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1.
In this paper we develop asymptotically optimal algorithms for fast computations with the discrete harmonic Poincaré–Steklov operators (Dirichlet–Neumann mapping) for interior and exterior problems in the presence of a nested mesh refinement. Our approach is based on the multilevel interface solver applied to the Schur complement reduction onto the nested refined interface associated with a nonmatching decomposition of a polygon by rectangular substructures. This paper extends methods from Khoromskij and Prössdorf (1995), where the finite element approximations of interior problems on quasi‐uniform grids have been considered. For both interior and exterior problems, the matrix–vector multiplication with the compressed Schur complement matrix on the interface is shown to have a complexity of the order O(N
r log3
N
u), where Nr = O((1 + p
r) N
u) is the number of degrees of freedom on the polygonal boundary under consideration, N
u is the boundary dimension of a finest quasi‐uniform level and p
r ⩾ 0 defines the refinement depth. The corresponding memory needs are estimated by O(N
r logq
N
u), where q = 2 or q = 3 in the case of interior and exterior problems, respectively. 相似文献
2.
The arclengths of the graphs Γ(s N(f)) of the partial sums s N(f) of the Fourier series of a piecewise smooth function f with a jump discontinuity grow at the rate O( logN). This problem does not arise if f is continuous, and can be removed by using the standard summability methods. 相似文献
3.
We formulate a modified nodal cubic spline collocation scheme for the solution of the biharmonic Dirichlet problem on the
unit square. We prove existence and uniqueness of a solution of the scheme and show how the scheme can be solved on an N × N uniform partition of the square at a cost O( N
2 log 2 N + mN
2) using fast Fourier transforms and m iterations of the preconditioned conjugate gradient method. We demonstrate numerically that m proportional to log 2
N guarantees the desired convergence rates. Numerical results indicate the fourth order accuracy of the approximations in the
global maximum norm and the fourth order accuracy of the approximations to the first order partial derivatives at the partition
nodes.
相似文献
4.
Summary An efficient algorithm for the solution of linear equations arising in a finite element method for the Dirichlet problem is given. The cost of the algorithm is proportional to N
2log 2
N ( N=1/h) where the cost of solving the capacitance matrix equations is Nlog 2
N on regular grids and N
3/2log 2
N on irregular ones. 相似文献
5.
Let P( n) denote the largest prime factor of an integer n ( N( x) = x (2+ O(?{log 2 x/log x} ) )ò 2xr(log x/log t) [(log t)/( t2)] d t,N(x) = x \left(2+O\left(\sqrt{\log_{2}\,x/\!\log x}\,\right) \right)\int_2^x\rho(\log x/\!\log t) {\log t\over t^2} {\rm d} t, 相似文献
6.
We present an algorithm to compute a Euclidean minimum spanning tree of a given set S of N points in E
d
in time O( F
d
( N,N) log
d
N), where F
d
( n,m) is the time required to compute a bichromatic closest pair among n red and m green points in E
d
. If F
d
( N,N)=Ω( N
1+ε), for some fixed ɛ>0, then the running time improves to O( F
d
( N,N)). Furthermore, we describe a randomized algorithm to compute a bichromatic closest pair in expected time O(( nm log n log m) 2/3+ m log 2
n+ n log 2
m) in E
3, which yields an O( N
4/3 log 4/3
N) expected time, algorithm for computing a Euclidean minimum spanning tree of N points in E
3. In d≥4 dimensions we obtain expected time O(( nm) 1−1/([d/2]+1)+ε+ m log n+n log m) for the bichromatic closest pair problem and O( N
2−2/([d/2]+1)ε) for the Euclidean minimum spanning tree problem, for any positive ɛ.
The first, second, and fourth authors acknowledge support from the Center for Discrete Mathematics and Theoretical Computer
Science (DIMACS), a National Science Foundation Science and Technology Center under NSF Grant STC 88-09648. The second author's
work was supported by the National Science Foundation under Grant CCR-8714565. The third author's work was supported by the
Deutsche Forschungsgemeinschaft under Grant A1 253/1-3, Schwerpunktprogramm “Datenstrukturen und effiziente Algorithmen”.
The last two authors' work was also partially supported by the ESPRIT II Basic Research Action of the EC under Contract No.
3075 (project ALCOM). 相似文献
7.
Using the coupled approach, we formulate a fourth order finite difference scheme for the solution of the Dirichlet biharmonic problem on the unit square. On an N × N uniform partition of the square the scheme is solved at a cost O( N 2 log 2 N)+ m8 N 2 using fast Fourier transforms and m iterations of the preconditioned conjugate gradient method. Numerical tests confirm the fourth order accuracy of the scheme at the partition nodes with m proportional to log 2 N. 相似文献
8.
