Interior Point Methods and Preconditioning for PDE-Constrained Optimization Problems Involving Sparsity Terms

PDE-constrained optimization problems with control or state constraints are challenging from an analytical as well as numerical perspective. The combination of these constraints with a sparsity-promoting $\rm L^1$ term within the objective function requires sophisticated optimization methods. We propose the use of an Interior Point scheme applied to a smoothed reformulation of the discretized problem, and illustrate that such a scheme exhibits robust performance with respect to parameter changes. To increase the potency of this method we introduce fast and efficient preconditioners which enable us to solve problems from a number of PDE applications in low iteration numbers and CPU times, even when the parameters involved are altered dramatically

Refined saddle-point preconditioners for discretized Stokes problems

This paper is concerned with the implementation of efficient solution algorithms for elliptic problems with constraints. We establish theory which shows that including a simple scaling within well-established block diagonal preconditioners for Stokes problems can result in significantly faster convergence when applying the preconditioned MINRES method. The codes used in the numerical studies are available online

Fourier Method for Approximating Eigenvalues of Indefinite Stekloff Operator

We introduce an efficient method for computing the Stekloff eigenvalues associated with the Helmholtz equation. In general, this eigenvalue problem requires solving the Helmholtz equation with Dirichlet and/or Neumann boundary condition repeatedly. We propose solving the related constant coefficient Helmholtz equation with Fast Fourier Transform (FFT) based on carefully designed extensions and restrictions of the equation. The proposed Fourier method, combined with proper eigensolver, results in an efficient and clear approach for computing the Stekloff eigenvalues.Comment: 12 pages, 4 figure

Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations

The Navier--Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretization. We analyse the semi-discrete equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index and quantify the numerical difficulties in the fully discrete schemes, that are induced by the strangeness of the system. By analyzing the Kronecker index of the difference-algebraic equations, that represent commonly and successfully used time stepping schemes for the Navier--Stokes equations, we show that those time-integration schemes factually remove the strangeness. The theoretical considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909, https://doi.org/10.5281/zenodo.99890

Effects of perceived cocaine availability on subjective and objective responses to the drug

<p>Abstract</p> <p>Rationale</p> <p>Several lines of evidence suggest that cocaine expectancy and craving are two related phenomena. The present study assessed this potential link by contrasting reactions to varying degrees of the drug's perceived availability.</p> <p>Method</p> <p>Non-treatment seeking individuals with cocaine dependence were administered an intravenous bolus of cocaine (0.2 mg/kg) under 100% ('unblinded'; N = 33) and 33% ('blinded'; N = 12) probability conditions for the delivery of drug. Subjective ratings of craving, high, rush and low along with heart rate and blood pressure measurements were collected at baseline and every minute for 20 minutes following the infusions.</p> <p>Results</p> <p>Compared to the 'blinded' subjects, their 'unblinded' counterparts had similar craving scores on a multidimensional assessment several hours before the infusion, but reported higher craving levels on a more proximal evaluation, immediately prior to the receipt of cocaine. Furthermore, the 'unblinded' subjects displayed a more rapid onset of high and rush cocaine responses along with significantly higher cocaine-induced heart rate elevations.</p> <p>Conclusion</p> <p>These results support the hypothesis that cocaine expectancy modulates subjective and objective responses to the drug. Provided the important public health policy implications of heavy cocaine use, health policy makers and clinicians alike may favor cocaine craving assessments performed in the settings with access to the drug rather than in more neutral environments as a more meaningful marker of disease staging and assignment to the proper level of care.</p

Fast interior point solution of quadratic programming problems arising from PDE-constrained optimization

Interior point methods provide an attractive class of approaches for solving linear, quadratic and nonlinear programming problems, due to their excellent efficiency and wide applicability. In this paper, we consider PDE-constrained optimization problems with bound constraints on the state and control variables, and their representation on the discrete level as quadratic programming problems. To tackle complex problems and achieve high accuracy in the solution, one is required to solve matrix systems of huge scale resulting from Newton iteration, and hence fast and robust methods for these systems are required. We present preconditioned iterative techniques for solving a number of these problems using Krylov subspace methods, considering in what circumstances one may predict rapid convergence of the solvers in theory, as well as the solutions observed from practical computations

Multilayer Modelling of Lubricated Contacts: A New Approach Based on a Potential Field Description

A first integral approach, derived in an analogous fashion to Maxwell’s use of potential fields, is employed to investigate the flow characteristics, with a view to minimising friction, of shear-driven fluid motion between rigid surfaces in parallel alignment as a model for a lubricated joint, whether naturally occurring or engineered replacement. For a viscous bilayer arrangement comprised of immiscible liquids, it is shown how the flow and the shear stress along the separating interface is influenced by the mean thickness of the layers and the ratio of their respective viscosities. Considered in addition, is how the method can be extended for application to the more challenging problem of when one, or both, of the layers is a viscoelastic material

Matching Schur complement approximations for certain saddle-point systems

The solution of many practical problems described by mathematical models requires approximation methods that give rise to linear(ized) systems of equations, solving which will determine the desired approximation. This short contribution describes a particularly effective solution approach for a certain class of so-called saddle-point linear systems which arises in different contexts