Roland Pulch

Stochastic collocation and stochastic Galerkin methods for linear differential algebraic equations

We consider time-invariant linear systems of differential algebraic equations, which include physical parameters or other parameters. Uncertainties of the parameters are modelled by random variables. We expand the corresponding random-dependent solutions in the polynomial chaos. Approximations of unknown coefficient functions can be obtained by quadrature or sampling schemes. Alternatively, stochastic collocation methods or the stochastic Galerkin approach yield larger coupled systems of differential algebraic equations. We show the equivalence of these types of numerical methods under certain assumptions. The index of the coupled systems is analysed in comparison to the original systems. Sufficient conditions for an identical index are derived. Furthermore, we present results of numerical simulations for an example.

Polynomial Chaos for the Computation of Failure Probabilities in Periodic Problems

Numerical simulation of electric circuits uses systems of differential algebraic equations (DAEs) in general. We examine forced oscillators, where the DAE models involve periodic solutions. Uncertainties in physical parameters can be described by random variables. We apply the strategy of the generalised polynomial chaos (gPC) to resolve the stochastic model. In particular, failure probabilities are determined using the approximation from gPC. We present results of numerical simulations for a system of DAEs modelling a Schmitt trigger.

Trajectory piecewise linear approach for nonlinear differential-algebraic equations in circuit simulation

In this paper we extend the Trajectory Piecewise Linear (TPWL) model order reduction (MOR) method for nonlinear differential algebraic equations (DAE). The TPWL method is based on combining several linear reduced models at different time points, which are created along a typical trajectory, to approximate the full nonlinear model. We discuss how to select the linearization tuples for linearization and the choice of linear MOR method. Then we study how to combine the local linearized reduced systems to create a global TPWL model. Finally, we show a numerical result.

Modelling and simulation of autonomous oscillators with random parameters

Abstract: We consider periodic problems of autonomous systems of ordinary differential equations or differential algebraic equations. To quantify uncertainties of physical parameters, we introduce random variables in the systems. Phase conditions are required to compute the resulting periodic random process. It follows that the variance of the process depends on the choice of the phase condition. We derive a necessary condition for a random process with a minimal total variance by the calculus of variations. A corresponding numerical method is constructed based on the generalised polynomial chaos. We present numerical simulations of two test examples.

Polynomial chaos for simulating random volatilities

In financial mathematics, the fair price of options can be achieved by solutions of parabolic differential equations. The volatility usually enters the model as a constant parameter. However, since this constant has to be estimated with respect to the underlying market, it makes sense to replace the volatility by an according random variable. Consequently, a differential equation with stochastic input occurs, whose solution determines the fair price in the refined model. Corresponding expected values and variances can be computed approximately via a Monte Carlo method. Alternatively, the generalised polynomial chaos yields an efficient approach for calculating the required data. Based on a parabolic equation modelling the fair price of Asian options, the technique is developed and corresponding numerical simulations are presented.

Variational Methods for Solving Warped Multirate Partial Differential Algebraic Equations

Signals exhibiting amplitude as well as frequency modulation at widely separated time scales arise in radio frequency (RF) applications. A multivariate model yields an adequate representation by decoupling the time scales of involved signals. Consequently, a system of differential algebraic equations (DAEs) modeling the electric circuit changes into a system of partial differential algebraic equations (PDAEs). The determination of an emerging local frequency function is crucial for the efficiency of this approach, since inappropriate choices produce many oscillations in the multivariate solution. Thus, the idea is to reduce oscillating behavior via minimizing the magnitude of partial derivatives. For this purpose, we apply variational calculus to obtain a necessary condition for a specific solution, which represents a minimum of an according functional. This condition can be included in numerical schemes computing the complete solution of the PDAE. Test results confirm that the used strategy ensures an efficient simulation of RF signals.

Initial-boundary value problems of warped MPDAEs including minimisation criteria

Electric circuits, which produce oscillations at widely separated time scales, cause a huge computational effort in a numerical simulation of the mathematical model based on differential-algebraic equations (DAEs). Alternatively, a multidimensional signal model yields a description via multirate partial differential-algebraic equations (MPDAEs). Initial-boundary value problems of the MPDAE system reproduce solutions of the underlying DAE system. In case of frequency modulation, an additional function occurs in the MPDAE model, which represents a degree of freedom in the multivariate description of the signals. We present two minimisation strategies, which are able to identify the additional parameters such that the resulting solutions exhibit an elementary structure. Thus numerical schemes can apply relatively coarse grids and an efficient simulation is achieved.

Uncertainty Quantification for Robust Topology Optimization of Power Transistor Devices

In this paper, we focus on incorporating a stochastic collocation method (SCM) into a topological shape optimization of a power semiconductor device, including material and geometrical uncertainties. This results in a stochastic direct problem and, in consequence, affects the formulation of an optimization problem. In particular, our aim is to minimize the current density overshoots, since the change of the shape and topology of a device layout is the proven technique for the reduction of a hotspot area. The gradient of a stochastic cost functional is evaluated using the topological asymptotic expansion and the continuous design sensitivity analysis with the SCM. Finally, we show the results of the robust optimization for the power transistor device, which is an example of a relevant problem in nanoelectronics, but which is also widely used in the automotive industry.

Nonlinear Model Order Reduction Based on Trajectory Piecewise Linear Approach: Comparing Different Linear Cores

Refined models for MOS-devices and increasing complexity of circuit designs cause the need for Model Order Reduction (MOR) techniques that are capable of treating nonlinear problems. In time-domain simulation the Trajectory PieceWise Linear (TPWL) approach is promising as it is designed to use MOR methodologies for linear problems as the core of the reduction process. We compare different linear approaches with respect to their performance when used as kernel for TPWL.

Multirate partial differential algebraic equations for simulating radio frequency signals

In radio frequency (RF) applications, electric circuits produce signals exhibiting fast oscillations, whereas the amplitude and frequency may change slowly in time. Thus solving a system of differential algebraic equations (DAEs), which describes the circuit’s transient behaviour, becomes inefficient, since the fast rate restricts the step sizes in time. A multivariate model is able to decouple the widely separated time scales of RF signals and thus provides an alternative approach. Consequently, a system of DAEs changes into a system of multirate partial differential algebraic equations (MPDAEs). The determination of multivariate solutions allows the exact reconstruction of corresponding time-dependent signals. Hence an efficient numerical simulation is obtained by exploiting the periodicities in fast time scales. We outline the theory of this multivariate approach with respect to the simulation of amplitude as well as frequency modulated signals. Furthermore, a survey of numerical methods for solving the arising problems of MPDAEs is given.