Optimality-Based Control Space Reduction for Infinite-Dimensional Control Spaces

Michael Kartmann; Stefan Volkwein

REVIEW 2 major objections 5 minor 27 references

Optimality-Based Control Space Reduction for Infinite-Dimensional Control Spaces

T0 review · 2 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For unconstrained linear-quadratic optimal control problems governed by time-varying parabolic PDEs, reducing the state space automatically induces a reduced structure in the optimal control, making the control-and-state-reduced problem equ

desk verdict Control/state reduction equivalence is clean and useful; the adaptive convergence proof has a real density gap, and Theorem 7's printed proof has fixable errors. read the letter →

arxiv 2510.14479 v2 pith:NL7JCTZT submitted 2025-10-16 math.OC cs.NAmath.NA

Michael Kartmann , Stefan Volkwein This is my paper · ORCID

classification math.OCcs.NAmath.NA MSC 49K2049M0549M4165G2093C20

keywords linear-quadraticoptimalcontrolparabolicPDEsadaptivemodel-orderreductionproperorthogonaldecompositionspaceaposteriorierrorestimatesvariationaldiscretization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that, for unconstrained linear-quadratic optimal control of parabolic PDEs, you never need to construct a separate reduced space for the control: the first-order optimality condition places the reduced optimal control in a space inherited from the reduced adjoint basis, so the control-and-state-reduced problem has exactly the same minimizer as the state-only-reduced problem. This equivalence, proven in Lemma 2, lets the authors import variational discretization into adaptive model-order reduction without introducing any extra approximation error. They supply two-sided a posteriori error bounds for the optimal control, an error representation for the objective value in which the control-reduction term vanishes, and an adaptive POD algorithm that provably converges to the full-order optimal control. If correct, this means a cheaper combined reduction is as accurate as state reduction alone, and numerical experiments show speed-ups up to roughly 34× over a full-order gradient method.

What carries the argument

The central mechanism is the variational-discretization identity: the first-order optimality condition for the state-reduced problem forces the optimal control into the finite-dimensional space U_r generated by the images of the reduced adjoint basis under the control operator B'. This identity (Lemma 2) makes the control- and state-reduced OCP equivalent to the solely state-reduced OCP. The convergence argument rests on Lemma 10, which shows that adding a snapshot with a genuinely new direction strictly increases the maximal POD rank, and on Lemma 11, the convergence of the reduced solutions to the full-order solution as the reduced spaces grow to V.

What would settle it

Run the adaptive algorithm on a parabolic optimal control problem whose true optimal state and adjoint generate snapshots that all lie in a fixed finite-dimensional subspace not reachable from the initial guess (e.g., a target state orthogonal to the span of all initially generated snapshots); if the algorithm terminates with a non-zero gradient while the POD rank stops increasing, the density premise behind Theorem 13 fails. Alternatively, construct a nested sequence of finite-dimensional POD spaces with strictly increasing dimensions whose union is a proper closed subspace of V and check whe

Watch

Extended reading notes

Core claim

The paper proves that a Galerkin reduction of the state space V_r ⊂ V induces, through the optimality condition ū = −1/β B' p̄, a reduced control space U_r = span{J_U^{-1} B' v_i} such that the minimizer of the control-and-state-reduced OCP equals that of the state-reduced OCP (Lemma 2). It further proves that the POD-based adaptive algorithm, which enriches V_r with the current state and adjoint snapshots, produces a sequence of reduced controls converging to the full-order optimal control, with rigorous lower and upper a posteriori bounds on the control error and a representation of the objective-value error in which the control-reduction contribution vanishes.

Load-bearing premise

The convergence proof assumes that every non-terminating iteration adds a snapshot with a genuinely new direction, so that the POD spaces eventually span all of V; the proof establishes only that the dimensions strictly increase, not that their union is dense, and the numerical experiments additionally use energy-truncated POD that falls outside the proof's assumptions.

