This paper presents an overview and comparison of iterative solvers for linear stochastic partial differential equations (PDEs). A stochastic Galerkin finite element discretization is applied to transform the PDE into a coupled set of deterministic PDEs. Specialized solvers are required to solve the very high-dimensional systems that result after a finite element discretization of the resulting set. This paper discusses one-level iterative methods, based on matrix splitting techniques; multigrid methods, which apply a coarsening in the spatial dimension; and multilevel methods, which make use of the hierarchical structure of the stochastic discretization. Also Krylov solvers with suitable preconditioning are addressed. A local Fourier analysis provides quantitative convergence properties. The efficiency and robustness of the methods are illustrated on two nontrivial numerical problems. The multigrid solver with block smoother yields the most robust convergence properties, though a cheaper point smoother performs as well in most cases. Multilevel methods based on coarsening the stochastic dimension perform in general poorly due to a large computational cost per iteration. Moderate size problems can be solved very quickly by a Krylov method with a mean-based preconditioner. For larger spatial and stochastic discretizations, however, this approach suffers from its nonoptimal convergence properties.

The optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed. The Lagrangian that defines the problem is postulated in terms of stochastic functions, with the control function possibly decomposed into an unknown deterministic component and a known zero-mean stochastic component. The extra freedom provided by the stochastic dimension in defining cost functionals is explored, demonstrating the scope for controlling statistical aspects of the system response. One-shot stochastic finite element methods are used to find approximate solutions to control problems. It is shown that applying the stochastic collocation finite element to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional, thereby obviating the primary advantage of the collocation method over the stochastic Galerkin method for the considered problem. The application of the presented methods is demonstrated through a number of numerical examples. The presented framework is sufficiently general to also consider a class of inverse problems, and numerical examples of this type are also presented

We consider the numerical solution of time-dependent partial differential equations (PDEs) with random coefficients. A spectral approach, called stochastic finite element method, is used to compute the statistical characteristics of the solution. This method transforms a stochastic PDE into a coupled system of deterministic equations by means of a Galerkin projection onto a generalized polynomial chaos. An algebraic multigrid (AMG) method is presented to solve the algebraic systems that result after discretization of this coupled system. High-order time integration schemes of an implicit Runge-Kutta type and spatial discretization on unstructured finite element meshes are considered. The convergence properties of the AMG method are demonstrated by a convergence analysis and by numerical tests.

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Whereas the introduction of 3D-dimensional devices such as FINFET's may be a solution for next generation technologies, they do represent significant challenges with respect to the doping strategies and the junction characterization.Aiming at a conformal doping of the source/drain regions in a FINFET in order to induce a conformal under diffusion and homogenous device operation, one can quickly recognize that classical beam implants fail to fulfill these needs, in particular when considering closely spaced fin's. Indeed the effects of different implant angles (top vs bottom) and the concurrent variation in projected range, dose retention and sputtering as well as the effect of the wafer rotation when tilting is used, all lead to a non-conformal doping. Alternative processes such as vapor phase deposition (VPD) or plasma doping are presently being considered, as they hold the promise of conformality. Using VPD or Atomic Layer Doping dopant atoms are deposited on the surface through thermal decomposition of typical chemical vapor deposition precursors and are subsequently in diffused. Good conformality (~ 93 % for sidewall vs. top dose), defect free junctions and high activation levels are the positive points of this process. Plasma immersion doping is an alternative approach which is easier to integrate (similar to ion implantation) and suitable for p-and n-type doping. Whereas it holds the promise of conformality when implanting large macroscopic features, the latter is far less obvious when trying to dope microscopic feature conformally. In fact the formation of conformal junctions in FINFET's with plasma based processes is quite challenging and relies on secondary processes such as resputtering, deposition and in diffusion etc. Their optimization is compromised by concurrent artifacts, sputter erosion being the most important one. In support of these developments the measurement and identification of conformality adequate metrology is required. For this purpose we have extensively used Scanning Spreading Resistance Microscopy (SSRM) as a means to characterize the vertical/lateral junction depths, the concentration levels and the degree of conformality. Characterization of the (3D)-underdiffusion can be achieved by a dedicated SSRM experiment and/or the Tomographic Atomprobe. As a complement to the SSRM technique we also developed a concept based on resistance measurements of fin's which allows to map the sidewall doping across the wafers and provides fast feedback on conformality.

The stochastic Galerkin and stochastic collocation method are two state-of-the-art methods for solving partial differential equations (PDE) containing random coefficients. While the latter method, which is based on sampling, can straightforwardly be applied to nonlinear stochastic PDEs, this is nontrivial for the stochastic Galerkin method and approximations are required. In this paper, both methods are used for constructing high-order solutions of a nonlinear stochastic PDE representing the magnetic vector potential in a ferromagnetic rotating cylinder. This model can be used for designing solid-rotor induction machines in various machining tools. A precise design requires to take ferromagnetic saturation effects into account and uncertainty on the nonlinear magnetic material properties. A numerical comparison of the computational cost and accuracy of both methods is performed. The stochastic Galerkin method requires in general less stochastic unknowns than the stochastic collocation approach to reach a certain level of accuracy, however at a higher computational cost. AbstractThe stochastic Galerkin and stochastic collocation method are two state-of-the-art methods for solving partial differential equations (PDE) containing random coefficients. While the latter method, which is based on sampling, can straightforwardly be applied to nonlinear stochastic PDEs, this is nontrivial for the stochastic Galerkin method and approximations are required. In this paper, both methods are used for constructing high-order solutions of a nonlinear stochastic PDE representing the magnetic vector potential in a ferromagnetic rotating cylinder. This model can be used for designing solid-rotor induction machines in various machining tools. A precise design requires to take ferromagnetic saturation effects into account and uncertainty on the nonlinear magnetic material properties. A numerical comparison of the computational cost and accuracy of both methods is performed. The stochastic Galerkin method requires in general less stochastic unknowns than the stochastic collocation approach to reach a certain level of accuracy, however at a higher computational cost.

This paper addresses the influence of uncertainty on ferromagnetic material properties on the torque yielded by ferromagnetic cylinders rotating at high speeds. To that end, two state-of-the-art high-order stochastic solution approaches, the stochastic collocation and stochastic Galerkin method, are applied to a nonlinear convection-diffusion problem with random coefficients and their performance is compared.

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