Research
I am a numerical analyst; I construct, analyze and implement numerical methods for approximating the solutions to various problems, typically partial differential equations. Most of my work concerns different types of splitting schemes, which can be applied to a large number of problem classes to great effect. During the last few years, I have been working in stochastic optimization from the viewpoint of time integration. This has led to several new optimization methods that are robust to parameter choices. I also have a focus on large-scale differential Riccati equations, which are used in optimal control applications. I have developed several low-rank splitting schemes for such problems. A standalone Matlab implementation is provided in the Software section, but they are also available in M.E.S.S..
Short CV
Born on February 13, 1986, in Myckleby, Sweden.
Education:
- Ph.D. in Numerical Analysis, Lund University, Lund, Sweden, 2015
- MSc. in Mathematics, Lund University, Lund, Sweden, 2010
- BSc. in Mathematics, Lund University, Lund, Sweden, 2009
Academic employment:
- Associate professor (universitetslektor), Lund University, Sweden, 2024-
- Assistant professor (biträdande universitetslektor), Lund University, Sweden, 2019-2024
- Postdoc, Max Planck Institute for Dynamics of Complex Technical Systems (CSC group), Germany, 2017-2019
- Postdoc, Chalmers and the University of Gothenburg, Sweden, 2015-2017
Preprints
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with M. Williamson:
Almost sure convergence of stochastic Hamiltonian descent methods
[arXiv]
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with M. Eisenmann, M. Williamson:
Analysis of a Class of Stochastic Component-Wise Soft-Clipping Schemes
[arXiv]
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with F. Tronarp:
Computing the matrix exponential and the Cholesky factor of a related finite horizon Gramian
[arXiv]
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Convergence analysis for the exponential Lie splitting scheme applied to the abstract differential Riccati equation
[preprint]
Publications
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with M. Eisenmann:
A randomized operator splitting scheme inspired by stochastic optimization methods
Numer. Math. 156 (2024), pp. 435-461
[journal]
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with M. Williamson:
SRKCD: a stabilized Runge-Kutta method for stochastic optimization
J. Comput. Appl. Math. 417 (2023), 114575
[preprint]
[journal]
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with M. Eisenmann:
Sub-linear convergence of a tamed stochastic gradient descent method in Hilbert space
SIAM J. Optim. 32(3) (2022), pp. 1642-1667
[preprint]
[journal]
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with M. Eisenmann, M. Williamson:
Sub-linear convergence for a stochastic proximal iteration method in Hilbert space
Comput. Optim. Appl. 83 (2022), pp. 181-210
[preprint]
[journal]
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with P. Benner, C. Trautwein:
A linear implicit Euler method for the finite element discretization of a controlled stochastic heat equation
IMA J. Numer. Anal. 42(3) (2021), pp. 2118-2150
[journal]
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with H. Mena, L.-M. Pfurtscheller:
GPU acceleration of splitting schemes applied to differential matrix equations
Numer. Algor. 83 (2020), pp. 395--419
[offprint]
[journal]
-
Singular value decay of operator-valued differential Lyapunov and Riccati equations
SIAM J. Control Optim. 56(5) (2018), pp. 3598–3618
[offprint]
[journal]
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with A. Målqvist, A. Persson:
Multiscale differential Riccati equations for linear quadratic regulator problems
SIAM J. Sci. Comput. 40(4) (2018), pp. A2406–A2426
[offprint]
[journal]
-
Adaptive high-order splitting schemes for large-scale differential Riccati equations
Numer. Algor. 78(4) (2018), pp. 1129–1151
[offprint]
[journal]
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with T. Damm, H. Mena:
Numerical Solution of the Finite Horizon Stochastic Linear Quadratic Control Problem
Numer. Linear Algebra Appl. 24(4) (2017), e2091
[preprint]
[journal]
-
with A. Målqvist:
Finite element convergence analysis for the thermoviscoelastic Joule heating problem
BIT 57(3) (2017), pp. 787-810
[preprint]
[journal]
-
Low-rank second-order splitting of large-scale differential Riccati equations
IEEE Trans. Automat. Control 60(10) (2015), pp. 2791-2796
[preprint]
[journal]
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with E. Hansen:
Convergence analysis for splitting of the abstract differential Riccati equation
SIAM J. Numer. Anal. 52(6) (2014), pp. 3128-3139
[preprint]
[journal]
-
with E. Hansen:
Implicit Euler and Lie splitting discretizations of nonlinear parabolic equations with delay
BIT 54(3) (2014), pp. 673-689
[preprint]
[journal]
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with E. Hansen:
Convergence of the implicit-explicit Euler scheme applied to perturbed dissipative evolution equations
Math. Comp. 82(284) (2013), pp. 1975-1985
[preprint]
[journal]
Thesis
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T. Stillfjord, Splitting schemes for nonlinear parabolic problems, 2015, LUP.
In addition to the LUP link, the thesis can be found here (with hyperlinks) or here (without hyperlinks). Both versions omit the included papers. For these, see above.
Peer review
I have refereed papers in the following journals:
Software
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DREsplit
MATLAB-code for approximating the solution to a differential Riccati equation, based on the methods described in the papers 4 and 7 (listed above), is available here: DREsplit. This newest version unifies the interface for the different solvers and reduces code duplication.
Please see the accompanying Readme and Changelog files for further information.
Last updated on: 2021-02-15
Previous versions:
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BST20_CODE
The code which was used to run the experiments in the preprint listed as number 2 above is available here: BST20_CODE.
Please see the accompanying Readme file for further information.
Last updated on: 2020-06-20
Other links
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Detailed contact information
Postal address:
Centre for Mathematical Sciences
Lund University
Box 118
SE-22100 Lund
Sweden
Office: room MH:562E
Email: my first name dot my surname at math dot lth dot se
URL: http://www.tonystillfjord.net
URL: http://www.maths.lu.se/staff/tony-stillfjord/
Phone: +46 46 222 4451 (office)
