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arxiv: 2604.24065 · v1 · submitted 2026-04-27 · 🧮 math.OC

An online adaptive finite-element method for nonsmooth PDE-constrained optimization

Harbir Antil , Robert J. Baraldi , Rohit Khandelwal , Drew P. Kouri This is my paper

Pith reviewed 2026-05-08 02:34 UTC · model grok-4.3

classification 🧮 math.OC
keywords adaptive finite element methodPDE-constrained optimizationnonsmooth optimizationtrust-region algorithma posteriori error estimationtopology optimizationcontrol optimizationsparsifying regularizers
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The pith

A trust-region adaptive finite-element algorithm solves nonsmooth PDE-constrained optimization by refining meshes via state and adjoint error estimators.

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

The paper develops an algorithm for PDE-constrained optimization problems whose objective is the sum of a smooth possibly nonconvex term and a nonsmooth convex term, including cases with sparsifying regularizers and convex constraints. It combines inexact trust-region steps, which tolerate nonsmoothness, with adaptive finite-element meshes that start coarse and refine only where error estimators for the state and adjoint equations indicate insufficient accuracy. This controls the quality of the objective value and gradient while keeping the total number of degrees of freedom modest. Readers would care because the method targets practical computation of problems that exhibit sparsity or localized features, where uniform fine meshes would be unnecessarily expensive.

Core claim

We present a trust-region-based adaptive finite-element algorithm for numerically solving a class of nonsmooth PDE-constrained optimization problems that includes problems with sparsifying regularizers and convex constraints. Our method combines the robustness of inexact trust-region algorithms for nonsmooth problems with the efficiency of adaptive finite-element discretizations. Starting from a coarse mesh, the algorithm automatically refines the discretization based on reliable a posteriori error estimators for both the state and adjoint equations, systematically controlling the accuracy of the computed smooth objective function value and gradient.

What carries the argument

Trust-region steps on an adaptively refined finite-element discretization whose mesh is driven by a posteriori error estimators for the state and adjoint equations, controlling the accuracy of the smooth part of the objective and its gradient.

If this is right

  • The algorithm automatically balances computational cost against accuracy by refining only where needed to resolve localized phenomena and sparsity patterns.
  • It applies directly to representative control and topology optimization problems with nonsmooth regularizers.
  • It enables high-resolution capture of sparse structures in the state and control variables without a uniformly fine mesh.
  • The inexact trust-region framework ensures robustness to the nonsmooth convex component while the adaptivity controls discretization error in the smooth component.

Where Pith is reading between the lines

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

  • The same error-driven refinement strategy could reduce degrees of freedom in larger-scale or three-dimensional instances where uniform refinement becomes prohibitive.
  • The approach may extend to other nonsmooth PDE optimization settings provided the state and adjoint estimators can be shown reliable.
  • Integration with parallel solvers or reduced-order models could further improve performance on problems with many local features.

Load-bearing premise

The a posteriori error estimators for the state and adjoint equations remain reliable and effective even when the objective is nonsmooth and the solution exhibits sparsity structures.

What would settle it

Numerical runs on a problem with strong sparsity in which the adaptive mesh produces objective values or gradients whose error does not decrease at the expected rate, or in which the trust-region iteration fails to converge, would falsify the claim that the estimators stay reliable.

Figures

Figures reproduced from arXiv: 2604.24065 by Drew P. Kouri, Harbir Antil, Robert J. Baraldi, Rohit Khandelwal.

Figure 1
Figure 1. Figure 1: Results of Algorithm 4 for the Poisson control problem. The last row represents the final view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the boundary conditions for the heat conduction topology optimization view at source ↗
Figure 3
Figure 3. Figure 3: Results of Algorithm 4 for the first heat conduction topology optimization example. Our view at source ↗
Figure 4
Figure 4. Figure 4: Results of Algorithm 4 for the second heat conduction topology optimization example. view at source ↗
read the original abstract

We present a trust-region-based adaptive finite-element algorithm for numerically solving a class of nonsmooth PDE-constrained optimization problems that includes problems with sparsifying regularizers and convex constraints. In particular, we consider the class of problems whose objective function is the sum of a smooth, possibly nonconvex, function and a nonsmooth extended real-valued convex function. Our method combines the robustness of inexact trust-region algorithms for nonsmooth problems with the efficiency of adaptive finite-element discretizations. Starting from a coarse mesh, the algorithm automatically refines the discretization based on reliable a posteriori error estimators for both the state and adjoint equations, systematically controlling the accuracy of the computed smooth objective function value and gradient. This adaptivity mechanism balances computational cost and solution accuracy, enabling high resolution of localized phenomena and sparsity structures in the state and control variables. We demonstrate the performance of our algorithm through numerical experiments on representative control and topology optimization examples.