Editorial extensions

If this is right

Because the control space is inherited, the combined reduced problem has the same minimizer as state-only reduction; control reduction introduces no additional approximation error.
The two-sided a posteriori estimates (Corollary 5) apply to any candidate control, so error certification does not require solving the reduced problem to optimality.
The adaptive POD Algorithm 1 converges: for ε = 0 the iterates converge to the FOM optimal control; for ε > 0 it terminates in finite steps with a certified error (Theorem 13).
The objective-value error representation (Theorem 7) isolates the state-reduction contribution, since the control-reduction term vanishes.
Numerically, the combined reduction achieves speed-ups up to 34× over the full-order model and outperforms state-only reduction, while the error estimators bracket the true error.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The equivalence suggests that control constraints would break the pure inheritance argument; handling bound constraints would require a separate projection step, since the reduced control must be forced into the admissible set.
The rank-growth proof (Lemma 10) ensures the dimensions increase, but not that the union of the POD spaces is dense in V; if snapshots accumulate in a proper closed subspace, the algorithm could stagnate at a non-optimal control. A testable extension would explicitly check density, e.g., by tracking the decay of the snapshot residual in the ambient space.
The energy-truncated POD used in the numerical experiments (r_k ≤ r̄_{S_k}) lies outside the formal convergence proof; a natural next step is to extend the proof to energy truncation or to construct counterexamples where truncation stalls the algorithm.
The error-representation identity (Theorem 7) suggests a goal-oriented enrichment strategy: use the residual terms in (25) to select which state directions to add rather than pure snapshot accumulation, potentially accelerating convergence.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: the central observation is correct and practically useful. Galerkin state reduction induces a control space through the optimality condition, and Lemma 2 — the control-and-state-reduced problem has the same minimizer as the state-reduced one — is a simple, right, and useful thing to lean on. The lower and upper a posteriori bounds in Corollary 5 are a clean repackaging of the standard residual estimate for an arbitrary control, and the numerics are honest: code is provided, the equivalence is confirmed to machine precision, and the speedups (up to 34x) are plausible because the inner problem is cheap and the FOM solves feed both the estimator and the snapshots.

The soft spots are real but localizable. The proof of Theorem 13 does not establish convergence as written. It shows r_k strictly increases and then asserts that iterating constructs an orthonormal basis of V. That inference is invalid: strictly increasing finite-dimensional subspaces can all lie in a proper closed subspace. To get density of the union of POD spaces you need an argument that the snapshots eventually span V. Lemma 12 only rules out both snapshots lying in the current W^r; it doesn't exclude all iterates and snapshots living in a fixed proper subspace. The numerics don't close this gap because Section 5.2 uses energy-truncated ranks, which Remark 14 explicitly puts outside the proof's assumptions. I suspect the algorithm genuinely converges for parabolic problems like the one tested, but the proof needs another idea.

There are also line-level errors in the proof of Theorem 7. The Lagrangian in (22) is missing the ∂_t y term, so the printed identity \hat J(\hat u_r)=L(\hat x_r) doesn't hold as stated. These look fixable.

Who this is for: people doing adaptive POD for linear-quadratic PDE-constrained optimal control, especially in online/realtime settings. The equivalence is a worthwhile result on its own; the certified adaptive algorithm is the intended contribution. I'd send this to a serious referee, explicitly asking them to check the convergence proof. The right outcome is major revision, not rejection.

Referee Report

2 major / 5 minor

Summary. The paper studies unconstrained linear-quadratic optimal control problems with parabolic PDE constraints and infinite-dimensional control/state spaces. Its main contributions are: (i) a proof that a Galerkin/POD reduction of the state space induces a reduced control space via the first-order optimality condition, so that the state-reduced OCP and the combined state/control-reduced OCP have the same minimizer (Lemma 2); (ii) lower and upper a posteriori error bounds for the optimal control with respect to an arbitrary control (Corollary 5); (iii) an error representation for the optimal objective value (Theorem 7); and (iv) an adaptive POD-based algorithm (Algorithm 1) with a claimed convergence theorem (Theorem 13). Numerical experiments on a 2D heat equation illustrate the equivalence, the sharpness of the estimators, and computational speedups, with code provided.

