The results are compared with known strong and weak coupling limits. Bound states and renormalization effects are studied. We find that the choice of bare masses which give specified physical masses can be non-unique once a critical couplingλ _μ has been exceeded.     

Hypernuclear binding energies in the α-particle model

The binding energies B_Λ for the hypernuclei with baryon number A = 4N + 1 up to_Λ²⁵Mg have been calculated by the variational method in the framework of Brink's α-particle model. The central ΛN potential is adjusted to the binding energy of_Λ⁵He. Due to their strong dependence on the nuclear density, the B_Λ energies are very sensitive to the choice of the effective nuclear interaction and have reasonable values for large A with the potential B1 of Brink and Boeker. The comparison with the B_Λ values obtained in the limiting shell model and the consideration of two different Λ-particle wave functions both indicate that the effect of nuclear clustering on B_Λ is significant only for_Λ⁹Be and_Λ¹³C. The nuclear distortion due to the Λ-particle binding is also evaluated in this calculation.     

Nonexponential dynamics and competition between quasi-trapped states in an N-atomic micromaser

Transient processes are examined for the reduced density matrix (RDM) of the quantum mode of a micromaser pumped by clusters of N two-level atoms. The RDM dynamics consists of the fast and slow stages. The hierarchy of time scales is explained by the fact that the spectrum of the operator of RDM evolution contains groups of eigenvalues concentrated near zero and unity. A convenient basis for describing the fast stage is provided by a set of eigenvectors and adjoint vectors of Jordan cells related to the degenerate zero eigenvalue. The dynamics of the adjoint vector as a function of the number of passed clusters is nonexponential. One adjoint vector transforms jumpwise into the neighboring vector upon the passage of the next cluster. The slow stage of dynamics is controlled by the quasi-trapped states (QTSs) of the field. These states are the generalization of the trapped states of the ideal one-atom model and can be represented as linear combinations of the long-lived eigenvectors of the evolution operator with eigenvalues close to unity. An important feature of the QTSs is their stability to fluctuations of the number of atoms in clusters and to the field relaxation rate.     

Diagonal structure of the S-matrix in the lee model

The diagonal structure of the S-matrix in the Lee model is studied. The N2θ- and N3θ-type eigenvalues are shown to factor exactly into the products of the corresponding Nθ eigenvalues, and it follows that these eigenvalues are free of production thresholds. The (implicit) eigenvalue equation for the V2θ-type eigenvalue is given, and this eigenvalue is shown to factor asymptotically into the product of the corresponding Vθ eigenvalues. These properties are conjectured to hold for Njθ- and Vjθ-type eigenvalues. It is conjectured that the two-body type eigenphase operator gives the dominant contribution to the S-matrix at high energy.     

Solutions of the Fokker-Planck equation for a double-well potential in terms of matrix continued fractions

Solutions of the Fokker-Planck (Kramers) equation in position-velocity space for the double-well potentiald ₂x²/2+d₄x⁴/4 in terms of matrix continued fractions are derived. It is shown that the method is also applicable to a Boltzmann equation with a BGK collision operator. Results of eigenvalues and of the Fourier transform of correlation functions are presented explicitly. The lowest nonzero eigenvalue is compared with the escape rate in the weak noise limit for various damping constants and the susceptibility is compared with the zero-friction-limit result.     

裂缝性介质多尺度深度学习模型

结合人工神经网络建立裂缝介质多尺度深度学习流动模型.基于一套粗网格和一套细网格,通过在粗网格上训练数据,多尺度神经网络能够以较少的自由度训练出准确的神经网络.并在粗网格上通过求解局部流动问题获得多尺度基函数,结合神经网络进一步得到精细网格的解.基于离散裂缝的流动方程可视为多层网络,网络层数依赖于求解时间步数.阐述裂缝介质多尺度机器学习数值计算格式的建立,介绍如何使用多尺度算法构建离散裂缝模型的多尺度基函数,并采用超样本技术进一步提高计算准确性.数值结果表明,多尺度有限元算法与机器学习结合是一种有效的流体流动模拟算法.     

An eigenvalue problem with limitations in a finite movable basis

An asymptotic method for taking into account the constraints of the orthogonality type when solving eigenvalue problems is developed. The features of the variational determination of eigenvalues are considered, with various finite-dimensional Hilbert subspaces being used to calculate different eigenvalues. A hydrogen molecular ion is used as an example to study the influence of the finite-dimensional approximation and basis optimization on the accuracy of determining the states of identical symmetry. Calculations have shown that the method proposed makes it possible to construct a flexible, consistent scheme for determining the energies of the ground and excited states. The total energy values obtained in a basis of 29 primitive Gaussian functions differ from exact values by 0.228 μH for the ground 1σ_g state and 0.413 μH (microhartree) for the excited 2σ_g state.     

The non-mesonic to π− mesonic decay ratio and Λ-neutron stimulation fraction for p-shell hypernuclei

Kinematic analysis of simple hypernuclear production reactions has produced a sample of hypernuclei of _ΛB, _ΛC and _ΛN with negligible background, and a much smaller sample of _ΛBe. The values of the non-mesonic to π^? mesonic ratio Q^? for the above samples are 5.5 ± 0.5 and 4.3 ± 1.1, respectively. A sub-sample of ¹¹_ΛB hypernuclei was separated on the basis of production via an excited state of _Λ¹²C, giving Q^? = 4.8 ± 1.1 for _Λ¹¹B. Assuming the Fermi gas model is applicable to nuclei of mass A ≈ 11, two independent means of analysis of the non-mesonic decays give values for the Λ-neutron stimulation fraction n of 0.41 ± 0.09 and 0.34 ± 0.07 for hypernuclei of charge 5 ≦ Z ≦ 7.