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

2 major / 2 minor

Summary. The paper presents a trust-region-based adaptive finite-element algorithm for a class of nonsmooth PDE-constrained optimization problems whose objective is the sum of a smooth (possibly nonconvex) function and a nonsmooth convex extended real-valued function. The method combines inexact trust-region steps with automatic mesh refinement driven by a posteriori error estimators for the state and adjoint equations, starting from a coarse mesh to control the accuracy of the smooth objective value and gradient while resolving sparsity structures. Performance is illustrated through numerical experiments on control and topology optimization examples.

Significance. If the reliability of the a posteriori estimators can be established for the nonsmooth regime, the work would provide a practical and efficient framework for high-resolution solution of PDE-constrained problems with sparsifying regularizers and convex constraints. The combination of robust trust-region globalization with adaptive discretization is a clear strength, and the reported numerical demonstrations on representative examples give concrete evidence of the method's ability to balance computational cost with solution accuracy.

major comments (2)
  1. [Abstract] Abstract and algorithm description: the manuscript asserts that the a posteriori error estimators for the state and adjoint equations remain reliable and enable systematic control of the smooth objective and gradient even when the objective contains nonsmooth convex terms that induce sparsity or reduced regularity. No additional analysis, modification, or reference establishing reliability bounds under these conditions is supplied; standard residual-based estimators for elliptic PDEs typically require H^2 regularity that may fail here. This assumption is load-bearing for the claimed adaptivity mechanism and efficiency gains.
  2. [Numerical experiments] Numerical experiments section: while the examples demonstrate the algorithm on control and topology optimization problems, no quantitative assessment is given of how the error estimators behave on solutions exhibiting sparsity (e.g., vanishing sets or discontinuities) or of the actual reduction in degrees of freedom versus a uniform refinement strategy. This weakens the efficiency argument that is central to the paper's contribution.
minor comments (2)
  1. The description of the problem class could be made more precise by explicitly stating the PDE type (e.g., elliptic) and the precise functional setting for the state and control variables.
  2. Figure captions and legends in the numerical section would benefit from additional detail on mesh sizes, iteration counts, and error decay rates to facilitate direct comparison with the claimed balancing of cost and accuracy.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each of the major comments below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: [Abstract] Abstract and algorithm description: the manuscript asserts that the a posteriori error estimators for the state and adjoint equations remain reliable and enable systematic control of the smooth objective and gradient even when the objective contains nonsmooth convex terms that induce sparsity or reduced regularity. No additional analysis, modification, or reference establishing reliability bounds under these conditions is supplied; standard residual-based estimators for elliptic PDEs typically require H^2 regularity that may fail here. This assumption is load-bearing for the claimed adaptivity mechanism and efficiency gains.

    Authors: The state and adjoint equations are linear second-order elliptic PDEs with fixed right-hand sides determined by the current control iterate. Consequently, the standard residual-based a posteriori error estimators from the adaptive FEM literature apply directly, provided the usual assumptions on the domain (e.g., convex or C^1 boundary) and data regularity hold. The nonsmooth convex term affects only the control variable and the optimality conditions, but does not alter the PDE regularity for each fixed control. We will revise the abstract, introduction, and add a new subsection in the analysis to explicitly state these regularity assumptions and cite standard references for the reliability of the estimators under H^2 regularity. This addresses the concern by making the scope of the reliability claim precise. revision: yes

  2. Referee: [Numerical experiments] Numerical experiments section: while the examples demonstrate the algorithm on control and topology optimization problems, no quantitative assessment is given of how the error estimators behave on solutions exhibiting sparsity (e.g., vanishing sets or discontinuities) or of the actual reduction in degrees of freedom versus a uniform refinement strategy. This weakens the efficiency argument that is central to the paper's contribution.