Significance. If the main results hold, the paper makes a useful contribution: control-space reduction is obtained for free from state-space reduction, which simplifies both analysis and implementation. The lower a posteriori bound in Corollary 5 is a genuine addition over the usual upper-only estimates, and the numerical verification is careful and reproducible. However, two load-bearing parts of the manuscript are not sound as written: the proof of Theorem 7 contains line-level errors, and the convergence proof of Theorem 13 has a serious logical gap. The equivalence result (Lemma 2) and the error estimator (Corollary 5) appear sound and are well supported by the numerics, so the paper's core idea is promising, but the stated theorems need substantial revision.

major comments (2)

[Sec. 3.2, Eq. (22) and proof of Theorem 7] The Lagrangian in (22) omits the time-derivative term ⟨∂_t y, p⟩; with the displayed L, the stationarity condition L'(\bar x)=0 is not equivalent to the optimality system (3b),(6),(7). Moreover, the proof asserts \hat J(\bar u)-\hat J(\hat u_r)=L(\bar x)-L(\hat x^r). This is false as printed: L(\hat x^r) uses the reduced state \bar y_r, whereas \hat J(\hat u_r) is evaluated at the full-order state y(\hat u_r). The correct statement would involve the reduced objective \hat J^r(\hat u_r), and the proof would need a Lagrangian that encodes the reduced state equation. As it stands, the proof does not establish the error representation (25).
[Sec. 4.2, proof of Theorem 13] The decisive step in the proof is the sentence: 'By iterating this procedure, we construct an orthonormal basis of V and Lemma 11 ensures convergence, since r_k → ∞.' This is not justified. Lemma 10 only guarantees that dim V^{r_k} strictly increases. A nested sequence of finite-dimensional subspaces with strictly increasing dimensions need not have a dense union; e.g., in V=ℓ², V^{r_k}=span{e_2,...,e_{k+1}} has r_k→∞ but the union is contained in the proper closed subspace {x_1=0}. Lemma 11 requires an orthonormal basis of all of V. The earlier statement that \bar r_{S_{k+1}}=∞ implies u_{k+1}=\bar u is equally unjustified, since an infinite-dimensional POD subspace need not be all of V. An additional argument is needed to prove that the union of the POD spaces is dense, or the theorem must be weakened or placed under extra assumptions (e.g., a controllability/persistence condition). Th

minor comments (5)

[Sec. 4.1, Lemma 10] The proof assumes 'W.l.o.g.' that exactly one snapshot in S_+\S satisfies (32). This is not a harmless reduction unless the argument is repeated for each such snapshot; otherwise the sum over S_+\S must be handled directly. The proof should be rewritten to treat the general case.
[Sec. 4.2, Lemma 11] The proof refers to [25, Theorem 3.11] and states that replacing finite-dimensional controls by infinite-dimensional U 'does not change the proof.' This is not obvious, since the control space appears in the cost, the optimality condition, and the compactness arguments. Please provide a self-contained argument or a more precise reference.
[Sec. 5.2 and Remark 14] The numerical experiments use energy-truncated POD with r_k ≤ \bar r_{S_k}, which Remark 14 explicitly places outside the assumptions of Theorem 13. The paper should state clearly that the numerical verification of convergence is for the modified algorithm, not for the theorem as proved.
[Table 2] The column headings are ambiguous: 'k' is not defined in the table, and the two error-estimator columns (overline and underline Delta) are easily confused. Please add a caption defining k, e_u, and the estimator notation.
[Sec. 3.2, notation] The proof of Theorem 7 switches between \hat J and \hat J^r without comment. Since these are different objects (FOM reduced cost vs. state-reduced cost), the notation should be fixed throughout Section 3.2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reductions follow from the optimality system and the error bounds are derived, not fitted.

