排序方式: 共有32条查询结果,搜索用时 15 毫秒
11.
Ivan P. Gavrilyuk Wolfgang Hackbusch Boris N. Khoromskij. 《Mathematics of Computation》2004,73(247):1297-1324
In previous papers the arithmetic of hierarchical matrices has been described, which allows us to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resolvent
In the present paper, we consider various operator functions, the operator exponential negative fractional powers , the cosine operator function and, finally, the solution operator of the Lyapunov equation. Using the Dunford-Cauchy representation, we get integrals which can be discretised by a quadrature formula which involves the resolvents mentioned above. We give error estimates which are partly exponentially, partly polynomially decreasing.
12.
In this paper, we will propose a boundary element method for solving classical boundary integral equations on complicated
surfaces which, possibly, contain a large number of geometric details or even uncertainties in the given data. The (small)
size of such details is characterised by a small parameter and the regularity of the solution is expected to be low in such zones on the surface (which we call the wire-basket zones).
We will propose the construction of an initial discretisation for such type of problems. Afterwards standard strategies for boundary element discretisations can be applied
such as the h, p, and the adaptive hp-version in a straightforward way.
For the classical boundary integral equations, we will prove the optimal approximation results of our so-called wire-basket boundary element method and discuss the stability aspects. Then, we construct the panel-clustering and -matrix approximations to the corresponding Galerkin BEM stiffness matrix. The method is shown to have an almost linear complexity
with respect to the number of degrees of freedom located on the wire basket. 相似文献
13.
Boris N. Khoromskij 《Constructive Approximation》2009,30(3):599-620
In the present paper we analyze a class of tensor-structured preconditioners for the multidimensional second-order elliptic operators in ? d , d≥2. For equations in a bounded domain, the construction is based on the rank-R tensor-product approximation of the elliptic resolvent ? R ≈(??λ I)?1, where ? is the sum of univariate elliptic operators. We prove the explicit estimate on the tensor rank R that ensures the spectral equivalence. For equations in an unbounded domain, one can utilize the tensor-structured approximation of Green’s kernel for the shifted Laplacian in ? d , which is well developed in the case of nonoscillatory potentials. For the oscillating kernels e ?i κ‖x‖/‖x‖, x∈? d , κ∈?+, we give constructive proof of the rank-O(κ) separable approximation. This leads to the tensor representation for the discretized 3D Helmholtz kernel on an n×n×n grid that requires only O(κ?|log?ε|2? n) reals for storage. Such representations can be applied to both the 3D volume and boundary calculations with sublinear cost O(n 2), even in the case κ=O(n). Numerical illustrations demonstrate the efficiency of low tensor-rank approximation for Green’s kernels e ?λ‖x‖/‖x‖, x∈?3, in the case of Newton (λ=0), Yukawa (λ∈?+), and Helmholtz (λ=i κ,?κ∈?+) potentials, as well as for the kernel functions 1/‖x‖ and 1/‖x‖ d?2, x∈? d , in higher dimensions d>3. We present numerical results on the iterative calculation of the minimal eigenvalue for the d-dimensional finite difference Laplacian by the power method with the rank truncation and based on the approximate inverse ? R ≈(?Δ)?1, with 3≤d≤50. 相似文献
14.
B.N. Khoromskij V. Khoromskaia S.R. Chinnamsetty H.-J. Flad 《Journal of computational physics》2009,228(16):5749-5762
In this paper, we investigate a novel approach based on the combination of Tucker-type and canonical tensor decomposition techniques for the efficient numerical approximation of functions and operators in electronic structure calculations. In particular, we study applicability of tensor approximations for the numerical solution of Hartree–Fock and Kohn–Sham equations on 3D Cartesian grids. We show that the orthogonal Tucker-type tensor approximation of electron density and Hartree potential of simple molecules leads to low tensor rank representations. This enables an efficient tensor-product convolution scheme for the computation of the Hartree potential using a collocation-type approximation via piecewise constant basis functions on a uniform n×n×n grid. Combined with the Richardson extrapolation, our approach exhibits O(h3) convergence in the grid-size h=O(n-1). Moreover, this requires O(3rn+r3) storage, where r denotes the Tucker rank of the electron density with r=O(logn), almost uniformly in n . For example, calculations of the Coulomb matrix and the Hartree–Fock energy for the CH4 molecule, with a pseudopotential on the C atom, achieved accuracies of the order of 10-6 hartree with a grid-size n of several hundreds. Since the tensor-product convolution in 3D is performed via 1D convolution transforms, our scheme markedly outperforms the 3D-FFT in both the computing time and storage requirements. 相似文献
15.
