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

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

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

Suicide risk in schizophrenia: learning from the past to change the future

Suicide is a major cause of death among patients with schizophrenia. Research indicates that at least 5–13% of schizophrenic patients die by suicide, and it is likely that the higher end of range is the most accurate estimate. There is almost total agreement that the schizophrenic patient who is more likely to commit suicide is young, male, white and never married, with good premorbid function, post-psychotic depression and a history of substance abuse and suicide attempts. Hopelessness, social isolation, hospitalization, deteriorating health after a high level of premorbid functioning, recent loss or rejection, limited external support, and family stress or instability are risk factors for suicide in patients with schizophrenia. Suicidal schizophrenics usually fear further mental deterioration, and they experience either excessive treatment dependence or loss of faith in treatment. Awareness of illness has been reported as a major issue among suicidal schizophrenic patients, yet some researchers argue that insight into the illness does not increase suicide risk. Protective factors play also an important role in assessing suicide risk and should also be carefully evaluated. The neurobiological perspective offers a new approach for understanding self-destructive behavior among patients with schizophrenia and may improve the accuracy of screening schizophrenics for suicide. Although, there is general consensus on the risk factors, accurate knowledge as well as early recognition of patients at risk is still lacking in everyday clinical practice. Better knowledge may help clinicians and caretakers to implement preventive measures. This review paper is the results of a joint effort between researchers in the field of suicide in schizophrenia. Each expert provided a brief essay on one specific aspect of the problem. This is the first attempt to present a consensus report as well as the development of a set of guidelines for reducing suicide risk among schizophenia patients