full rationale

The derivation chain is self-contained. Lemma 2 defines U^r as span(J_U^{-1}B'v_i) and then derives from the state-reduced optimality condition (9) that the state-reduced minimizer satisfies (13), so equality with the minimizer of the control- and state-reduced OCP follows from strict convexity/uniqueness, not from assuming the conclusion. Corollary 5 is the coercivity/residual equivalence of Lemma 4 applied to the FOM optimality condition, with no fitted constants. Theorem 7 is a Lagrange-identity error representation requiring only the FOM and ROM optimality systems. The adaptive loop's rank-growth criterion (Lemma 10) is a genuine characterization of new POD directions, and the snapshots are FOM states and adjoints, not copies of the reduced solution. The self-citations, including the deferred proof of Lemma 11 to [25], supply background technique rather than importing the paper's conclusion: Lemma 11 is a standard Galerkin convergence statement with stated assumptions, not an unverified uniqueness claim that already contains the target result. The reviewer-flagged issue in Theorem 13 — that strictly increasing r_k does not by itself guarantee that the union of V^{r_k} is dense in V — is a mathematical correctness gap, not a circular reduction, because the proof does not use the target as an input. Remark 14's admission that numerical energy truncation lies outside the proof is an acknowledged limitation, not evidence that the estimators were fitted to their targets. Hence no prediction reduces by construction to its inputs.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central derivation adds no fitted constants or hypothetical entities. It relies on standard functional-analytic assumptions, the POD spectral theorem, and prior residual-based error estimation results. The mathematical content is self-contained in the sense that the control space is defined directly from the state-reduced optimality condition, not fitted to the target solution.

free parameters (2)

POD energy truncation threshold = 10^{-12}
Chosen by hand in Section 5.2 to cap basis size in numerics; not part of the convergence theorem, which assumes maximal POD rank (Remark 14).
Gradient tolerance tau = 10^{-8} (reference solve 10^{-12})
Section 5.2; stopping tolerance for FOM and ROM iterations; algorithmic parameter, not fitted to data.

assumptions (6)

domain assumption Assumptions (A1)-(A2): A is uniformly coercive in L^infty(0,T;L(V,V')), C in L^infty(L(H,H)), B in L(U,V'), beta>0, y_d in L^2(H), y0 in H.
Section 2.1; guarantees well-posedness of state/adjoint equations and a unique minimizer; the whole theory operates in this setting.
domain assumption U, V, H are real separable Hilbert spaces with compact dense embedding V subset H.
Section 1.1; needed for POD spectral theory and Galerkin convergence.
standard math POD correlation operator R_S is compact, nonnegative, self-adjoint and its eigenfunctions give the optimal POD basis.
Theorem 8, taken from [11]; standard spectral theorem for compact self-adjoint operators.
standard math Error-residual equivalence Lemma 4 for coercive operator Q.
Standard residual-based a posteriori bound; used to prove Corollary 5.
domain assumption ROM convergence Lemma 11: as r -> infinity, Galerkin state/adjoint/control minimizers converge to FOM minimizers (based on [25, Thm 3.11]).
Section 4.2; assumes reduced spaces form a complete approximation of V; this is exactly what the convergence proof needs to establish for the adaptively chosen spaces.
standard math Dual weighted residual error representation of Becker-Rannacher [24] underlies Theorem 7.
Section 3.2; the derivation of the optimal-value error formula uses this framework plus the state-reduced optimality condition.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Optimality-Based Control Space Reduction for Infinite-Dimensional Control Spaces." pith.science (2026). https://pith.science/paper/NL7JCTZT

@misc{pith2026251014479,
  author       = {Pith},
  title        = {Pith review of: Optimality-Based Control Space Reduction for Infinite-Dimensional Control Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NL7JCTZT}},
  note         = {Machine review of arXiv:2510.14479}
}

read the original abstract

We consider linear model reduction in both the control and state variables for unconstrained linear-quadratic optimal control problems subject to time-varying parabolic PDEs. The first-order optimality condition for a state-space reduced model naturally leads to a reduced structure of the optimal control. Thus, we consider a control- and state-reduced problem that admits the same minimizer as the solely state-reduced problem. Lower and upper \emph{a posteriori} error bounds for the optimal control and a representation for the error in the optimal function value are provided. These bounds are used in an adaptive algorithm to solve the control problem. We prove its convergence and numerically demonstrate the advantage of combined control and state space reduction.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 8 canonical work pages

[1]

Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints vol. 23. Springer, Dordrecht (2009). https://doi.org/10.1007/ 978-1-4020-8839-1

2009
[2]

Computational Optimization and Applications 30(1), 45–61 (2005) https://doi.org/10.1007/s10589-005-4559-5

Hinze, M.: A variational discretization concept in control constrained optimiza- tion: the linear-quadratic case. Computational Optimization and Applications 30(1), 45–61 (2005) https://doi.org/10.1007/s10589-005-4559-5

work page doi:10.1007/s10589-005-4559-5 2005
[3]

