Order reduction methods for solving large-scale differential matrix Riccati equations

We consider the numerical solution of large-scale symmetric differential matrix Riccati equations. Under certain hypotheses on the data, reduced order methods have recently arisen as a promising class of solution strategies, by forming low-rank approximations to the sought after solution at selected timesteps. We show that great computational and memory savings are obtained by a reduction process onto rational Krylov subspaces, as opposed to current approaches. By specifically addressing the solution of the reduced differential equation and reliable stopping criteria, we are able to obtain accurate final approximations at low memory and computational requirements. This is obtained by employing a two-phase strategy that separately enhances the accuracy of the algebraic approximation and the time integration. The new method allows us to numerically solve much larger problems than in the current literature. Numerical experiments on benchmark problems illustrate the effectiveness of the procedure with respect to existing solvers

A numerical algorithm for optimal feedback gains in high dimensional LQR problems

A hybrid method for computing the feedback gains in linear quadratic regulator problems is proposed. The method, which combines the use of a Chandrasekhar type system with an iteration of the Newton-Kleinman form with variable acceleration parameter Smith schemes, is formulated so as to efficiently compute directly the feedback gains rather than solutions of an associated Riccati equation. The hybrid method is particularly appropriate when used with large dimensional systems such as those arising in approximating infinite dimensional (distributed parameter) control systems (e.g., those governed by delay-differential and partial differential equations). Computational advantage of the proposed algorithm over the standard eigenvector (Potter, Laub-Schur) based techniques are discussed and numerical evidence of the efficacy of our ideas presented

Forward diffraction modelling : analysis and application to grating reconstruction

The semiconductor industry uses lithography machines for manufacturing complex integrated circuits (also called ICs) onto wafers. Because an IC is built up layer by layer and feature sizes get smaller and smaller, tight control of the lithography process is required to guarantee a fast production of working ICs. Typically a lot of information on the lithography process can be obtained by measuring test structures or gratings which are scattered over the wafer. These gratings are tiny periodic structures much smaller than ICs. First these gratings are illuminated and its response (a scattered intensity) is measured. For certain applications like overlay metrology the asymmetry in this measured signal (due to an offset between two gratings) can be used to align the lithographic process. For other applications like critical dimension (CD) metrology one is interested in the shape of the grating lines that produced the measured signal. Since this information is not directly available but encrypted in the measurement, a reconstruction algorithm is used to extract it. The reconstructed values like height, width and sidewall angle can then be related to machine settings like dose and focus which control the lithographic process. In particular the CD metrology application requires rigorous mathematical models that solve optical diffraction problems for periodic gratings in combination with advanced reconstruction algorithms. This thesis focuses on the optical diffraction problem for 1D periodic gratings. Starting from Maxwell's equations a reduced model is derived by simplifying both the grating and the incident electromagnetic field. The former is approximated with an infinitely periodic layered structure with isotropic non-magnetic materials. The latter is approximated with a time-harmonic incident plane wave. The reduced model is discretised using two different mode expansion methods, Bloch and the Rigorous Coupled-Wave Analysis (RCWA). Bloch expands the electromagnetic field in each layer in terms of the exact eigenfunctions whereas RCWA only uses approximate eigenfunctions. After truncation of the involved series a transmission problem is derived by matching the fields at the layer interfaces. Having solved the resulting linear system, the scattered field can be computed easily. Both mode expansion methods solve a similar linear system containing a large but sparse block-structured coefficient matrix. However, special care needs to be taken when solving this system stably and efficiently. Therefore a stable condensation algorithm is derived based on Riccati transformations that decouples the exponentially growing and decaying terms that are present in the solution. This separation or decoupling is the key feature explaining the stability which is not always clear in alternative condensation algorithms. Furthermore the algorithm is optimised for speed by using a two-stage approach. Finally it is shown that the resulting stable recursions are identical to those used in the 'enhanced transmittance matrix approach" (a frequently used condensation algorithm), thereby confirming its stability as well. This thesis also examines and extends both mode expansions methods. The Bloch method is generalised to deal with multiple material transitions inside a grating layer covering a wider range of applications. However, lossy or fully asymmetric gratings are still hard to solve. On the other hand the Fourier discretisation used in RCWA is much more exible but only approximates the more exact discretisation of Bloch. Therefore two RCWA modifications have been investigated to improve the accuracy while keeping its exibility and relatively straightforward implementation. Adaptive Spatial Resolution applies an additional layer specific coordinate transformation before Fourier discretising the problem again. A good transformation not only refines near a material interface but also does this in a smooth way. A significant improvement in accuracy is observed that approaches and sometimes outperforms the results obtained with the Bloch method. The second modification removes the Fourier discretisation completely and uses a finite difference approximation in the periodic direction. Although this approach allows for a better discretisation near a material interface, the sparsity of the resulting matrices could not be exploited to make a competitive implementation within the standard RCWA framework. Finally the integration of the forward diffraction model in the CD reconstruction application is discussed. Either a library based or real-time regressions approach can be used for this reconstruction. Both approaches rely heavily on having an accurate and fast forward model. By exploiting additional symmetries and smart reuse of information, acceptable library fill times and real-time reconstructions are now feasible