    Authors: We agree that quantitative comparisons would better support the efficiency claims. In the revised manuscript, we will augment the numerical experiments with tables reporting the degrees of freedom, estimator effectivity indices, and CPU times for both the adaptive algorithm and a uniform refinement strategy achieving comparable accuracy on the same test problems. Additional plots will illustrate the local refinement patterns near sparsity sets and discontinuities, together with the observed convergence rates of the estimators. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algorithm integrates established trust-region and a posteriori estimation methods

full rationale

The paper's central contribution is an adaptive FEM algorithm for nonsmooth PDE-constrained optimization that starts from a coarse mesh and refines using a posteriori error estimators for state and adjoint equations, combined with inexact trust-region methods. No equations or steps in the provided description reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the reliability of the estimators is invoked as an external assumption from numerical PDE analysis rather than derived internally. The derivation chain remains self-contained against standard benchmarks in optimization and FEM theory, with no renaming of known results or smuggling of ansatzes via self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions for PDE-constrained optimization and numerical analysis; no free parameters or invented entities are introduced beyond the algorithmic framework itself.

axioms (2)
  • domain assumption The objective is the sum of a smooth (possibly nonconvex) function and a nonsmooth extended real-valued convex function.
    This defines the problem class handled by the algorithm.
  • domain assumption Reliable a posteriori error estimators exist for the state and adjoint equations in the presence of nonsmoothness.
    This underpins the adaptivity mechanism and accuracy control.

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discussion (0)

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Works this paper leans on

63 extracted references · 35 canonical work pages

  1. [1]

    Ainsworth and J

    M. Ainsworth and J. T. Oden.A posteriori error estimation in finite element analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York,

  2. [2]

    URLhttps://doi.org/10.1002/9781118032824

    ISBN 0-471-29411-X. URLhttps://doi.org/10.1002/9781118032824

    work page doi:10.1002/9781118032824

  3. [3]

    Allendes, F

    A. Allendes, F. Fuica, and E. Ot´ arola. Adaptive finite element methods for sparse PDE- constrained optimization.IMA J. Numer. Anal., 40(3):2106–2142, 2020. ISSN 0272-4979,1464-

    2020

  4. [4]

    URLhttps://doi.org/10.1093/imanum/drz025

    work page doi:10.1093/imanum/drz025

  5. [5]

    Alshehri, H

    M. Alshehri, H. Antil, E. Herberg, and D. P. Kouri. An inexact semismooth newton method with application to adaptive randomized sketching for dynamic optimization.Finite Elem. Anal. Des., 228:104052, 2024. ISSN 0168-874X. doi: https://doi.org/10.1016/j.finel.2023. 104052

    work page doi:10.1016/j.finel.2023 2024

  6. [6]

    Antil, D

    H. Antil, D. P. Kouri, D. Ridzal, D. B. Robinson, and M. Salloum. Uniform flow in ax- isymmetric devices through permeability optimization.Optim. Eng., 2023. doi: 10.1007/ s11081-023-09820-0

    2023

  7. [7]

    Bangerth and R

    W. Bangerth and R. Rannacher.Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics, ETH Z¨ urich. Birkh¨ auser Basel, Basel, 2003. ISBN 978-3-0348-76-5-6

    2003

  8. [8]

    R. J. Baraldi and D. P. Kouri. A proximal trust-region method for nonsmooth optimization with inexact function and gradient evaluations.Math. Program., 201(1-2):559–598, 2023. ISSN 0025-5610,1436-4646. URLhttps://doi.org/10.1007/s10107-022-01915-3. 18 Ndof k Gridz u h 4225 26 5·10 4 48 1.1·10 5 55 1.5·10 5 90 Figure 3: Results of Algorithm 4 for the first...

    work page doi:10.1007/s10107-022-01915-3 2023

  9. [9]

    R. J. Baraldi and D. P. Kouri. Local convergence analysis of an inexact trust-region method for nonsmooth optimization.Optim. Lett., 18(3):663–680, 2024

    2024

  10. [10]

    R. J. Baraldi and D. P. Kouri. Efficient proximal subproblem solvers for a nonsmooth trust- region method.Comput. Optim. Appl., 90(1):193–226, 2025. ISSN 0926-6003,1573-2894. URL https://doi.org/10.1007/s10589-024-00628-x

    work page doi:10.1007/s10589-024-00628-x 2025

  11. [11]

    R. J. Baraldi, D. P. Kouri, and D. Ridzal.Trust-Region Methods with Inexact and Adaptive 19 Ndof k Gridz u h 4193 21 2·10 4 52 5·10 4 76 1.5·10 5 147 Figure 4: Results of Algorithm 4 for the second heat conduction topology optimization example. Our algorithm terminated after 147 iterations. Note theN dofs are approximate. Computations, pages 1–8. Springer...

