Type II balanced truncation for deterministic bilinear control systems

When solving partial differential equations numerically, usually a high order spatial discretisation is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretised systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is balanced truncation (BT), a method which has been extensively studied for deterministic linear systems. As so-called type I BT it has already been extended to bilinear equations, an important subclass of nonlinear systems. We provide an alternative generalisation of the linear setting to bilinear systems which is called type II BT. The Gramians that we propose in this context contain information about the control. It turns out that the new approach delivers energy bounds which are not just valid in a small neighbourhood of zero. Furthermore, we provide an $H_\infty$-error bound which so far is not known when applying type I BT to bilinear systems

Towards Time-Limited H2\mathcal H_2-Optimal Model Order Reduction

In order to solve partial differential equations numerically and accurately, a high order spatial discretization is usually needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized systems and hence reduce computational complexity. A particular class of MOR techniques are $\mathcal H_2$-optimal methods such as the iterative rational Krylov subspace algorithm (IRKA) and related schemes. However, these methods are used to obtain good approximations on a infinite time-horizon. Thus, in this work, our main goal is to discuss MOR schemes for time-limited linear systems. For this, we propose an alternative time-limited $\mathcal H_2$-norm and show its connection with the time-limited Gramians. We then provide first-order optimality conditions for an optimal reduced order model (ROM) with respect to the time-limited $\mathcal H_2$-norm. Based on these optimality conditions, we propose an iterative scheme, which, upon convergence, aims at satisfying these conditions approximately. Then, we analyze how far away the obtained ROM due to the proposed algorithm is from satisfying the optimality conditions. We test the efficiency of the proposed iterative scheme using various numerical examples and illustrate that the newly proposed iterative method can lead to a better reduced-order compared to the unrestricted IRKA in the finite time interval of interest

Type II singular perturbation approximation for linear systems with Lévy noise

When solving linear stochastic partial differential equations numerically, usually a high order spatial discretisation is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretised systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is singular perturbation approximation (SPA), a method which has been extensively studied for deterministic systems. As so-called type I SPA it has already been extended to stochastic equations. We provide an alternative generalisation of the deterministic setting to linear systems with Lévy noise which is called type II SPA. It turns out that the ROM from applying type II SPA has better properties than the one of using type I SPA. In this paper, we provide new energy interpretations for stochastic reachability Gramians, show the preservation of mean square stability in the ROM by type II SPA and prove two different error bounds for type II SPA when applied to Lévy driven systems

Dynamic programming for optimal stopping via pseudo-regression

We introduce new variants of classical regression-based algorithms for optimal stopping problems based on computation of regression coefficients by Monte Carlo approximation of the corresponding $L^2$ inner products instead of the least-squares error functional. Coupled with new proposals for simulation of the underlying samples, we call the approach "pseudo regression". A detailed convergence analysis is provided and it is shown that the approach asymptotically leads to less computational cost for a pre-specified error tolerance, hence to lower complexity. The method is justified by numerical examples

Type II balanced truncation for deterministic bilinear control systems

When solving partial differential equations numerically, usually a high order spatial discretisation is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretised systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is balanced truncation (BT), a method which has been extensively studied for deterministic linear systems. As so-called type I BT it has already been extended to bilinear equations, an important subclass of nonlinear systems. We provide an alternative generalisation of the linear setting to bilinear systems which is called type II BT. The Gramians that we propose in this context contain information about the control. It turns out that the new approach delivers energy bounds which are not just valid in a small neighbourhood of zero. Furthermore, we provide an H1-error bound which so far is not known when applying type I BT to bilinear systems

Energy estimates and model order reduction for stochastic bilinear systems

In this paper, we investigate a large-scale stochastic system with bilinear drift and linear diffusion term. Such high dimensional systems appear for example when discretizing a stochastic partial differential equations in space. We study a particular model order reduction technique called balanced truncation (BT) to reduce the order of spatially-discretized systems and hence reduce computational complexity. We introduce suitable Gramians to the system and prove energy estimates that can be used to identify states which contribute only very little to the system dynamics. When BT is applied the reduced system is obtained by removing these states from the original system. The main contribution of this paper is an $L^2$-error bound for BT for stochastic bilinear systems. This result is new even for deterministic bilinear equations. In order to achieve it, we develop a new technique which is not available in the literature so far

Energy estimates and model order reduction for stochastic bilinear systems

In this paper, we investigate a large-scale stochastic system with bilinear drift and linear diffusion term. Such high dimensional systems appear for example when discretizing a stochastic partial differential equations in space. We study a particular model order reduction technique called balanced truncation (BT) to reduce the order of spatially-discretized systems and hence reduce computational complexity. We introduce suitable Gramians to the system and prove energy estimates that can be used to identify states which contribute only very little to the system dynamics. When BT is applied the reduced system is obtained by removing these states from the original system. The main contribution of this paper is an L2-error bound for BT for stochastic bilinear systems. This result is new even for deterministic bilinear equations. In order to achieve it, we develop a new technique which is not available in the literature so far