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 largescale differential Riccati equations, which are used in optimal control applications. I have developed several lowrank 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, 20192024
 Postdoc, Max Planck Institute for Dynamics of Complex Technical Systems (CSC group), Germany, 20172019
 Postdoc, Chalmers and the University of Gothenburg, Sweden, 20152017
Preprints

with M. Williamson:
Almost sure convergence of stochastic Hamiltonian descent methods
[arXiv]

with M. Eisenmann, M. Williamson:
Analysis of a Class of Stochastic ComponentWise SoftClipping Schemes
[arXiv]

with F. Tronarp:
Computing the matrix exponential and the Cholesky factor of a related finite horizon Gramian
[arXiv]

Convergence analysis for the exponential Lie splitting scheme applied to the abstract differential Riccati equation
[preprint]
Publications

with M. Eisenmann:
A randomized operator splitting scheme inspired by stochastic optimization methods
Numer. Math. 156 (2024), pp. 435461
[journal]

with M. Williamson:
SRKCD: a stabilized RungeKutta method for stochastic optimization
J. Comput. Appl. Math. 417 (2023), 114575
[preprint]
[journal]

with M. Eisenmann:
Sublinear convergence of a tamed stochastic gradient descent method in Hilbert space
SIAM J. Optim. 32(3) (2022), pp. 16421667
[preprint]
[journal]

with M. Eisenmann, M. Williamson:
Sublinear convergence for a stochastic proximal iteration method in Hilbert space
Comput. Optim. Appl. 83 (2022), pp. 181210
[preprint]
[journal]

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. 21182150
[journal]

with H. Mena, L.M. Pfurtscheller:
GPU acceleration of splitting schemes applied to differential matrix equations
Numer. Algor. 83 (2020), pp. 395419
[offprint]
[journal]

Singular value decay of operatorvalued differential Lyapunov and Riccati equations
SIAM J. Control Optim. 56(5) (2018), pp. 3598–3618
[offprint]
[journal]

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 highorder splitting schemes for largescale differential Riccati equations
Numer. Algor. 78(4) (2018), pp. 1129–1151
[offprint]
[journal]

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. 787810
[preprint]
[journal]

Lowrank secondorder splitting of largescale differential Riccati equations
IEEE Trans. Automat. Control 60(10) (2015), pp. 27912796
[preprint]
[journal]

with E. Hansen:
Convergence analysis for splitting of the abstract differential Riccati equation
SIAM J. Numer. Anal. 52(6) (2014), pp. 31283139
[preprint]
[journal]

with E. Hansen:
Implicit Euler and Lie splitting discretizations of nonlinear parabolic equations with delay
BIT 54(3) (2014), pp. 673689
[preprint]
[journal]

with E. Hansen:
Convergence of the implicitexplicit Euler scheme applied to perturbed dissipative evolution equations
Math. Comp. 82(284) (2013), pp. 19751985
[preprint]
[journal]
Thesis

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

DREsplit
MATLABcode 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: 20210215
Previous versions:

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: 20200620
Other links

Detailed contact information
Postal address:
Centre for Mathematical Sciences
Lund University
Box 118
SE22100 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/tonystillfjord/
Phone: +46 46 222 4451 (office)