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arxiv: 2606.17744 · v1 · pith:KAJQEJB7new · submitted 2026-06-16 · 🧮 math.NA · cs.NA

On the Solution of Large-scale Non-autonomous Differential Riccati Equations: a Numerical Study

Eugenio Mancinelli , Davide Palitta , Jens Saak This is my paper

Pith reviewed 2026-06-26 23:58 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords differential Riccati equationsgeneralized algebraic Riccati equationsNewton-Kleinman methodRADI iterationBDF discretizationnumerical methodslarge-scale systemstime-stepping
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The pith

Numerical experiments compare Newton-Kleinman and RADI solvers on sequences of generalized algebraic Riccati equations from BDF-discretized differential Riccati equations.

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

The paper studies the numerical treatment of large-scale non-autonomous differential Riccati equations. Time discretization via backward differentiation formulas converts the differential problem into a sequence of generalized algebraic Riccati equations. Each equation in the sequence is solved by either the inexact Newton-Kleinman method with line search or the low-rank RADI iteration, and each solver is tested with both a zero initial guess and a warm-start from the solution at the previous time step. The work presents numerical results that reveal how these solvers behave inside the full time-stepping pipeline rather than when applied to an isolated algebraic equation. This distinction matters because many control and filtering applications require the solution of the differential equation over an interval, not just a single static algebraic problem.

Core claim

When the backward differentiation formula is applied to a non-autonomous differential Riccati equation, the resulting sequence of generalized algebraic Riccati equations is solved by the inexact Newton-Kleinman method with line search and by the low-rank RADI iteration; both methods are examined under zero initialization and warm-start initialization, and a panel of numerical tests illustrates the potential and limitations that appear only when the solvers operate inside the complete differential-equation pipeline.

What carries the argument

BDF time discretization of the differential Riccati equation followed by iterative solution of each resulting generalized algebraic Riccati equation via Newton-Kleinman or RADI, using either zero or warm-start initialization at successive steps.

If this is right

  • Algebraic Riccati solvers previously benchmarked only on single equations must be re-evaluated when applied repeatedly across time steps.
  • The choice between zero initialization and warm-start affects the total computational effort required to integrate the differential Riccati equation to a final time.
  • Limitations observed for one solver in the pipeline may not appear when the same solver is applied to an isolated algebraic equation.

Where Pith is reading between the lines

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

  • The same pipeline structure could be used to test other iterative solvers or other time discretizations such as implicit Runge-Kutta methods.
  • If conditioning of the algebraic equations degrades over many time steps, the observed performance gaps between solvers could widen or reverse.
  • The numerical results may guide the design of adaptive switching strategies that choose the solver or initialization based on the current time-step residual.

Load-bearing premise

The generalized algebraic Riccati equations produced by BDF time-stepping remain well-conditioned and solvable by the chosen iterative methods without additional stabilization or preconditioning that would alter the comparison.

What would settle it

A collection of test cases in which both solvers require the same number of iterations with warm-start and with zero initialization, or in which one solver diverges on every warm-started step while converging on every zero-started step, would contradict the reported distinction in behavior inside the pipeline.

read the original abstract

We explore the numerical solution of large-scale non-autonomous Differential Riccati Equations (DREs). While we assume to discretize the differential operator using a Backward Differentiation Formula (BDF) of order s, we solve the generalized Algebraic Riccati Equation (gARE) resulting at each time step by different state-of-the-art methods. In particular, we compare the performance of the inexact Newton- Kleinman method with line search and the low-rank RADI iteration, considering for both methods two different initialization strategies: zero initialization and warm-start. A comprehensive panel of numerical results illustrate the potential and limitations of these methods when employed within a numerical pipeline for the solution of DREs, rather than for the isolated solution of a single gARE, as commonly considered in the existing literature.

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

0 major / 4 minor

Summary. The paper conducts a numerical study of large-scale non-autonomous Differential Riccati Equations (DREs) discretized in time via Backward Differentiation Formulas (BDF) of order s. At each time step the resulting generalized Algebraic Riccati Equation (gARE) is solved by the inexact Newton-Kleinman method with line search and by the low-rank RADI iteration; both methods are tested with zero initialization and warm-start. The central claim is that a panel of experiments demonstrates the practical behavior, potential, and limitations of these solvers when embedded in the full DRE time-stepping pipeline rather than applied to isolated gAREs.

Significance. If the reported timings, iteration counts, and accuracy metrics hold, the study supplies useful engineering guidance for practitioners who must solve sequences of gAREs arising from time-dependent control or PDE-constrained problems. The explicit focus on the coupled pipeline (BDF + gARE) rather than single gARE solves is a modest but genuine contribution to the existing literature on large-scale Riccati solvers.

minor comments (4)
  1. [Abstract] Abstract: the claim of a 'comprehensive panel of numerical results' is not supported by any quantitative statement (problem dimensions, number of time steps, observed iteration counts, or residual tolerances). Adding one or two concrete figures would strengthen the abstract.
  2. [Section 3] Section 3 (or wherever the algorithmic descriptions appear): the precise stopping criteria, line-search parameters, and low-rank truncation tolerances used for Newton-Kleinman and RADI should be stated explicitly so that the experiments are reproducible.
  3. [Numerical experiments] Numerical experiments section: all tables and figures must list the matrix dimensions (n, m), the BDF order s, the time interval, and the number of time steps; without these the reader cannot judge the scale or the fairness of the comparison.
  4. [Numerical experiments] The manuscript should clarify whether the same preconditioner or shift strategy is used for both solvers at every time step; any difference would affect the reported performance comparison.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript. The report recommends minor revision but lists no specific major comments. We have therefore prepared no point-by-point replies and stand ready to address any editorial or minor issues that may arise.

Circularity Check

0 steps flagged

No significant circularity

full rationale

This is an empirical numerical performance study comparing inexact Newton-Kleinman and low-rank RADI methods (with zero and warm-start initializations) for gAREs that arise from BDF time-stepping of DREs. No mathematical derivation, first-principles prediction, uniqueness theorem, or fitted parameter is claimed or presented; the contribution consists solely of reported runtimes, iteration counts, and residual histories on test problems. The abstract and scope contain no equations that could reduce to their own inputs by construction, and no self-citation is invoked as load-bearing justification for any result. The work is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The study rests on the domain assumption that BDF discretization produces a sequence of standard gAREs that can be attacked by the cited iterative solvers; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption BDF discretization of order s converts the non-autonomous DRE into a sequence of solvable generalized algebraic Riccati equations
    Invoked in the first sentence of the abstract to justify the numerical pipeline

pith-pipeline@v0.9.1-grok · 5668 in / 1152 out tokens · 31979 ms · 2026-06-26T23:58:59.561998+00:00 · methodology

discussion (0)

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Reference graph

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