Computational Science and Engineering1(1), 3 (2024) https://doi.org/10.1007/ s44207-024-00002-z

Kartmann, M., Keil, T., Ohlberger, M., Volkwein, S., Kaltenbacher, B.: Adap- tive reduced basis trust region methods for parameter identification problems. Computational Science and Engineering1(1), 3 (2024) https://doi.org/10.1007/ s44207-024-00002-z

2024
[4]

Adaptive Reduced Basis Trust Region Methods for Parabolic Inverse Problems

Kartmann, M., Klein, B., Ohlberger, M., Schuster, T., Volkwein, S.: Adaptive reduced basis trust region methods for parabolic inverse problems. arXiv (2025) https://doi.org/10.48550/arXiv.2507.11130

work page Pith review arXiv doi:10.48550/arxiv.2507.11130 2025
[5]

SIAM Jour- nal on Scientific Computing42(1), 271–291 (2020) https://doi.org/10.1137/ 18M1227147

Ali, A.A., Hinze, M.: Reduced basis methods – an application to variational discretization of parametrized elliptic optimal control problems. SIAM Jour- nal on Scientific Computing42(1), 271–291 (2020) https://doi.org/10.1137/ 18M1227147

2020
[6]

Journal of Scientific Computing75(1), 276–307 (2018) https://doi.org/10.1007/ s10915-017-0539-z

K¨ archer, M., Tokoutsi, Z., Grepl, M.A., Veroy, K.: Certified reduced basis meth- ods for parametrized elliptic optimal control problems with distributed controls. Journal of Scientific Computing75(1), 276–307 (2018) https://doi.org/10.1007/ s10915-017-0539-z

2018
[7]

Optimization and Engineering19(3), 663–695 (2018) https://doi.org/10.1007/s11081-018-9389-2 23

K¨ archer, M., Boyaval, S., Grepl, M.A., Veroy, K.: Reduced basis approximation and a posteriori error bounds for 4d-var data assimilation. Optimization and Engineering19(3), 663–695 (2018) https://doi.org/10.1007/s11081-018-9389-2 23

work page doi:10.1007/s11081-018-9389-2 2018
[8]

SIAM Journal on Scientific Computing38(6), 3921–3946 (2016) https://doi.org/10.1137/16M1059898

Bader, E., K¨ archer, M., Grepl, M.A., Veroy, K.: Certified reduced basis meth- ods for parametrized distributed elliptic optimal control problems with control constraints. SIAM Journal on Scientific Computing38(6), 3921–3946 (2016) https://doi.org/10.1137/16M1059898

doi:10.1137/16m1059898 2016

Show all 27 references

[9]

SIAM Journal on Scientific Computing 39(5), 434–460 (2017) https://doi.org/10.1137/16M1081981

Qian, E., Grepl, M., Veroy, K., Willcox, K.: A certified trust region reduced basis approach to pde-constrained optimization. SIAM Journal on Scientific Computing 39(5), 434–460 (2017) https://doi.org/10.1137/16M1081981

2017 doi
[10]

ESAIM: Mathematical Modelling and Numerical Analysis55(3), 1239–1269 (2021) https://doi.org/10.1051/m2an/ 2021019

Keil, T., Mechelli, L., Ohlberger, M., Schindler, F., Volkwein, S.: A non- conforming dual approach for adaptive trust-region reduced basis approximation of PDE-constrained parameter optimization. ESAIM: Mathematical Modelling and Numerical Analysis55(3), 1239–1269 (2021) http...

2021 doi
[11]

In: Model Reduction and Approximation: Theory and Algo- rithms, pp

Gubisch, M., Volkwein, S.: Proper orthogonal decomposition for linear-quadratic optimal control. In: Model Reduction and Approximation: Theory and Algo- rithms, pp. 3–63. Society for Industrial and Applied Mathematics, ??? (2017). https://doi.org/10.1137/1.9781611974829.ch1

2017 doi
[12]

Open Journal of Fluid Dynamics7(04), 525 (2017) https://doi.org/10.4236/ojfd.2017