    work page doi:10.1007/978-3-030-54621-2 2025

  12. [12]

    R. J. Baraldi, D. P. Kouri, and H. Antil. Memory-efficient nonsmooth dynamic opti- mization using adaptive randomized compression.Optim. Eng., pages 1–32, 2026. doi: 10.1007/s11081-025-10061-6

    work page doi:10.1007/s11081-025-10061-6 2026

  13. [13]

    Becker and B

    R. Becker and B. Vexler. A posteriori error estimation for finite element discretization 20 of parameter identification problems.Numer. Math., 96:435–459, 2004. doi: 10.1007/ s00211-003-0482-9

    2004

  14. [14]

    Becker and B

    R. Becker and B. Vexler. Mesh refinement and numerical sensitivity analysis for parameter calibration of partial differential equations.J. Comput. Phys., 206(1):95–110, 2005. doi: 10.1016/j.jfcp.2004.12.018

    work page doi:10.1016/j.jfcp.2004.12.018 2005

  15. [15]

    Becker, H

    R. Becker, H. Kapp, and R. Rannacher. Adaptive finite element methods for optimal control of partial differential equations: basic concept.SIAM J. Control Optim., 39(1):113–132, 2000. doi: 10.1127/S0363012999351097

    work page doi:10.1127/s0363012999351097 2000

  16. [16]

    Bellavia, S

    S. Bellavia, S. Gratton, and E. Riccietti. A Levenberg–Marquardt method for large nonlinear least-squares problems with dynamic accuracy in functions and gradients.Numer. Math., 140: 791–825, 2018. doi: 10.1007/s00211-018-0977-z

    work page doi:10.1007/s00211-018-0977-z 2018

  17. [17]

    Besier and R

    M. Besier and R. Rannacher. Goal-oriented space–time adaptivity in the finite element galerkin method for the computation of nonstationary incompressible flow.Int. J. Numer. Methods Fluids, 70(9):1139–1166, 2012. doi: 10.1002/fld.2735

    work page doi:10.1002/fld.2735 2012

  18. [18]

    Bonito, C

    A. Bonito, C. Canuto, R. H. Nochetto, and A. Veeser. Adaptive finite element methods.Acta Numer., 33:163–485, 2024. ISSN 0962-4929,1474-0508. doi: 10.1017/S0962492924000011

    work page doi:10.1017/s0962492924000011 2024

  19. [19]

    C. J. Budd, W. Huang, and R. D. Russell. Adaptivity with moving grids.Acta Numer., 18: 111–241, 2009

    2009

  20. [20]

    R. H. Byrd, F. E. Curtis, and J. Nocedal. An inexact SQP method for equality constrained optimization.SIAM J. Optim., 19(1):351–369, 2008. doi: 10.1137/060674004

    work page doi:10.1137/060674004 2008

  21. [21]

    Carstensen, M

    C. Carstensen, M. Feischl, M. Page, and Praetorius. Axioms and adaptivity.Computers & Mathematics with Applications, 67(6):1195–1253, 2014

    2014

  22. [22]

    R. G. Carter. On the global convergence of trust region algorithms using inexact gradient information.SIAM J. Numer. Anal., 28:251–265, 1991

    1991

  23. [23]

    R. G. Carter. Numerical experience with a class of algorithms for nonlinear optimization using inexact function and gradient information.SIAM J. Sci. Comput., 14(2):368–388, 1993. doi: 10.1137/0914023

    work page doi:10.1137/0914023 1993

  24. [24]

    A. R. Conn, N. I. M. Gould, and P. L. Toint. Global convergence of a class of trust region algorithms for optimization with simple bounds.SIAM J. Numer. Anal., 25(2):433–460, 1988. ISSN 0036-1429. URLhttps://doi.org/10.1137/0725029

    work page doi:10.1137/0725029 1988

  25. [25]

    Demkowicz, W

    L. Demkowicz, W. Rachowicz, and P. Devloo. A fully automatichp-adaptivity.SIAM J. Sci. Comput., 17:117–142, 2002. doi: 10.1023/A:1015192312705

    work page doi:10.1023/a:1015192312705 2002

  26. [26]

    Demlow and E

    A. Demlow and E. H. Georgoulis. Pointwise a posteriori error control for discontinuous Galerkin methods for elliptic problems.SIAM J. Numer. Anal., 50(5):2159–2181, 2012. ISSN 0036- 1429,1095-7170. doi: 10.1137/110846397

    work page doi:10.1137/110846397 2012

  27. [27]

