Harbir Antil

Domain decomposition and balanced truncation model reduction for shape optimization of the Stokes system

The optimal design of structures and systems described by partial differential equations (PDEs) often gives rise to large-scale optimization problems, in particular if the underlying system of PDEs represents a multi-scale, multi-physics problem. Therefore, reduced-order modelling techniques such as balanced truncation model reduction (BTMR), proper orthogonal decomposition, or reduced basis methods are used to significantly decrease the computational complexity while maintaining the desired accuracy of the approximation. In this paper, we are interested in such shape optimization problems where the design issue is restricted to a relatively small portion of the computational domain. In this case, it appears to be natural to rely on a full-order model only in that specific part of the domain and to use a reduced-order model elsewhere. A convenient methodology to realize this idea consists of a suitable combination of domain decomposition techniques and BTMR. We will consider such an approach for shape optimization problems associated with the time-dependent Stokes system and derive explicit error bounds for the modelling error. As an application in life sciences, we will be concerned with the optimal design of capillary barriers as part of a network of microchannels and reservoirs on microfluidic biochips that are used in clinical diagnostics, pharmacology, and forensics for high-throughput screening and hybridization in genomics and protein profiling in proteomics.

A FEM for an Optimal Control Problem of Fractional Powers of Elliptic Operators

We study solution techniques for a linear-quadratic optimal control problem involving fractional powers of elliptic operators. These fractional operators can be realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. Thus, we consider an equivalent formulation with a nonuniformly elliptic operator as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We discretize the proposed truncated state equation using first degree tensor product finite elements on anisotropic meshes. For the control problem we analyze two approaches: one that is semi-discrete based on the so-called variational approach, where the control is not discretized, and the other one is fully discrete via the discretization of the control by piecewise constant functions. For both approaches, we derive a priori error estimates with respect to the degrees of freedom. Numerical experiments validate the derived error estimates and reveal a competitive performance of anisotropic over quasi-uniform refinement.

Application of the Discrete Empirical Interpolation Method to Reduced Order Modeling of Nonlinear and Parametric Systems

Projection based methods lead to reduced order models (ROMs) with dramatically reduced numbers of equations and unknowns. However, for nonlinear or parametrically varying problems the cost of evaluating these ROMs still depends on the size of the full order model and therefore is still expensive. The Discrete Empirical Interpolation Method (DEIM) further approximates the nonlinearity in the projection based ROM. The resulting DEIM ROM nonlinearity depends only on a few components of the original nonlinearity. If each component of the original nonlinearity depends only on a few components of the argument, the resulting DEIM ROM can be evaluated efficiently at a cost that is independent of the size of the original problem. For systems obtained from finite difference approximations, the ith component of the original nonlinearity often depends only on the ith component of the argument. This is different for systems obtained using finite element methods, where the dependence is determined by the mesh and by the polynomial degree of the finite element subspaces. This paper describes two approaches of applying DEIM in the finite element context, one applied to the assembled and the other to the unassembled form of the nonlinearity. We carefully examine how the DEIM is applied in each case, and the substantial efficiency gains obtained by the DEIM. In addition, we demonstrate how to apply DEIM to obtain ROMs for a class of parameterized system that arises, e.g., in shape optimization. The evaluations of the DEIM ROMs are substantially faster than those of the standard projection based ROMs. Additional gains are obtained with the DEIM ROMs when one has to compute derivatives of the model with respect to the parameter.

A Space-Time Fractional Optimal Control Problem: Analysis and Discretization

We study a linear-quadratic optimal control problem involving a parabolic equation with fractional diffusion and Caputo fractional time derivative of orders

Two-Step Greedy Algorithm for Reduced Order Quadratures

s \in (0,1)

Modeling and Simulation of Piezoelectrically Agitated Acoustic Streaming on Microfluidic Biochips

and

Adaptive Path Following Primal Dual Interior Point Methods for Shape Optimization of Linear and Nonlinear Stokes Flow Problems

\gamma \in (0,1]

Path-following primal-dual interior-point methods for shape optimization of stationary flow problems

, respectively. The spatial fractional diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic operator. Thus, we consider an equivalent formulation with a quasi-stationary elliptic problem with a dynamic boundary condition as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We consider a fully discrete scheme: piecewise constant functions for the control and, for the state, first-degree tensor product finite elements in space and a finite difference discretization in time. We show convergence of this scheme and, under additional data regularity, derive a priori error estimates for the case

Optimal Control of a Free Boundary Problem with Surface Tension Effects: A Priori Error Analysis

s \in (0,1)

Adaptive Multilevel Interior-Point Methods in PDE Constrained Optimization

and