Kone, T.C., Marchesse, Y., Panneton, R.,et al.: A numerical approach to possible identification of the noisiest zones of a wall surface with a flow interaction. Open Journal of Fluid Dynamics7(04), 525 (2017) https://doi.org/10.4236/ojfd.2017. 74036

2017 doi
[13]

Afanasiev, K., Hinze, M.: Adaptive control of a wake flow using proper orthogonal decomposition. Lect. Notes Pure Appl. Math.216(2001) https://doi.org/10. 1201/9780203904169.ch13

2001
[14]

ESAIM: Mathematical Modelling and Numerical Analysis42, 1–23 (2008) https://doi.org/10.1051/m2an:2007054

Kunisch, K., Volkwein, S.: Proper orthogonal decomposition for optimality sys- tems. ESAIM: Mathematical Modelling and Numerical Analysis42, 1–23 (2008) https://doi.org/10.1051/m2an:2007054

2008 doi
[15]

Control and Cybernetics40(4), 1109–1124 (2011)

Volkwein, S.: Optimality system POD and a-posteriori error analysis for linear- quadratic problems. Control and Cybernetics40(4), 1109–1124 (2011)

2011
[16]

Computational Optimization and Applications44(1), 83–115 (2009) https://doi.org/10.1007/s10589-008-9224-3

Tr¨ oltzsch, F., Volkwein, S.: POD a-posteriori error estimates for linear-quadratic optimal control problems. Computational Optimization and Applications44(1), 83–115 (2009) https://doi.org/10.1007/s10589-008-9224-3

2009 doi
[17]

arXiv (2024) https://doi.org/10.48550/arXiv.2412.12930

Kartmann, M., Manucci, M., Unger, B., Volkwein, S.: Certified model predic- tive control for switched evolution equations using model order reduction. arXiv (2024) https://doi.org/10.48550/arXiv.2412.12930

2024 doi arXiv
[18]

arXiv (2025) https://doi.org/10

Delavari, E., Khanzada, F.K., Kwon, J.: Action space reduction strategies for reinforcement learning in autonomous driving. arXiv (2025) https://doi.org/10. 48550/arXiv.2507.05251 24

2025 doi arXiv
[19]

Computers & Mathematics with Applications157, 215–234 (2024) https://doi.org/10.1016/j.camwa.2024.01.007

Chun, M.M., Edwards, B.L., Bukshtynov, V.: Multiscale optimization via enhanced multilevel PCA-based control space reduction for electrical impedance tomography imaging. Computers & Mathematics with Applications157, 215–234 (2024) https://doi.org/10.1016/j.camwa.2024.01.007

2024 doi
[20]

IEEE Robotics and Automation Letters 3(1), 328–335 (2018) https://doi.org/10.1109/LRA.2017.2743100

Sadati, S.M.H., Naghibi, S.E., Walker, I.D., Althoefer, K., Nanayakkara, T.: Con- trol space reduction and real-time accurate modeling of continuum manipulators using Ritz and Ritz–Galerkin methods. IEEE Robotics and Automation Letters 3(1), 328–335 (2018) https://doi.org/10.1...

2018
[21]

17356821

Kartmann, M., Volkwein, S.: Code for ”Optimality-Based Control Reduction for Infinite-Dimensional Control Spaces” (2025) https://doi.org/10.5281/zenodo. 17356821

2025 doi
[22]

Graduate Studies in Mathemat- ics, vol

Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathemat- ics, vol. 19. American Mathematical Society, Providence, Rhode Island (2010). https://bookstore.ams.org/gsm-19-r

2010
[23]

Model reduction and Approximation: Theory and Algorithms15, 65 (2017) https://doi.org/10.1137/ 1.9781611974829.ch

Haasdonk, B.: Reduced basis methods for parametrized PDEs–a tutorial introduction for stationary and instationary problems. Model reduction and Approximation: Theory and Algorithms15, 65 (2017) https://doi.org/10.1137/ 1.9781611974829.ch

2017
[24]

Acta numerica10, 1–102 (2001) https: //doi.org/10.1017/S0962492901000010

Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta numerica10, 1–102 (2001) https: //doi.org/10.1017/S0962492901000010

