Abstract
It has been an open problem to derive a necessary and sufficient condition for a linear tensor product problem S=Sd in the average case setting to be weakly tractable but not polynomially tractable. As a result of the tensor product structure, the eigenvalues of the covariance operator of the induced measure in the one-dimensional problem characterize the complexity of approximating Sd d≥1, with accuracy ε. If ∑j=j=1 ∞λj ≥ 1 and λ2 > 0, we know that S is not polynomially tractable iff lim sup j→∞λjjp =∞ for all p > 1. Thus we settle the open problem by showing that S is weakly tractable iff ∑j>n λj=o(ln-2n). In particular, assume that ℓ=lim/j→∞λjjln 3(j+1), exists. Then S is weakly tractable iff ℓ = 0.
Original language | English (US) |
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Pages (from-to) | 273-280 |
Number of pages | 8 |
Journal | Journal of Complexity |
Volume | 27 |
Issue number | 3-4 |
DOIs | |
State | Published - 2011 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Statistics and Probability
- Numerical Analysis
- General Mathematics
- Control and Optimization
- Applied Mathematics
Keywords
- Complexity
- Multivariate problem
- Tractability