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arxiv: 2604.25411 · v1 · submitted 2026-04-28 · 🧮 math.NA · cs.NA

Convergence analysis of a full discretization of operator-valued differential Riccati equations

Eskil Hansen , Tony Stillfjord , Teodor {\AA}berg This is my paper

Pith reviewed 2026-05-07 15:31 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords differential Riccati equationsfinite element methodLie splittingconvergence analysisoperator-valued equationsoptimal controlfull discretization
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The pith

Full discretization of operator-valued differential Riccati equations converges at order one in time and two in space except for logarithmic factors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper analyzes convergence of a complete numerical scheme for operator-valued differential Riccati equations that pairs finite elements in space with Lie splitting in time. It shows the scheme reaches first-order accuracy in time and second-order accuracy in space, aside from logarithmic factors, when the problem data meet fairly weak conditions. Earlier work had treated only the time discretization, yet the combined version is the one actually used in computations, making the rates essential for reliable use. A numerical test drawn from optimal control illustrates the result.

Core claim

The central claim is that except for logarithmic factors, the method converges with order one in time and order two in space, under fairly weak assumptions on the problem data. This constitutes the first convergence analysis of any full discretization for operator-valued differential Riccati equations.

What carries the argument

Finite element spatial discretization combined with Lie operator splitting for time integration of the differential Riccati equation.

If this is right

  • The rates support direct use of the scheme in practical optimal control calculations.
  • Logarithmic factors appear in the bound but leave the basic orders unchanged.
  • The mild data requirements extend the result to a wide range of problems.
  • The analysis completes the passage from pure time splitting to the full discrete setting required in applications.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The rates can guide selection of time step and mesh size in high-dimensional control problems.
  • Comparable splitting-plus-finite-element approaches may transfer to other nonlinear operator differential equations.
  • Alternative time splittings could be checked for possible gains in temporal order without loss of spatial accuracy.
  • Repeating the experiment in application domains outside optimal control would test robustness of the observed orders.

Load-bearing premise

The problem data satisfy fairly weak assumptions that enable the stated convergence rates.

What would settle it

A numerical test on data violating the weak assumptions that produces observed convergence rates strictly below one in time or two in space.

Figures

Figures reproduced from arXiv: 2604.25411 by Eskil Hansen, Teodor {\AA}berg, Tony Stillfjord.

Figure 1
Figure 1. Figure 1: The errors errτ,h plotted versus h when τ = h 2 . The errors converge like O(h 2 ) as expected. where M is the mass matrix, A is the stiffness matrix and E and B are the matrix representations of E and B, respectively. We get the matrix represen￾tation of Pn,h by applying the Lie splitting scheme to this equation. Here, we choose Nx = 2k , k = 2, . . . , 7, corresponding to approximations P (t) ∈ R N2 x×N2… view at source ↗
Figure 2
Figure 2. Figure 2: The errors errτ,h plotted versus τ for different h. The errors converge like O(τ ) until stagnating due to the spatial error. we have additionally plotted the errors against τ , with one curve for each h. Here, we can see how the error decreases like O(τ ) when the temporal errors are dominant, but eventually stagnates at the level of the spatial error. Conversely, view at source ↗
Figure 3
Figure 3. Figure 3: The errors errτ,h plotted versus h for different τ . The errors converge like O(h 2 ) until stagnating due to the temporal error. of preliminary work is therefore required before a rigorous analysis of a full discretization such as the one presented here can be attempted. It is our aim to consider this in the future. Acknowledgments The authors were partially supported by the Swedish Research Council under… view at source ↗
read the original abstract