    Dontchev and R

    A. Dontchev and R. T. Rockafellar. Convergence of inexact Newton methods for generalized equations.Math. Program., 139:115–137, 2013. doi: 10.1007/s10107-013-0664-x

    work page doi:10.1007/s10107-013-0664-x 2013

  28. [28]

    D¨ orfler

    W. D¨ orfler. A convergent adaptive algorithm for Poisson’s equation.SIAM J. Numer. Anal., 33(3):1106–1124, 1996. ISSN 0036-1429. doi: 10.1137/0733054. 21

    work page doi:10.1137/0733054 1996

  29. [29]

    Ern and J

    A. Ern and J. Guermond.Theory and Practice of Finite Elements. Applied Mathematical Sciences. Springer New York, 2004. ISBN 9780387205748

    2004

  30. [30]

    Fahl and E

    M. Fahl and E. W. Sachs. Reduced order modelling approaches to pde-constrained optimiza- tion based on proper orthogonal decomposition. InLarge-scale PDE-constrained optimization, pages 268–280. Springer, 2003

    2003

  31. [31]

    Feischl, T

    M. Feischl, T. F¨ uhrer, M. Karkulik, and D. Praetorius. ZZ-type a posteriori error estimators for adaptive boundary element methods on a curve.Eng. Anal. Bound. Elem., 38:49–60, 2014. doi: 10.1015/j.enganabound.2013.10.008

    work page doi:10.1015/j.enganabound.2013.10.008 2014

  32. [32]

    Garreis and M

    S. Garreis and M. Ulbrich. Constrained optimization with low-rank tensors and applications to parametric problems with PDEs.SIAM J. Sci. Comput., 39(1):A25–A54, 2017

    2017

  33. [33]

    Gersborg-Hansen, M

    A. Gersborg-Hansen, M. P. Bendsøe, and O. Sigmund. Topology optimization of heat con- duction problems using the finite volume method.Struct. Multidiscip. Optim., 31(4):251–259,

  34. [34]

    URLhttps://doi.org/10.1007/s00158-005-0584-3

    ISSN 1615-147X,1615-1488. URLhttps://doi.org/10.1007/s00158-005-0584-3

    work page doi:10.1007/s00158-005-0584-3

  35. [35]

    Heinkenschloss and D

    M. Heinkenschloss and D. Ridzal. A matrix-free trust-region sqp method for equality con- strained optimization.SIAM J. Optim., 24(3):1507–1541, 2014

    2014

  36. [36]

    Heinkenschloss and L

    M. Heinkenschloss and L. N. Vicente. Analysis of inexact trust-region sqp algorithms.SIAM J. Optim., 12(2):283–302, 2002. doi: 10.1137/S1052623499361543

    work page doi:10.1137/s1052623499361543 2002

  37. [37]

    Hille and R

    E. Hille and R. Phillips.Functional Analysis and Semi-groups. Number v. 27 in Colloquium publications. American Mathematical Society, 1957

    1957

  38. [38]

    Hinterm¨ uller, R

    M. Hinterm¨ uller, R. H. Hoppe, and C. L¨ obhard. Dual-weighted goal-oriented adaptive finite elements for optimal control of elliptic variational inequalities.ESAIM: COCV, 20:524–546,

  39. [39]

    doi: 10.1051/cocv/2013074

    work page doi:10.1051/cocv/2013074

  40. [40]

    Springer Dordrecht, 2009.doi:10.1007/978-1-4020-8839-1

    M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich.Optimization with PDE Constraints. Math- ematical Modelling: Theory and Applications. Springer Dordrecht, 2008. ISBN 978-1-4020- 8838-4. URLhttps://doi.org/10.1007/978-1-4020-8839-1

    work page doi:10.1007/978-1-4020-8839-1 2008

  41. [41]

    D. P. Kouri. A matrix-free trust-region Newton algorithm for convex-constrained optimization. Optimization Letters, 2020. In Review

    2020

  42. [42]

    D. P. Kouri, M. Heinkenschloss, D. Ridzal, and B. G. van Bloemen Waanders. A trust-region algorithm with adaptive stochastic collocation for PDE optimization under uncertainty.SIAM J. Sci. Comput., 35(4):A1847–A1879, 2013

    2013

  43. [43]

    D. P. Kouri, M. Heinkenschloss, D. Ridzal, and B. G. van Bloemen Waanders. Inexact objec- tive function evaluations in a trust-region algorithm for PDE-constrained optimization under uncertainty.SIAM J. Sci. Comput., 36(6):A3011–A3029, 2014

