| 1. |
- Målqvist, Axel, 1978, et al.
(author)
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Multiscale Differential Riccati Equations for Linear Quadratic Regulator Problems
- 2018
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In: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 40:4
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Journal article (peer-reviewed)abstract
- We consider approximations to the solutions of differential Riccati equations in the context of linear quadratic regulator problems, where the state equation is governed by a multiscale operator. Similarly to elliptic and parabolic problems, standard finite element discretizations perform poorly in this setting unless the grid resolves the fine-scale features of the problem. This results in unfeasible amounts of computation and high memory requirements. In this paper, we demonstrate how the localized orthogonal decomposition method may be used to acquire accurate results also for coarse discretizations, at the low cost of solving a series of small, localized elliptic problems. We prove second-order convergence (except for a logarithmic factor) in the $L^2$ operator norm and first-order convergence in the corresponding energy norm. These results are both independent of the multiscale variations in the state equation. In addition, we provide a detailed derivation of the fully discrete matrix-valued equations and show how they can be handled in a low-rank setting for large-scale computations. In connection to this, we also show how to efficiently compute the relevant operator-norm errors. Finally, our theoretical results are validated by several numerical experiments.
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| 2. |
- Målqvist, Axel, 1978, et al.
(author)
-
Multiscale differential riccati equations for linear quadratic regulator problems
- 2018
-
In: SIAM Journal of Scientific Computing. - 1064-8275 .- 1095-7197. ; 40:4, s. A2406-A2426
-
Journal article (peer-reviewed)abstract
- We consider approximations to the solutions of differential Riccati equations in the context of linear quadratic regulator problems, where the state equation is governed by a multiscale operator. Similarly to elliptic and parabolic problems, standard finite element discretizations perform poorly in this setting unless the grid resolves the fine-scale features of the problem. This results in unfeasible amounts of computation and high memory requirements. In this paper, we demonstrate how the localized orthogonal decomposition method may be used to acquire accurate results also for coarse discretizations, at the low cost of solving a series of small, localized elliptic problems. We prove second-order convergence (except for a logarithmic factor) in the L2operator norm and first-order convergence in the corresponding energy norm. These results are both independent of the multiscale variations in the state equation. In addition, we provide a detailed derivation of the fully discrete matrix-valued equations and show how they can be handled in a low-rank setting for large-scale computations. In connection to this, we also show how to efficiently compute the relevant operator-norm errors. Finally, our theoretical results are validated by several numerical experiments.
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| 3. |
- Damm, Tobias, et al.
(author)
-
Numerical solution of the finite horizon stochastic linear quadratic control problem
- 2017
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In: Numerical Linear Algebra with Applications. - : Wiley. - 1070-5325 .- 1099-1506. ; 24:4
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Journal article (peer-reviewed)abstract
- © 2017 John Wiley & Sons, Ltd. The treatment of the stochastic linear quadratic optimal control problem with finite time horizon requires the solution of stochastic differential Riccati equations. We propose efficient numerical methods, which exploit the particular structure and can be applied for large-scale systems. They are based on numerical methods for ordinary differential equations such as Rosenbrock methods, backward differentiation formulas, and splitting methods. The performance of our approach is tested in numerical experiments.
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| 4. |
- Målqvist, Axel, 1978, et al.
(author)
-
Finite element convergence analysis for the thermoviscoelastic Joule heating problem
- 2017
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In: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 57:3, s. 787-810
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Journal article (peer-reviewed)abstract
- © 2017 The Author(s) We consider a system of equations that model the temperature, electric potential and deformation of a thermoviscoelastic body. A typical application is a thermistor; an electrical component that can be used e.g. as a surge protector, temperature sensor or for very precise positioning. We introduce a full discretization based on standard finite elements in space and a semi-implicit Euler-type method in time. For this method we prove optimal convergence orders, i.e. second-order in space and first-order in time. The theoretical results are verified by several numerical experiments in two and three dimensions.
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| 5. |
- Stillfjord, Tony, 1986
(author)
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Adaptive high-order splitting schemes for large-scale differential Riccati equations
- 2018
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In: Numerical Algorithms. - : Springer Science and Business Media LLC. - 1017-1398 .- 1572-9265. ; 78:4, s. 1129-1151
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Journal article (peer-reviewed)abstract
- We consider high-order splitting schemes for large-scale differential Riccati equations. Such equations arise in many different areas and are especially important within the field of optimal control. In the large-scale case, it is critical to employ structural properties of the matrix-valued solution, or the computational cost and storage requirements become infeasible. Our main contribution is therefore to formulate these high-order splitting schemes in an efficient way by utilizing a low-rank factorization. Previous results indicated that this was impossible for methods of order higher than 2, but our new approach overcomes these difficulties. In addition, we demonstrate that the proposed methods contain natural embedded error estimates. These may be used, e.g., for time step adaptivity, and our numerical experiments in this direction show promising results.
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