2001 doi
[25]

arXiv (2025) https: //doi.org/10.48550/arXiv.2508.16801

Azmi, B., Kartmann, M., Volkwein, S.: Stabilization of parabolic time-varying PDEs using certified reduced-order receding horizon control. arXiv (2025) https: //doi.org/10.48550/arXiv.2508.16801

2025 doi arXiv
[26]

SIAM Journal on Scientific Computing40(5), 3267–3292 (2018) https://doi.org/10.1137/16M1085413

Himpe, C., Leibner, T., Rave, S.: Hierarchical approximate proper orthogonal decomposition. SIAM Journal on Scientific Computing40(5), 3267–3292 (2018) https://doi.org/10.1137/16M1085413

2018 doi
[27]

IMA Journal of Numerical Analysis42(4), 2984–3021 (2022) https://doi.org/10.1093/imanum/ drab056 25

Azmi, B., Kunisch, K.: On the convergence and mesh-independent property of the Barzilai–Borwein method for PDE-constrained optimization. IMA Journal of Numerical Analysis42(4), 2984–3021 (2022) https://doi.org/10.1093/imanum/ drab056 25

2022 doi

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints vol. 23. Springer, Dordrecht (2009). https://doi.org/10.1007/ 978-1-4020-8839-1

2009

[2] [2]

Computational Optimization and Applications 30(1), 45–61 (2005) https://doi.org/10.1007/s10589-005-4559-5

Hinze, M.: A variational discretization concept in control constrained optimiza- tion: the linear-quadratic case. Computational Optimization and Applications 30(1), 45–61 (2005) https://doi.org/10.1007/s10589-005-4559-5

work page doi:10.1007/s10589-005-4559-5 2005

[3] [3]

Computational Science and Engineering1(1), 3 (2024) https://doi.org/10.1007/ s44207-024-00002-z

Kartmann, M., Keil, T., Ohlberger, M., Volkwein, S., Kaltenbacher, B.: Adap- tive reduced basis trust region methods for parameter identification problems. Computational Science and Engineering1(1), 3 (2024) https://doi.org/10.1007/ s44207-024-00002-z

2024

[4] [4]

Adaptive Reduced Basis Trust Region Methods for Parabolic Inverse Problems

Kartmann, M., Klein, B., Ohlberger, M., Schuster, T., Volkwein, S.: Adaptive reduced basis trust region methods for parabolic inverse problems. arXiv (2025) https://doi.org/10.48550/arXiv.2507.11130

work page Pith review arXiv doi:10.48550/arxiv.2507.11130 2025

[5] [5]

SIAM Jour- nal on Scientific Computing42(1), 271–291 (2020) https://doi.org/10.1137/ 18M1227147

Ali, A.A., Hinze, M.: Reduced basis methods – an application to variational discretization of parametrized elliptic optimal control problems. SIAM Jour- nal on Scientific Computing42(1), 271–291 (2020) https://doi.org/10.1137/ 18M1227147

2020

[6] [6]

Journal of Scientific Computing75(1), 276–307 (2018) https://doi.org/10.1007/ s10915-017-0539-z

K¨ archer, M., Tokoutsi, Z., Grepl, M.A., Veroy, K.: Certified reduced basis meth- ods for parametrized elliptic optimal control problems with distributed controls. Journal of Scientific Computing75(1), 276–307 (2018) https://doi.org/10.1007/ s10915-017-0539-z

2018

[7] [7]

Optimization and Engineering19(3), 663–695 (2018) https://doi.org/10.1007/s11081-018-9389-2 23

K¨ archer, M., Boyaval, S., Grepl, M.A., Veroy, K.: Reduced basis approximation and a posteriori error bounds for 4d-var data assimilation. Optimization and Engineering19(3), 663–695 (2018) https://doi.org/10.1007/s11081-018-9389-2 23

work page doi:10.1007/s11081-018-9389-2 2018

[8] [8]

SIAM Journal on Scientific Computing38(6), 3921–3946 (2016) https://doi.org/10.1137/16M1059898

Bader, E., K¨ archer, M., Grepl, M.A., Veroy, K.: Certified reduced basis meth- ods for parametrized distributed elliptic optimal control problems with control constraints. SIAM Journal on Scientific Computing38(6), 3921–3946 (2016) https://doi.org/10.1137/16M1059898

doi:10.1137/16m1059898 2016