Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces
Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces
复制标题
稀疏张量积空间上各向异性积分微分算子的小波压缩
DOI：
10.1051/m2an/2009039
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
N. Reich
中科院分区：
文献类型：
--
作者：
N. Reich
对于一类作为Markov过程半群生成元的各向异性积分微分算子B，我们给出了相应的积分微分方程Bu = f在（0，1）n（n可能很大）上的Galerkin有限元离散的稀疏张量积小波压缩格式.在B上的一定条件下，该方案具有本质最优和维数无关的复杂度O（h-1|坯料h| 2（n-1）），而不会破坏原始稀疏张量有限元格式的收敛性或光滑性要求。如果不满足B上的条件，则复杂度可以限制为O（h −（1+e）），其中e − 1随着小波消失矩的增加而趋于零。这里h表示相应有限元网格的宽度。所考虑的算子被假设为非负（各向异性）阶，并且允许非标准核κ（·，·）在所有次对角线上都是奇异的。从数学金融等运营商的实际例子给出了一些数值结果。
For a class of anisotropic integrodifferential operators B arising as semigroup generators of Markov processes, we present a sparse tensor product wavelet compression scheme for the Galerkin finite element discretization of the corresponding integrodifferential equations Bu = f on (0, 1) n with possibly large n. Under certain conditions on B, the scheme is of essentially optimal and dimension independent complexity O(h −1 | log h| 2(n−1) ) without corrupting the convergence or smoothness requirements of the original sparse tensor finite element scheme. If the conditions on B are not satisfied, the complexity can be bounded by O(h −(1+e) ), where e � 1 tends to zero with increasing number of the wavelets' vanishing moments. Here h denotes the width of the corresponding finite element mesh. The operators under consideration are assumed to be of non-negative (anisotropic) order and admit a non-standard kernel κ(·, ·) that can be singular on all secondary diagonals. Practical examples of such operators from Mathematical Finance are given and some numerical results are presented.
DOI：
10.1090/s0025-5718-09-02248-0
发表时间：
2009
期刊：
Math. Comput.
影响因子：
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作者：
M. Griebel;S. Knapek
通讯作者：
S. Knapek