A Snapshot Algorithm for Linear Feedback Flow Control Design

The control of fluid flows has many applications. For micro air vehicles, integrated flow control designs could enhance flight stability by mitigating the effect of destabilizing air flows in their low Reynolds number regimes. However, computing model based feedback control designs can be challenging due to high dimensional discretized flow models. In this work, we investigate the use of a snapshot algorithm proposed in Ref. 1 to approximate the feedback gain operator for a linear incompressible unsteady flow problem on a bounded domain. The main component of the algorithm is obtaining solution snapshots of certain linear flow problems. Numerical results for the example flow problem show convergence of the feedback gains

A general framework for the perturbation theory of matrix equations

A general framework is presented for the local and non-local perturbation analysis of general real and complex matrix equations in the form $F(P,X) = 0$, where $F$ is a continuous, matrix valued function, $P$ is a collection of matrix parameters and $X$ is the unknown matrix. The local perturbation analysis produces condition numbers and improved first order homogeneous perturbation bounds for the norm \|\de X\| or the absolute value |\de X| of \de X. The non-local perturbation analysis is based on the method of Lyapunov majorants and fixed point principles. % for the operator $\pi(p,\cdot)$. It gives rigorous non-local perturbation bounds as well as conditions for solvability of the perturbed equation. The general framework can be applied to various matrix perturbation problems in science and engineering. We illustrate the procedure with several simple examples. Furhermore, as a model problem for the new framework we derive a new perturbation theory for continuous-time algebraic matrix Riccati equations in descriptor form, $Q + A^HXE + E^HXA - E^HXSXE = 0$. The associated equation $Q + A^HXE + E^HX^HA - E^HX^HSXE = 0$ is also briefly considered

Order reduction of semilinear differential matrix and tensor equations

In this thesis, we are interested in approximating, by model order reduction, the solution to large-scale matrix- or tensor-valued semilinear Ordinary Differential Equations (ODEs). Under specific hypotheses on the linear operators and the considered domain, these ODEs often stem from the space discretization on a tensor basis of semilinear Partial Differential Equations (PDEs) with a dimension greater than or equal to two. The bulk of this thesis is devoted to the case where the discrete system is a matrix equation. We consider separately the cases of general Lipschitz continuous nonlinear functions and the Differential Riccati Equation (DRE) with a quadratic nonlinear term. In both settings, we construct a pair of left-right approximation spaces that leads to a reduced semilinear matrix differential equation with the same structure as the original problem, which can be more rapidly integrated with matrix-oriented integrators. For the DRE, under certain assumptions on the data, we show that a reduction process onto rational Krylov subspaces obtains significant computational and memory savings as opposed to current approaches. In the more general setting, a challenging difference lies in selecting and constructing the two approximation bases to handle the nonlinear term effectively. In addition, the nonlinear term also needs to be approximated for efficiency. To this end, in the framework of the Proper Orthogonal Decomposition (POD) methodology and the Discrete Empirical Interpolation Method (DEIM), we derive a novel matrix-oriented reduction process leading to a practical, structure-aware low order approximation of the original problem. In the final part of the thesis, we consider the multidimensional setting. Here we extend the matrix-oriented POD-DEIM algorithm to the tensor setting and illustrate how we can apply it to systems of such equations. Moreover, we discuss how to integrate the reduced-order model and, in particular, how to solve the resulting tensor-valued linear systems

Differential-Algebraic Equations

Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed