Wei Gong

Wei Gong, Hehu Xie, Ningning Yan

We propose in this paper a multilevel correction method to solve optimal control problems constrained by elliptic equations with the finite element method. In this scheme, solving optimization problem on the finest finite element space is transformed to a series of solutions of linear boundary value problems by the multigrid method on multilevel meshes and a series of solutions of optimization problems on the coarsest finite element space. Our proposed scheme, instead of solving a large scale optimization problem in the finest finite element space, solves only a series of linear boundary value problems and the optimization problems in a very low dimensional finite element space, and thus can improve the overall efficiency for the solution of optimal control problems governed by PDEs.

Wei Gong, Ningning Yan, Zhaojie Zhou

This paper aims to study the convergence of adaptive finite element method for control constrained elliptic optimal control problems under $L^2$-norm. We prove the contraction property and quasi-optimal complexity for the $L^2$-norm errors of both the control, the state and adjoint state variables with $L^2$-norm based AFEM, this is in contrast to and improve our previous work [13] where convergence of AFEM based on energy norm had been studied and suboptimal convergence for the control variable was obtained and observed numerically. For the discretization we use variational discretization for the control and piecewise linear and continuous finite elements for the state and adjoint state. Under mild assumptions on the initial mesh and the mesh refinement algorithm to keep the adaptive meshes sufficiently mildly graded we prove the optimal convergence of AFEM for the control problems, numerical results are provided to support our theoretical findings.

Echo Diyun LU, Charles Findling, Marianne Clausel et al.

Regime transitions routinely break stationarity in time series, making calibrated uncertainty as important as point accuracy. We study distribution-free uncertainty for regime-switching forecasting by coupling Deep Switching State Space Models with Adaptive Conformal Inference (ACI) and its aggregated variant (AgACI). We also introduce a unified conformal wrapper that sits atop strong sequence baselines including S4, MC-Dropout GRU, sparse Gaussian processes, and a change-point local model to produce online predictive bands with finite-sample marginal guarantees under nonstationarity and model misspecification. Across synthetic and real datasets, conformalized forecasters achieve near-nominal coverage with competitive accuracy and generally improved band efficiency.