QTT-finite-element approximation for multiscale problems I: model problems in one dimension

Tensor-compressed numerical solution of elliptic multiscale-diffusion and high frequency scattering problems is considered. For either problem class, solutions exhibit multiple length scales governed by the corresponding scale parameter: the scale of oscillations of the diffusion coefficient or smallest wavelength, respectively. As is well-known, this imposes a scale-resolution requirement on the number of degrees of freedom required to accurately represent the solutions in standard finite-element (FE) discretizations. Low-order FE methods are by now generally perceived unsuitable for high-frequency coefficients in diffusion problems and high wavenumbers in scattering problems. Accordingly, special techniques have been proposed instead (such as numerical homogenization, heterogeneous multiscale method, oversampling, etc.) which require, in some form, a-priori information on the microstructure of the solution. We analyze the approximation properties of tensor-formatted, conforming first-order FE methods for scale resolution in multiscale problems without a-priori information. The FE methods are based on the dynamic extraction of principal components from stiffness matrices, load and solution vectors by the quantized tensor train (QTT) decomposition. For prototypical model problems, we prove that this approach, by means of the QTT reparametrization of the FE space, allows to identify effective degrees of freedom to replace the degrees of freedom of a uniform “virtual” (i.e. never directly accessed) mesh, whose number may be prohibitively large to realize computationally. Precisely, solutions of model elliptic homogenization and high-frequency acoustic scattering problems are proved to admit QTT-structured approximations whose number of effective degrees of freedom required to reach a prescribed approximation error scales polylogarithmically with respect to the reciprocal of the target Sobolev-norm accuracy ε with only a mild dependence on the scale parameter. No a-priori information on the nature of the problems and intrinsic length scales of the solution is required in the numerical realization of the presently proposed QTT-structured approach. Although only univariate model multiscale problems are analyzed in the present paper, QTT structured algorithms are applicable also in several variables. Detailed numerical experiments confirm the theoretical bounds. As a corollary of our analysis, we prove that for the mentioned model problems, the Kolmogorov n-widths of solution sets are exponentially small for analytic data, independently of the problems’ scale parameters. That implies, in particular, the exponential convergence of reduced basis techniques which is scale-robust, i.e., independent of the scale parameter in the problem.     

Analysis of some numerical methods on layer adapted meshes for singularly perturbed quasilinear systems

We consider a coupled system of first-order singularly perturbed quasilinear differential equations with given initial conditions. The leading term of each equation is multiplied by a distinct small positive parameter, which induces overlapping layers. The quasilinear system is discretized by using first and second order accurate finite difference schemes for which we derive general error estimates in the discrete maximum norm. As consequences of these error estimates we establish nodal convergence of O((N ^?1 lnN) ^p ),p=1,2, on the Shishkin mesh and O(N ^?p ),p=1,2, on the Bakhvalov mesh, where N is the number of mesh intervals and the convergence is robust in all of the parameters. Numerical computations are included which confirm the theoretical results.     

A weak Galerkin finite element method for the Oseen equations

In this paper, a weak Galerkin finite element method for the Oseen equations of incompressible fluid flow is proposed and investigated. This method is based on weak gradient and divergence operators which are designed for the finite element discontinuous functions. Moreover, by choosing the usual polynomials of degree i ≥ 1 for the velocity and polynomials of degree i ? 1 for the pressure and enhancing the polynomials of degree i ? 1 on the interface of a finite element partition for the velocity, this new method has a lot of attractive computational features: more general finite element partitions of arbitrary polygons or polyhedra with certain shape regularity, fewer degrees of freedom and parameter free. Stability and error estimates of optimal order are obtained by defining a weak convection term. Finally, a series of numerical experiments are given to show that this method has good stability and accuracy for the Oseen problem.     

