In large-scale applications including medical imaging, collocation differential equation solvers, and estimation with differential privacy, the underlying linear inverse problem can be reformulated as a streaming problem. In theory, the streaming problem can be effectively solved using memory-efficient, exponentially-converging streaming solvers. In practice, a streaming solver's effectiveness is undermined if it is stopped before, or well-after, the desired accuracy is achieved. In special cases when the underlying linear inverse problem is finite-dimensional, streaming solvers can periodically evaluate the residual norm at a substantial computational cost. When the underlying system is infinite dimensional, streaming solver can only access noisy estimates of the residual. While such noisy estimates are computationally efficient, they are useful only when their accuracy is known. In this work, we rigorously develop a general family of computationally-practical residual estimators and their uncertainty sets for streaming solvers, and we demonstrate the accuracy of our methods on a number of large-scale linear problems. Thus, we further enable the practical use of streaming solvers for important classes of linear inverse problems.

翻译：在大规模应用（包括医学成像、配置法微分方程求解器以及差分隐私估计）中，底层线性逆问题可重新表述为流式问题。理论上，流式问题可通过内存高效且指数收敛的流式求解器有效求解。实践中，若流式求解器在达到期望精度之前或之后被终止，其有效性将受到削弱。在底层线性逆问题为有限维的特殊情形下，流式求解器能以显著计算成本定期评估残差范数。当底层系统为无限维时，流式求解器只能访问含噪声的残差估计值——此类估计虽计算高效，但仅在已知其精度时才有实际价值。本研究严谨构建了适用于流式求解器的一般性实用残差估计器族及其不确定集，并在多个大规模线性问题上验证了方法的准确性。由此，我们进一步推动了流式求解器在关键线性逆问题类别中的实用化进程。