Spherical harmonics arise on the sphere S 2 in the same way that the (Fourier) exponential functions {e ik}k arise on the circle. Spherical harmonic series have many of the same wonderful properties as Fourier series, but have lacked one important thing: a numerically stable fast transform analogous to the Fast Fourier Transform (FFT). Without a fast transform, evaluating (or expanding in) spherical harmonic series on the computer is slow—for large computations probibitively slow. This paper provides a fast transform.For a grid of O(N 2) points on the sphere, a direct calculation has computational complexity O(N 4), but a simple separation of variables and FFT reduce it to O(N 3) time. Here we present algorithms with times O(N 5/2
log N) and O(N 2( log N) 2).
The problem quickly reduces to the fast application of matrices of associated Legendre functions of certain orders. The essential insight is that although these matrices are dense and oscillatory, locally they can be represented efficiently in trigonometric series. 相似文献
9.
Recently, fast and reliable algorithms for the evaluation of spherical harmonic expansions have been developed. The corresponding
sampling problem is the computation of Fourier coefficients of a function from sampled values at scattered nodes. We consider
a least squares approximation and an interpolation of the given data. Our main result is that the rate of convergence of the
two proposed iterative schemes depends only on the mesh norm and the
separation distance of the nodes. In conjunction with the nonequispaced FFT on the sphere, the reconstruction of N 2 Fourier coefficients from M reasonably distributed samples is shown to take O(N 2 log 2 N + M) floating point operations. Numerical results support our theoretical findings. 相似文献
10.
In this paper we present algorithms to calculate the fast Fourier synthesis and its adjoint on the rotation group SO(3) for arbitrary sampling sets. They are based on the fast Fourier transform for nonequispaced nodes on the three-dimensional
torus. Our algorithms evaluate the SO(3) Fourier synthesis and its adjoint, respectively, of B-bandlimited functions at M arbitrary input nodes in O( M+ B4)\mathcal O(M+B^4) or even O( M + B3 log 2 B)\mathcal O(M + B^3 \log^2 B) flops instead of O( MB3)\mathcal O(MB^3). Numerical results will be presented establishing the algorithm’s numerical stability and time requirements. 相似文献
11.
Earlier work by Driscoll and Healy has produced an efficient algorithm for
computing the Fourier transform of band-limited functions on the 2-sphere. In this article we present
a reformulation and variation of the original algorithm which results in a greatly improved inverse
transform, and consequent improved convolution algorithm for such functions. All require at most
O( N log 2
N) operations where N is the number of sample points. We also address implementation
considerations and give heuristics for allowing reliable and computationally efficient floating point
implementations of slightly modified algorithms. These claims are supported by extensive numerical
experiments from our implementation in C on DEC, HP, SGI and Linux Pentium platforms. These
results indicate that variations of the algorithm are both reliable and efficient for a large range of
useful problem sizes. Performance appears to be architecture-dependent. The article concludes
with a brief discussion of a few potential applications. 相似文献
12.
1.IntroductionLetG=(V,E,W)beaconnected,weightedandundirectedgraph,VeEE,w(e)(相似文献
13.
This paper presents a simple dictionary structure designed for a hierarchical memory. The proposed data structure is cache-oblivious and locality-preserving. A cache-oblivious data structure has memory performance optimized for all levels of the memory hierarchy even though it has no memory-hierarchy-specific parameterization. A locality-preserving dictionary maintains elements of similar key values stored close together for fast access to ranges of data with consecutive keys.The data structure presented here is a simplification of the cache-oblivious B-tree of Bender, Demaine, and Farach-Colton. The structure supports search operations on N data items using O(log BN+1) block transfers at a level of the memory hierarchy with block size B. Insertion and deletion operations use O(log BN+log 2N/ B+1) amortized block transfers. Finally, the data structure returns all k data items in a given search range using O(log BN+ k/ B+1) block transfers.This data structure was implemented and its performance was evaluated on a simulated memory hierarchy. This paper presents the results of this simulation for various combinations of block and memory sizes. 相似文献
14.