Equiconvergence of expansions in multiple Fourier series and in fourier integrals with “lacunary sequences of partial sums”

We investigate the equiconvergence on T^N = [?π, π)^N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L_p(T^N) and g ∈ L_p(R^N), p > 1, N ≥ 3, g(x) = f(x) on T^N, in the case where the “partial sums” of these expansions, i.e., S_n(x; f) and J_α(x; g), respectively, have “numbers” n ∈ Z^N and α ∈ R^N (n_j = [α_j], j = 1,..., N, [t] is the integral part of t ∈ R¹) containing N ? 1 components which are elements of “lacunary sequences.”     

Stability and convergence of a fully discrete local discontinuous Galerkin method for multi-term time fractional diffusion equations

In this paper, a fully discrete local discontinuous Galerkin method for a class of multi-term time fractional diffusion equations is proposed and analyzed. Using local discontinuous Galerkin method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established. By choosing the numerical flux carefully, we prove that the method is unconditionally stable and convergent with order O(h ^k+1 + (Δt)^2?α ), where k, h, and Δt are the degree of piecewise polynomial, the space, and time step sizes, respectively. Numerical examples are carried out to illustrate the effectiveness of the numerical scheme.     

<Emphasis Type="Italic">G</Emphasis>-Convergence of systems of generalized Beltrami equations

Many problems of mathematical physics lead to problems of G-convergence of differential operators and, in particular, to the problem of homogenization of partial differential operators. Similar problems arise in elasticity theory, electrodynamics, and other fields of physics and mechanics. In this paper, we consider the problem of G-convergence of systems of Beltrami operators. We prove that the class of such systems is G-compact and study the properties of G-convergence.     

Asymptotic equalities for best approximations for classes of infinitely differentiable functions defined by the modulus of continuity

We obtain asymptotic estimates for best approximations by trigonometric polynomials in the metric of the space C(L_p) for classes of periodic functions expressible as convolutions of kernels Ψ_β with Fourier coefficients decreasing to zero faster than any power sequence, and with functions ? ∈ C (? ∈ L_p) whose moduli of continuity do not exceed the given majorant of ω(t). It is proved that, in the spaces C and L₁, for convex moduli of continuity ω(t), the obtained estimates are asymptotically sharp.     

Randomness of the square root of 2 and the giant leap,part 2

We prove that the “quadratic irrational rotation” exhibits a central limit theorem. More precisely, let α be an arbitrary real root of a quadratic equation with integer coefficients; say, \(\alpha = \sqrt 2\). Given any rational number 0 < x < 1 (say, x = 1/2) and any positive integer n, we count the number of elements of the sequence α, 2α, 3α, ..., nα modulo 1 that fall into the subinterval [0, x]. We prove that this counting number satisfies a central limit theorem in the following sense. First, we subtract the “expected number” nx from the counting number, and study the typical fluctuation of this difference as n runs in a long interval 1 ≤ n ≤ N. Depending on α and x, we may need an extra additive correction of constant times logarithm of N; furthermore, what we always need is a multiplicative correction: division by (another) constant times square root of logarithm of N. If N is large, the distribution of this renormalized counting number, as n runs in 1 ≤ n ≤ N, is very close to the standard normal distribution (bell shaped curve), and the corresponding error term tends to zero as N tends to infinity. This is the main result of the paper (see Theorem 1.1).     

Weighted stationary phase of higher orders

The subject matter of this paper is an integral with exponential oscillation of phase f(x) weighted by g(x) on a finite interval [α β]: When the phase f(x) has a single stationary point in (α β), an nth-order asymptotic expansion of this integral is proved for n ≥ 2: This asymptotic expansion sharpens the classical result for n = 1 by M. N. Huxley. A similar asymptotic expansion was proved by V. Blomer, R. Khan and M. Young under the assumptions that f(x) and g(x) are smooth and g(x) is compactly supported on R: In the present paper, however, these functions are only assumed to be continuously differentiable on [α β] 2n + 3 and 2n + 1 times, respectively. Because there are no requirements on the vanishing of g(x) and its derivatives at the endpoints α and β, the present asymptotic expansion contains explicit boundary terms in the main and error terms. The asymptotic expansion in this paper is thus applicable to a wider class of problems in analysis, analytic number theory, and other fields.     

