Graduate Students Seminar

Speaker 1: Alex Sheranko

Title

Multigrid Preconditioner for Optimal Control of the Heat Equation

Abstract

Inverse problems for differential equations arise across many fields and can often be reformulated as optimal control problems. This presentation addresses the inversion of the stationary heat equation to recover the spatially varying thermal conductivity. Our research develops a multigrid-based preconditioner that is used to accelerate the application of the Gauss-Newton method. The PDE is discretized using the Finite Element method and the multigrid method analyzes the problem on coarser mesh grids. The proposed preconditioner exhibits optimal-order performance and substantially reduces the number of Conjugate Gradient iterations required at each Newton step.

Speaker 2: Yuxin Zhang

Title

Adaptive Neyman Allocation

Abstract

Neyman allocation proposed an unequal allocation method based on stratum size and the within-stratum standard deviation to minimize the variance of the stratified sample mean. In practice, however, such prior knowledge for the population variance is often unavailable, which limits the direct applicability of this approach. To overcome this issue, we introduce a sequential allocation algorithm that does not rely on prior variance information. The procedure begins by allocating two initial units to each stratum. At each subsequent step, one additional unit is assigned to the stratum that produces the lowest estimated total variance. This adaptive rule continues until the overall sample size is reached. This sequential allocation method enables our allocation decision to incorporate the most recent insights into stratum variability. Additionally, this procedure also strikes a balance between exploration and exploitation. Under the unknown variance setting, sequential allocation allows for updating the estimator continually, which makes the estimator more reliable. Numerical experiments indicate that the proposed method closely approximates the efficiency of Neyman allocation when variances are known, and provides a practical and robust alternative when variances must be learned from the data.