We show that, using the L
∞ metric, the minimum Hausdorff distance under translation between two point sets of cardinality n in d -dimensional space can be computed in time O(n
(4d-2)/3
log
2
n) for 3 < d
8, and in time O(n
5d/4
log
2
n) for any d > 8 . Thus we improve the previous time bound of O(n
2d-2
log
2
n) due to Chew and Kedem. For d=3 we obtain a better result of O(n
3
log
2
n) time by exploiting the fact that the union of n axis-parallel unit cubes can be decomposed into O(n) disjoint axis-parallel boxes. We prove that the number of different translations that achieve the minimum Hausdorff distance
in d -space is . Furthermore, we present an algorithm which computes the minimum Hausdorff distance under the L
2
metric in d -space in time , for any δ > 0.
Received March 17, 1997, and in revised form January 19, 1998. 相似文献
15.
Let N(n) and N
*
(n) denote, respectively, the number of unlabeled and labeled N-free posets with n elements. It is proved that N( n)=2
n logn+o(n logn) and N
*( n)=2 2n logn+o(n logn). This is obtained by considering the class of N-free interval posets which can be easily counted. 相似文献
16.
We present new strongly polynomial algorithms for special cases of convex separable quadratic minimization over submodular constraints. The main results are: an O( NM log( N
2/ M)) algorithm for the problem Network defined on a network on M arcs and N nodes; an O( n log n) algorithm for the tree problem on n variables; an O( n log n) algorithm for the Nested problem, and a linear time algorithm for the Generalized Upper Bound problem. These algorithms are the best known so far for these problems. The status of the general problem and open questions are presented as well.This research has been supported in part by ONR grant N00014-91-J-1241.Corresponding author. 相似文献
17.
This paper proposes new methods to answer approximate nearest neighbor queries on a set of n points in d -dimensional Euclidean space. For any fixed constant d , a data structure with O(
(1-d)/2
n log n) preprocessing time and O(
(1-d)/2
log n) query time achieves an approximation factor 1+
for any given 0 <
< 1; a variant reduces the -dependence by a factor of
-1/2
. For any arbitrary d , a data structure with O(d
2
n log n) preprocessing time and O(d
2
log n) query time achieves an approximation factor O(d
3/2
) . Applications to various proximity problems are discussed.
Received May 28, 1997, and in revised form March 4, 1998. 相似文献
18.
In this paper a method for fast computations with the inverse to weakly singular, hypersingular and double layer potential boundary integral operators associated with the Laplacian on Lipschitz domains is proposed and analyzed. It is based on the representation formulae suggested for above-mentioned boundary operations in terms of the Poincare-Steklov interface mappings generated by the special decompositions of the interior and exterior domains. Computations with the discrete counterparts of these formulae can be efficiently performed by iterative substructuring algorithms provided some asymptotically optimal techniques for treatment of interface operators on subdomain boundaries. For both two- and three-dimensional cases the computation cost and memory needs are of the order O(N log p N) and O( N log 2 N), respectively, with 1 ≤ p ≤ 3, where N is the number of degrees of freedom on the boundary under consideration (some kinds of polygons and polyhedra). The proposed algorithms are well suited for serial and parallel computations. 相似文献
19.
Summary A generalized linear rank statistic is introduced to include, as special cases, both signed as well as unsigned linear rank
statistics. For this statistic, the rate of convergence to asymptotic normality is investigated. It is shown that this rate
is of order O(N
−1/2 log N) if the score generating function ϕ is twice differentiable, and it is of order O(N
−1/2) if the second derivative of ϕ satisfies Lipschitz's condition of order ≧1/2. The results obtained extend as well as generalize
most of the earlier results obtained in this direction. 相似文献
20.
Our goal in this paper is to analyze carry propagation in addition using only elementary methods (that is, those not involving residues, contour integration, or methods of complex analysis). Our results concern the length of the longest carry chain when two independent uniformly distributed n-bit numbers are added. First, we show using just first- and second-moment arguments that the expected length Cn of the longest carry chain satisfies Cn = log 2n + O(1). Second, we use a sieve (inclusion–exclusion) argument to give an exact formula for Cn. Third, we give an elementary derivation of an asymptotic formula due to Knuth, Cn = log 2n + Φ(log 2 n) + O((log n) 4/ n), where Φ(ν) is a bounded periodic function of ν, with period 1, for which we give both a simple integral expression and a Fourier series. Fourth, we give an analogous asymptotic formula for the variance Vn of the length of the longest carry chain: Vn = Ψ(log 2 n) + O((log n) 5/ n), where Ψ(ν) is another bounded periodic function of ν, with period 1. Our approach can be adapted to addition with the “end-around” carry that occurs in the sign-magnitude and 1s-complement representations. Finally, our approach can be adapted to give elementary derivations of some asymptotic formulas arising in connection with radix-exchange sorting and collision-resolution algorithms, which have previously been derived using contour integration and residues. 相似文献
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