Speaker
Description
The goal underlying our work is to develop provably accurate and data-efficient learning algorithms for non-self-adjoint operators using only input-output pairs. State-of-the-art approximation techniques with fast convergence rates either apply only to self-adjoint operators or require access to the adjoint operator, which is unavailable in experimental settings and often difficult to access computationally. Recent work has shown that provably accurate adjoint-free learning is possible but uses an amount of data that is likely excessive. In this talk, we discuss an adjoint-free learning algorithm for asymptotically smooth integral operators which is based on an extension of the fixed sparsity matrix approximation algorithm [Amsel et al., arXiv, 2024] to the case where a fixed subspace - determined by asymptotic smoothness and the resulting exponentially-convergent separable expansion - is known for each row. The algorithm queries the action of the underlying operator on functions with standard Gaussian random coefficients in a finely discretized finite element space, yielding an approximation of the unknown finite element discretization of the operator. While the number of degrees of freedom in this discretization grows polynomially with the inverse error tolerance, asymptotic smoothness and the hierarchical decomposition of each row ensure that only a polylogarithmic number of queries $\mathcal{O}(\text{polylog}(\varepsilon^{-1}))$ is required to achieve the desired accuracy $\varepsilon$ with high probability.