Tractability Analysis of Multivariate Approximation Problems

Tractability analysis addresses the fundamental challenge of approximating multivariate functions as both the number of variables and the desired accuracy increase. In high dimensions, naive algorithms often suffer from the curse of dimensionality, with computational cost growing exponentially. To overcome this, researchers introduce weight structures that reflect the relative importance of coordinate directions and develop complexity frameworks categorising how information cost scales with problem dimension and error tolerance. Central notions include strong polynomial, polynomial and weak tractability, as well as refined (s,t)-weak tractability, each describing specific bounds on growth rates. Investigations span worst-case and average-case error criteria, and cover L2-approximation and numerical integration in weighted Hilbert, Korobov and Hermite spaces. Advances have yielded necessary and sufficient weight conditions that guarantee various convergence regimes, guiding the design of efficient algorithms for uncertainty quantification, high-dimensional modelling and machine-learning applications.

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Recent studies have advanced the understanding of exponential convergence in weighted Hilbert spaces by establishing necessary and sufficient conditions on coordinate-dependent weight sequences to achieve exponential convergence-weak tractability in the worst-case L2-approximation setting. Complementary work has investigated the average-case L2-approximation problem in Banach spaces equipped with weighted Gaussian covariance kernels, deriving matching conditions on parameter sequences to ensure (s,t)-weak tractability under both absolute and normalized error criteria. Another line of enquiry has compared multiple families of weighted Hermite spaces of finite smoothness and provided characterisations of strong polynomial, polynomial and weak tractability for L2-approximation and numerical integration, demonstrating how the rate of weight decay directly influences the growth of information complexity as dimensionality and precision requirements grow.

Tractability Analysis of Multivariate Approximation Problems publication trend

The graph below shows the total number of articles in tractability analysis of multivariate approximation problems across all publications each year (not limited to Nature Index journals).

Technical terms

Tractability: Study of how computational cost scales with problem dimension and error tolerance in approximation tasks.

Polynomial tractability: Existence of constants ensuring complexity grows at most polynomially in dimension and inverse error tolerance.

Strong polynomial tractability: Complexity bound independent of dimension, growing only polynomially in inverse error tolerance.

Weak tractability: Absence of exponential dependence on dimension or inverse error tolerance in complexity growth.

(s,t)-weak tractability: Condition that complexity grows slower than any polynomial in d^t and ε^−s for specified positive s and t.

Weighted Hilbert space: Function space equipped with coordinate-specific weights modelling varying input importance.

Worst-case setting: Error measured as the maximum deviation over all functions in the prescribed space.

Average-case setting: Error measured as the expected deviation with respect to a given probability distribution.

References

1. Exponential Convergence-(t,s)-Weak Tractability of Approximation in Weighted Hilbert Spaces. Mathematics (2024).
2. Average Case (s, t)-Weak Tractability of L2-Approximation with Weighted Covariance Kernels. Symmetry (2024).
3. Tractability of L 2-approximation and integration in weighted Hermite spaces of finite smoothness. Journal of Complexity (2023).