Solving Equations of Free Vibration for a Cylindrical Shell Rotating on Rollers by the Fourier Method

The small free vibrations of an infinite circular cylindrical shell rotating about its axis at a constant angular velocity are considered. The shell is supported on n absolutely rigid cylindrical rollers equispaced on its circle. The roller-supported shell is a model of an ore benefication centrifugal concentrator with a floating bed. The set of linear differential equations of vibrations is sought in the form of a truncated Fourier series containing N terms along the circumferential coordinate. A system of 2N–n linear homogeneous algebraic equations with 2N–n unknowns is derived for the approximate estimation of vibration frequencies and mode shapes. The frequencies ω _k , k = 1, 2, …, 2N–n, are positive roots of the (2N–n)th-order algebraic equation D(ω²) = 0, where D is the determinant of this set. It is shown that the system of 2N–n equations is equivalent to several independent systems with a smaller number of unknowns. As a consequence, the (2N–n)th-order determinant D can be written as a product of lower-order determinants. In particular, the frequencies at N = n are the roots of algebraic equations of an order is lower than 2 and can be found in an explicit form. Some frequency estimation algorithms have been developed for the case of N > n. When N increases, the number of found frequencies also grows, and the frequencies determined at N = n are refined. However, in most cases, the vibration frequencies can not be found for N > n in an explicit form.     

Two-Sided Estimates of Fourier Sums Lebesgue Functions with Respect to Polynomials Orthogonal on Nonuniform Grids

Let Ω = {t₀, t₁, …, t_N} and Ω_N = {x₀, x₁, …, x_N–1}, where x_j = (t_j + t_{j + 1})/2, j = 0, 1, …, N–1 be arbitrary systems of distinct points of the segment [–1, 1]. For each function f(x) continuous on the segment [–1, 1], we construct discrete Fourier sums S_{n, N}( f, x) with respect to the system of polynomials {p?_k,N(x)} _k=0 ^N–1 , forming an orthonormal system on nonuniform point systems Ω_N consisting of finite number N of points from the segment [–1, 1] with weight Δt_j = t_{j + 1}–t_j. We find the growth order for the Lebesgue function L_n,N (x) of the considered partial discrete Fourier sums S_n,N ( f, x) as n = O(δ _N ^?2/7 ), δ_N = max_{0≤ j≤N?1} Δt_j More exactly, we have a two-sided pointwise estimate for the Lebesgue function L_{n, N}(x), depending on n and the position of the point x from [–1, 1].     

Regular variation of a random length sequence of random variables and application to risk assessment

When assessing risks on a finite-time horizon, the problem can often be reduced to the study of a random sequence C(N) = (C ₁,…,C _N ) of random length N, where C(N) comes from the product of a matrix A(N) of random size N × N and a random sequence X(N) of random length N. Our aim is to build a regular variation framework for such random sequences of random length, to study their spectral properties and, subsequently, to develop risk measures. In several applications, many risk indicators can be expressed from the extremal behavior of ∥C(N)∥, for some norm ∥?∥. We propose a generalization of Breiman’s Lemma that gives way to a tail estimate of ∥C(N)∥ and provides risk indicators such as the ruin probability and the tail index for Shot Noise Processes on a finite-time horizon. Lastly, we apply our main result to a model used in dietary risk assessment and in non-life insurance mathematics to illustrate the applicability of our method.     

Boundary-value problems for the Schröinger equation with rapidly oscillating and delta-liked potentials

This paper deals with boundary-value problems on the closed interval [a, b] for the Schrödinger equation with potential of the form q(x, μ ^?1 x) + ε ^?1 Q(ε ^?1 x), where q(x, ζ) is a 1-periodic (in ζ) function, Q(ξ) is a compactly supported function, 0 ∈ (a, b), and μ, ε are small positive parameters. The solutions of these boundary-value problemsup to O(ε +μ) are constructed by combining the homogenization method and the method of matching asymptotic expansions.     