This paper investigates best rank-(r
1,..., r
d
) Tucker tensor approximation of higher-order tensors arising from the discretization of linear operators and functions in
ℝ
d
. Super-convergence of the best rank-(r
1,..., r
d
) Tucker-type decomposition with respect to the relative Frobenius norm is proven. Dimensionality reduction by the two-level
Tucker-to-canonical approximation is discussed. Tensor-product representation of basic multi-linear algebra operations is
considered, including inner, outer and Hadamard products. Furthermore, we focus on fast convolution of higher-order tensors
represented by the Tucker/canonical models. Optimized versions of the orthogonal alternating least-squares (ALS) algorithm
is presented taking into account the different formats of input data. We propose and test numerically the mixed CT-model, which is based on the additive splitting of a tensor as a sum of canonical and Tucker-type representations. It allows to
stabilize the ALS iteration in the case of “ill-conditioned” tensors. The best rank-(r
1,..., r
d
) Tucker decomposition is applied to 3D tensors generated by classical potentials, for example
and
with x, y ∈ ℝ
d
. Numerical results for tri-linear decompositions illustrate exponential convergence in the Tucker rank, and robustness of
the orthogonal ALS iteration.
相似文献
16.
In recent papers tensor-product structured Nyström and Galerkin-type approximations of certain multi-dimensional integral operators have been introduced and analysed. In the present paper, we focus on the analysis of the collocation-type schemes with respect to the tensor-product basis in a high spatial dimension d. Approximations up to an accuracy are proven to have the storage complexity with q independent of d, where N is the discrete problem size. In particular, we apply the theory to a collocation discretisation of the Newton potential with the kernel , , d3. Numerical illustrations are given in the case of d=3. 相似文献
17.
Chinnamsetty SR Espig M Khoromskij BN Hackbusch W Flad HJ 《The Journal of chemical physics》2007,127(8):084110
Tensor product decompositions with optimal separation rank provide an interesting alternative to traditional Gaussian-type basis functions in electronic structure calculations. We discuss various applications for a new compression algorithm, based on the Newton method, which provides for a given tensor the optimal tensor product or so-called best separable approximation for fixed Kronecker rank. In combination with a stable quadrature scheme for the Coulomb interaction, tensor product formats enable an efficient evaluation of Coulomb integrals. This is demonstrated by means of best separable approximations for the electron density and Hartree potential of small molecules, where individual components of the tensor product can be efficiently represented in a wavelet basis. We present a fairly detailed numerical analysis, which provides the basis for further improvements of this novel approach. Our results suggest a broad range of applications within density fitting schemes, which have been recently successfully applied in quantum chemistry. 相似文献
18.
We have investigated the superconducting properties of the Bi1.7 Pb0.3Sr2Ca2−xCe
x
Cu3O10+δ system with x=0.00, 0.02, 0.04, 0.08 and 0.1 by X-ray diffraction and magnetic susceptibility. The substitution of Ce for Ca has been found
to drastically change the superconducting properties of the system. X-ray diffraction studies on these compounds indicate
decrease in the c-parameter with increased substitution of Ce at Ca site and volume fraction of high T
c (2 : 2 : 2 : 3) phase decreases and low T
c phase increases. The magnetic susceptibility of this compound shows that the diamagnetic on set superconducting transition
temperature (onset) varies from 109 K to 51 K for x=0.00, 0.02, 0.04, 0.08 and 0.1. These results suggest the possible existence of Ce in a tetravalent state rather than a trivalent
state in this system; that is, Ca2+ → Ce4+ replacement changes the hole carrier concentration. Hole filling is the cause of lowering T
c of the system. 相似文献
19.
Summary.
In this paper we introduce a class of robust multilevel
interface solvers for two-dimensional
finite element discrete elliptic problems with highly
varying coefficients corresponding to geometric decompositions by a
tensor product of strongly non-uniform meshes.
The global iterations convergence rate is shown to be of
the order
with respect to the number of degrees
of freedom on the single subdomain boundaries, uniformly upon the
coarse and fine mesh sizes, jumps in the coefficients
and aspect ratios of substructures.
As the first approach, we adapt the frequency filtering techniques
[28] to construct robust smoothers
on the highly non-uniform coarse grid. As an alternative, a multilevel
averaging procedure for successive coarse grid correction is
proposed and analyzed.
The resultant multilevel coarse grid
preconditioner is shown to have (in a two level case) the condition
number independent
of the coarse mesh grading and
jumps in the coefficients related to the coarsest refinement level.
The proposed technique exhibited high serial and parallel
performance in the skin diffusion processes modelling [20]
where the high dimensional coarse mesh problem inherits a strong geometrical
and coefficients anisotropy.
The approach may be also applied to magnetostatics problems
as well as in some composite materials simulation.
Received December 27, 1994 相似文献
20.
Boris N. Khoromskij. 《Mathematics of Computation》2007,76(259):1291-1315
We develop efficient data-sparse representations to a class of high order tensors via a block many-fold Kronecker product decomposition. Such a decomposition is based on low separation-rank approximations of the corresponding multivariate generating function. We combine the interpolation and a quadrature-based approximation with hierarchically organised block tensor-product formats. Different matrix and tensor operations in the generalised Kronecker tensor-product format including the Hadamard-type product can be implemented with the low cost. An application to the collision integral from the deterministic Boltzmann equation leads to an asymptotical cost - in the one-dimensional problem size (depending on the model kernel function), which noticeably improves the complexity of the full matrix representation.