The article introduces a concept of tractability for linear ill-posed operator equations in Hilbert space, which aims to capture the computational difficulty of solving such problems. The key points are:
For ill-posed operator equations Ax = y, the optimal reconstruction rates in terms of the noise level δ are often known, but the relevant question is the level of discretization required to achieve this optimal accuracy.
The authors propose a notion of tractability adapted from Information-based Complexity, which considers the cardinality k* of the discretization required to reach the optimal reconstruction rate.
Several examples are discussed to illustrate the relevance of this concept, particularly in the context of the "curse of dimensionality" where the discretization level k* can depend exponentially on the spatial dimension d, even when the optimal reconstruction rate is dimension-independent.
The authors show that the tractability of the ill-posed inverse problem is equivalent to the tractability of a related family of direct problems, providing a connection to the existing theory of tractability in Information-based Complexity.
For operators with power-type decay of singular values, the authors analyze the impact of the leading constant on tractability, distinguishing cases where the problem is tractable, intractable in the noise level δ, or intractable in the dimension d.
As a specific example, the article studies the tractability of the multivariate integration operator, showing that this class of ill-posed problems is weakly tractable despite the curse of dimensionality.
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소스 콘텐츠 기반
arxiv.org
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