Unexpected spectral asymptotics for wave equations on certain compact spacetimes

We study the spectral asymptotics of wave equations on certain compact spacetimes, where some variant of the Weyl asymptotic law is valid. The simplest example is the spacetime S¹×S². For the Laplacian on S¹×S², theWeyl asymptotic law gives a growth rate O(s^3/2) for the eigenvalue counting function N(s) = #{λ_j: 0 ≤ λ_j ≤ s}. For the wave operator, there are two corresponding eigenvalue counting functions: N^±(s) = #{λ_j: 0 < ±λ_j ≤ s}, and they both have a growth rate of O(s²). More precisely, there is a leading term π²s²/4 and a correction term of as^3/2, where the constant a is different for N^±. These results are not robust in that if we include a speed of propagation constant to the wave operator, the result depends on number theoretic properties of the constant, and generalizations to S¹ × S^q are valid for q even but not q odd. We also examine some related examples.     

Locally Unmixed Modules and Linearly Equivalent Ideal Topologies

Let R be a commutative Noetherian ring, and let N be a non-zero finitely generated R-module. The purpose of this paper is to show that N is locally unmixed if and only if, for any N-proper ideal I of R generated by ht _N I elements, the topology defined by (I N)⁽ⁿ⁾, n ≥ 0, is linearly equivalent to the I-adic topology.     

Galerkin finite element method and error analysis for the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. These equations can be derived from the Nernst-Planck equation for electro-diffusion in smooth homogeneous cylinders. Fractional cable equations are introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, a Galerkin finite element method(GFEM) is presented for the numerical simulation of the fractional cable equation(FCE) involving two integro-differential operators. The proposed method is based on a semi-discrete finite difference approximation in time and Galerkin finite element method in space. We prove that the numerical solution converges to the exact solution with order O(τ+h^l+1) for the lth-order finite element method. Further, a novel Galerkin finite element approximation for improving the order of convergence is also proposed. Finally, some numerical results are given to demonstrate the theoretical analysis. The results show that the numerical solution obtained by the improved Galerkin finite element approximation converges to the exact solution with order O(τ²+h^l+1).     

Stability and absence of binding for multi-polaron systems

We resolve several longstanding problems concerning the stability and the absence of multi-particle binding for N≥2 polarons. Fröhlich’s 1937 polaron model describes non-relativistic particles interacting with a scalar quantized field with coupling \(\sqrt{\alpha}\), and with each other by Coulomb repulsion of strength U. We prove the following: (i) While there is a known thermodynamic instability for U<2α, stability of matter does hold for U>2α, that is, the ground state energy per particle has a finite limit as N→∞. (ii) There is no binding of any kind if U exceeds a critical value that depends on α but not on N. The same results are shown to hold for the Pekar-Tomasevich model.     

On the evaluation of highly oscillatory finite Hankel transform using special functions

In this paper, an efficient Clenshaw–Curtis–Filon–type method is presented for approximation of the highly oscillatory finite Hankel transform \({{\int }_{0}^{1}}f(x)H_{\nu }^{(1)}(\omega x)dx\), which arises in acoustic and electromagnetic scattering problems. This method is based on Fast Fourier Transform (FFT) and fast computation of the modified moments by using Meijer G–function and Lommel function. Moreover, the method shares the property that the higher the frequency ω, the higher the precision. In particular, for each fixed ω the method is uniformly convergent as N tends to infinity, where (N+1) is the number of Clenshaw–Curtis points c_i=(1+ cos(iπ/N))/2,i=0,? ,N. Also, the corresponding error bound in inverse powers of ω for this method for the integral is presented. The efficiency and accuracy of the proposed method are illustrated by numerical examples.