1,624 research outputs found
Reduced basis method for computational lithography
A bottleneck for computational lithography and optical metrology are long
computational times for near field simulations. For design, optimization, and
inverse scatterometry usually the same basic layout has to be simulated
multiple times for different values of geometrical parameters. The reduced
basis method allows to split up the solution process of a parameterized model
into an expensive offline and a cheap online part. After constructing the
reduced basis offline, the reduced model can be solved online very fast in the
order of seconds or below. Error estimators assure the reliability of the
reduced basis solution and are used for self adaptive construction of the
reduced system. We explain the idea of reduced basis and use the finite element
solver JCMsuite constructing the reduced basis system. We present a 3D
optimization application from optical proximity correction (OPC).Comment: BACUS Photomask Technology 200
Reduced basis method for source mask optimization
Image modeling and simulation are critical to extending the limits of leading
edge lithography technologies used for IC making. Simultaneous source mask
optimization (SMO) has become an important objective in the field of
computational lithography. SMO is considered essential to extending immersion
lithography beyond the 45nm node. However, SMO is computationally extremely
challenging and time-consuming. The key challenges are due to run time vs.
accuracy tradeoffs of the imaging models used for the computational
lithography. We present a new technique to be incorporated in the SMO flow.
This new approach is based on the reduced basis method (RBM) applied to the
simulation of light transmission through the lithography masks. It provides a
rigorous approximation to the exact lithographical problem, based on fully
vectorial Maxwell's equations. Using the reduced basis method, the optimization
process is divided into an offline and an online steps. In the offline step, a
RBM model with variable geometrical parameters is built self-adaptively and
using a Finite Element (FEM) based solver. In the online step, the RBM model
can be solved very fast for arbitrary illumination and geometrical parameters,
such as dimensions of OPC features, line widths, etc. This approach
dramatically reduces computational costs of the optimization procedure while
providing accuracy superior to the approaches involving simplified mask models.
RBM furthermore provides rigorous error estimators, which assure the quality
and reliability of the reduced basis solutions. We apply the reduced basis
method to a 3D SMO example. We quantify performance, computational costs and
accuracy of our method.Comment: BACUS Photomask Technology 201
A nonintrusive Reduced Basis Method applied to aeroacoustic simulations
The Reduced Basis Method can be exploited in an efficient way only if the
so-called affine dependence assumption on the operator and right-hand side of
the considered problem with respect to the parameters is satisfied. When it is
not, the Empirical Interpolation Method is usually used to recover this
assumption approximately. In both cases, the Reduced Basis Method requires to
access and modify the assembly routines of the corresponding computational
code, leading to an intrusive procedure. In this work, we derive variants of
the EIM algorithm and explain how they can be used to turn the Reduced Basis
Method into a nonintrusive procedure. We present examples of aeroacoustic
problems solved by integral equations and show how our algorithms can benefit
from the linear algebra tools available in the considered code.Comment: 28 pages, 7 figure
Inverse Problems Using Reduced Basis Method
Inverse Problems is a field of great interest for many applications, such as parameter identification and image reconstruction. The underlying models of inverse problems in many applications often involve Partial Differential Equations (PDEs). A Reduced Basis (RB) method for solving PDE based inverse problems is introduced in this thesis. The RB has been rigorously established as an efficient approach for solving PDEs in recent years. In this work, we investigate whether the RB method can be used as a regularization for solving ill-posed and nonlinear inverse problems using iterative methods. We rigorously analyze the RB method and prove convergence of the RB approximation to the exact solution. Furthermore, an iterative algorithm is proposed based on gradient method with RB regularization. We also implement the proposed method numerically and apply the algorithm to the inverse problem of Electrical Impedance Tomography (EIT) which is known to be a notoriously ill-posed and nonlinear. For the EIT example, we provide all necessary details and carefully explain each step of the RB method. We also investigate the limitations of the RB method for solving nonlinear inverse problems in general. We conclude that the RB method can be used to solve nonlinear inverse problems with appropriate assumptions however the assumptions are somewhat restrictive and may not be applicable for a wide range of problems
A weighted reduced basis method for parabolic PDEs with random data
This work considers a weighted POD-greedy method to estimate statistical
outputs parabolic PDE problems with parametrized random data. The key idea of
weighted reduced basis methods is to weight the parameter-dependent error
estimate according to a probability measure in the set-up of the reduced space.
The error of stochastic finite element solutions is usually measured in a root
mean square sense regarding their dependence on the stochastic input
parameters. An orthogonal projection of a snapshot set onto a corresponding POD
basis defines an optimum reduced approximation in terms of a Monte Carlo
discretization of the root mean square error. The errors of a weighted
POD-greedy Galerkin solution are compared against an orthogonal projection of
the underlying snapshots onto a POD basis for a numerical example involving
thermal conduction. In particular, it is assessed whether a weighted POD-greedy
solutions is able to come significantly closer to the optimum than a
non-weighted equivalent. Additionally, the performance of a weighted POD-greedy
Galerkin solution is considered with respect to the mean absolute error of an
adjoint-corrected functional of the reduced solution.Comment: 15 pages, 4 figure
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