A practical factorization of a Schur complement for PDE-constrained Distributed Optimal Control

A distributed optimal control problem with the constraint of a linear elliptic partial differential equation is considered. A necessary optimality condition for this problem forms a saddle point system, the efficient and accurate solution of which is crucial. A new factorization of the Schur complement for such a system is proposed and its characteristics discussed. The factorization introduces two complex factors that are complex conjugate to each other. The proposed solution methodology involves the application of a parallel linear domain decomposition solver---FETI-DPH---for the solution of the subproblems with the complex factors. Numerical properties of FETI-DPH in this context are demonstrated, including numerical and parallel scalability and regularization dependence. The new factorization can be used to solve Schur complement systems arising in both range-space and full-space formulations. In both cases, numerical results indicate that the complex factorization is promising

Full-waveform inversion in three-dimensional PML-truncated elastic media

We are concerned with high-fidelity subsurface imaging of the soil, which commonly arises in geotechnical site characterization and geophysical explorations. Specifically, we attempt to image the spatial distribution of the Lame parameters in semi-infinite, three-dimensional, arbitrarily heterogeneous formations, using surficial measurements of the soil's response to probing elastic waves. We use the complete waveform response of the medium to derive the inverse problem, by using a partial-differential-equation (PDE)-constrained optimization approach, directly in the time-domain, to minimize the misfit between the observed response of the medium at select measurement locations, and a computed response corresponding to a trial distribution of the Lame parameters. We discuss strategies that lend algorithmic robustness to our proposed inversion scheme. To limit the computational domain to the size of interest, we employ perfectly-matched-layers (PMLs). In order to resolve the forward problem, we use a recently developed hybrid finite element approach, where a displacement-stress formulation for the PML is coupled to a standard displacement-only formulation for the interior domain, thus leading to a computationally cost-efficient scheme. Time-integration is accomplished by using an explicit Runge-Kutta scheme, which is well-suited for large-scale problems on parallel computers. We verify the accuracy of the material gradients obtained via our proposed scheme, and report numerical results demonstrating successful reconstruction of the two Lame parameters for both smooth and sharp profiles.Comment: Submitted Journal Pape

Space-time least-squares Petrov-Galerkin projection for nonlinear model reduction

This work proposes a space-time least-squares Petrov-Galerkin (ST-LSPG) projection method for model reduction of nonlinear dynamical systems. In contrast to typical nonlinear model-reduction methods that first apply (Petrov-)Galerkin projection in the spatial dimension and subsequently apply time integration to numerically resolve the resulting low-dimensional dynamical system, the proposed method applies projection in space and time simultaneously. To accomplish this, the method first introduces a low-dimensional space-time trial subspace, which can be obtained by computing tensor decompositions of state-snapshot data. The method then computes discrete-optimal approximations in this space-time trial subspace by minimizing the residual arising after time discretization over all space and time in a weighted $\ell^2$-norm. This norm can be defined to enable complexity reduction (i.e., hyper-reduction) in time, which leads to space-time collocation and space-time GNAT variants of the ST-LSPG method. Advantages of the approach relative to typical spatial-projection-based nonlinear model reduction methods such as Galerkin projection and least-squares Petrov-Galerkin projection include: (1) a reduction of both the spatial and temporal dimensions of the dynamical system, (2) the removal of spurious temporal modes (e.g., unstable growth) from the state space, and (3) error bounds that exhibit slower growth in time. Numerical examples performed on model problems in fluid dynamics demonstrate the ability of the method to generate orders-of-magnitude computational savings relative to spatial-projection-based reduced-order models without sacrificing accuracy.Comment: Accepted to the SIAM Journal on Scientific Computin

Accelerating design optimization using reduced order models

Although design optimization has shown its great power of automatizing the whole design process and providing an optimal design, using sophisticated computational models, its process can be formidable due to a computationally expensive large-scale linear system of equations to solve, associated with underlying physics models. We introduce a general reduced order model-based design optimization acceleration approach that is applicable not only to design optimization problems, but also to any PDE-constrained optimization problems. The acceleration is achieved by two techniques: i) allowing an inexact linear solve and ii) reducing the number of iterations in Krylov subspace iterative methods. The choice between two techniques are made, based on how close a current design point to an optimal point. The advantage of the acceleration approach is demonstrated in topology optimization examples, including both compliance minimization and stress-constrained problems, where it achieves a tremendous reduction and speed-up when a traditional preconditioner fails to achieve a considerable reduction in the number of linear solve iterations.Comment: 20 pages, 5 figures, 5 tables, 5 algorithm

Efficient space-time reduced order model for linear dynamical systems in Python using less than 120 lines of code

A classical reduced order model (ROM) for dynamical problems typically involves only the spatial reduction of a given problem. Recently, a novel space-time ROM for linear dynamical problems has been developed, which further reduces the problem size by introducing a temporal reduction in addition to a spatial reduction without much loss in accuracy. The authors show an order of a thousand speed-up with a relative error of less than 0.00001 for a large-scale Boltzmann transport problem. In this work, we present for the first time the derivation of the space-time Petrov-Galerkin projection for linear dynamical systems and its corresponding block structures. Utilizing these block structures, we demonstrate the ease of construction of the space-time ROM method with two model problems: 2D diffusion and 2D convection diffusion, with and without a linear source term. For each problem, we demonstrate the entire process of generating the full order model (FOM) data, constructing the space-time ROM, and predicting the reduced-order solutions, all in less than 120 lines of Python code. We compare our Petrov-Galerkin method with the traditional Galerkin method and show that the space-time ROMs can achieve O(100) speed-ups with O(0.001) to O(0.0001) relative errors for these problems. Finally, we present an error analysis for the space-time Petrov-Galerkin projection and derive an error bound, which shows an improvement compared to traditional spatial Galerkin ROM methods.Comment: 24 pages, 18 figure