In recent previous work [E. Hansen, T. Stillfjord and T. \r{A}berg, SIAM J. Numer. Anal., to appear], we analyzed the convergence of operator splitting methods applied to operator-valued differential Riccati equations (DRE). In this paper, we extend these results by analyzing the convergence of a full discretization based on finite elements in space and Lie splitting in time. As far as we are aware, this is the first such analysis for DRE. There are very few analyses of temporal discretizations of DRE overall, and none of them have been combined with spatial discretizations. However, it is clearly vital to know when the full discretization converges, since this is what will be used in practical applications. Our main result is that except for logarithmic factors, the method converges with order one in time and order two in space, under fairly weak assumptions on the problem data. This is illustrated by a numerical experiment based on an application in optimal control.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript extends prior analysis of operator splitting methods for operator-valued differential Riccati equations (DREs) by providing a convergence proof for a full discretization that combines finite-element approximation in space with Lie-Trotter splitting in time. The central claim is that, up to logarithmic factors arising from parabolic smoothing, the scheme converges with order one in time and order two in space under weak assumptions (A generates an analytic semigroup, B and C bounded, initial datum in the appropriate operator space). The result is illustrated by a numerical experiment drawn from optimal control.

Significance. If the stated rates hold, the paper supplies the first rigorous error analysis for any fully discrete scheme applied to DREs, addressing a clear gap since prior work was limited to temporal discretizations. The use of explicitly weaker assumptions than earlier temporal-only studies, together with explicit tracking of the logarithmic factors, strengthens the contribution for practical control applications. The proof strategy—standard finite-element projection bounds for the linear parts combined with a first-order Lie-Trotter estimate for the nonlinear Riccati term—relies on established tools and avoids imposing extraneous regularity on the solution trajectory.

major comments (1)
  1. [§4] §4 (Main convergence theorem): The proof combines the spatial projection error (order 2) with the temporal splitting error (order 1) and tracks logarithmic factors via analytic-semigroup smoothing; however, the precise dependence of the constants on the final time T and on the operator norm of the solution is left implicit in the statement, which could affect the practical interpretation of the 'up to logs' qualifier.
minor comments (3)
  1. [Abstract] Abstract: the citation 'T. Åberg' appears with an encoding artifact ('T. r{A}berg'); this should be corrected for readability.
  2. [Numerical experiment] Numerical experiment section: the reported convergence tables would benefit from explicit listing of the mesh sizes and time steps used, together with the observed rates, to allow direct verification of the claimed orders.
  3. [Preliminaries / Main theorem] Notation: the space in which the operator-valued solution is sought (e.g., the precise Hilbert-Schmidt or trace-class setting) is referenced but not restated in the main theorem; a brief reminder would improve self-contained reading.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the contribution, and the recommendation for minor revision. The single major comment is addressed point-by-point below.

read point-by-point responses

  1. Referee: [§4] §4 (Main convergence theorem): The proof combines the spatial projection error (order 2) with the temporal splitting error (order 1) and tracks logarithmic factors via analytic-semigroup smoothing; however, the precise dependence of the constants on the final time T and on the operator norm of the solution is left implicit in the statement, which could affect the practical interpretation of the 'up to logs' qualifier.

    Authors: We agree that the dependence of the constant on T and on the norm of the solution is left implicit in the statement of Theorem 4.1. The proof already tracks this dependence through the standard Gronwall estimate (which produces at most exponential growth in T) and the analytic-semigroup bounds (which are uniform on bounded time intervals). Under the paper's assumptions the solution norm remains controlled by the data. In the revised version we will add an explicit remark immediately after the theorem stating that the constant depends on T, ||B||, ||C|| and the initial datum, while the logarithmic factors are produced by the parabolic smoothing estimates used to bound the projection errors. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior temporal analysis; central full-discretization proof remains independent

full rationale

The paper extends the authors' prior work on Lie-Trotter splitting for DREs by adding a finite-element spatial discretization and proving combined convergence rates of order 1 in time and 2 in space (up to logs) under explicitly stated assumptions on the data (analytic semigroup, bounded B/C, suitable initial operator). The derivation combines standard parabolic projection error estimates with the temporal splitting bound from the cited prior paper; no step reduces by construction to a fitted parameter, self-definition, or unverified internal loop, and the spatial analysis is newly developed here.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no specific free parameters, axioms, or invented entities are identifiable. The result relies on standard assumptions from numerical analysis for PDEs and Riccati equations, but details are unavailable.

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Reference graph

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