Slide 3
Slide 3 text
Introduction Algorithm Optimality Tractability Alternate Cone Summary References
Context
Linear solution operator SOL : F → G, input space F contains functions defined on Ω ⊆ Rd
e.g., SOL(f) = f, −∇2 SOL(f) = f, SOL(f) = 0 on boundary
Approximation APP(f, n) =
n
i=1
Li
(f)gi
, can sample linear functionals
SOL(f) − APP(f, n)
G
SOL − APP(·, n) F→G
f
F
Algorithm ALG(f, ε) = APP f, n∗(f, ε) satisfying SOL(f) − ALG(f, ε)
G
ε for all f ∈ C ⊂ F
Solvability: what C? how to determine n∗(f, ε) ∈ N?
n∗(f, ε) = min{n : SOL − APP(·, n)
F→G
ε/R} if f ∈ of radius R
Alternatively, assume f ∈ and bound f
F
or equivalent
What you do not see is not much worse than what you see?
H., F. J., Jiménez Rugama, & Li, D. in Contemporary Computational Mathematics — a celebration of the 80th birthday of Ian
Sloan (eds Dick, J., Kuo, F. Y. & Woźniakowski, H.) 597–619 (Springer-Verlag, 2018), Kunsch, R. J., Novak, E. & Rudolf, D.
Solvable Integration Problems and Optimal Sample Size Selection. Journal of Complexity. To appear (2018), Ding, Y., H., F. J. &
Jiménez Rugama, An Adaptive Algorithm Employing Continuous Linear Functionals. in Monte Carlo and Quasi-Monte Carlo
Methods: MCQMC, Rennes, France, July 2018 (eds L’Ecuyer, P. & Tuffin, B.) Under revision (Springer-Verlag, Berlin, 2019+). 2/14