    2014

  44. [44]

    Lang.Undergraduate Analysis

    S. Lang.Undergraduate Analysis. Undergraduate Texts in Mathematics. Springer New York,

  45. [45]

    Leconte and D

    G. Leconte and D. Orban. Complexity of trust-region methods with unbounded hessian approximations for smooth and nonsmooth optimization.Math. Program., 2026. doi: 10.1007/s10107-025-02304-2. 22

    work page doi:10.1007/s10107-025-02304-2 2026

  46. [46]

    J. Li, M. Andersen, and L. Vandenberghe. Inexact proximal Newton methods for self-concordant functions.Math. Meth. Oper. Res., 85:19–41, 2017. doi: 10.1007/ s00186-016-0566-9

    2017

  47. [47]

    J. J. Mor´ e.Recent developments in algorithms and software for trust region methods, pages 258–

  48. [48]

    doi: 10.1007/978-3-642-68874-1 11

    Springer Berlin Heidelberg, Berlin Heidelberg, 1983. doi: 10.1007/978-3-642-68874-1 11

    work page doi:10.1007/978-3-642-68874-1 1983

  49. [49]

    Muthukumar, D

    R. Muthukumar, D. P. Kouri, and M. Udell. Randomized sketching algorithms for low-memory dynamic optimization.SIAM J. Optim., 31(2):1242–1275, 2021

    2021

  50. [50]

    R. H. Nochetto. Pointwise a posteriori error estimates for elliptic problems on highly graded meshes.Math. Comp., 64(209):1–22, 1995. ISSN 0025-5718,1088-6842. URLhttps://doi. org/10.2307/2153320

    work page doi:10.2307/2153320 1995

  51. [51]

    D. Pardo. Multigoal-oriented adaptivity forhp-finite element methods.Procedia Computer Science, 1(1):1953–1961, 2010

    1953

  52. [52]

    Power and T

    J. Power and T. Pryer. Adaptive regularisation for PDE-constrained optimal control.J. Comput. Appl. Math., 470:Paper No. 116651, 22, 2025. ISSN 0377-0427,1879-1778. URL https://doi.org/10.1016/j.cam.2025.116651

    work page doi:10.1016/j.cam.2025.116651 2025

  53. [53]

    Shi and J

    Z. Shi and J. Shen. New inexact line search method for unconstrained optimization.J. Optim. Theory Appl., 127:425–446, 2005. doi: 10.1007/s10957-005-6553-6

    work page doi:10.1007/s10957-005-6553-6 2005

  54. [54]

    P. L. Toint. Global convergence of a class of trust-region methods for nonconvex minimization in Hilbert space.IMA J. Numer. Anal., 8(2):231–252, 1988. ISSN 0272-4979. URLhttps: //doi.org/10.1093/imanum/8.2.231

    work page doi:10.1093/imanum/8.2.231 1988

  55. [55]

    Ulbrich and J

    S. Ulbrich and J. C. Ziems. Adaptive multilevel trust-region methods for time-dependent PDE-constrained optimization.Port. Math., 74(1):37–67, 2017. ISSN 0032-5155,1662-2758. URLhttps://doi.org/10.4171/PM/1992

    work page doi:10.4171/pm/1992 2017

  56. [56]

    Verf¨ urth

    R. Verf¨ urth. A posteriori error estimation and adaptive mesh-refinement techniques. InPro- ceedings of the Fifth International Congress on Computational and Applied Mathematics (Leu- ven, 1992), volume 50, pages 67–83, 1994. URLhttps://doi.org/10.1016/0377-0427(94) 90290-9

    work page doi:10.1016/0377-0427(94 1992

  57. [57]

    B. Vexler. Adaptive finite elements for output-oriented model calibration. In H. G. Bock, H. X. Phu, E. Kostina, and R. Rannacher, editors,Modeling, Simulation and Optimization of Complex Processes, pages 523–538, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg. ISBN 978-3-540-27170-3

    2005

  58. [58]

    Vexler and W

    B. Vexler and W. Wollner. Adaptive finite elements for elliptic optimization problems with control constraints.SIAM J. Control Optim., 47(1):509–534, 2008. doi: 10.1137/070683416

    work page doi:10.1137/070683416 2008

  59. [59]

    M. J. Zahr, K. T. Carlberg, and D. P. Kouri. An efficient, globally convergent method for optimization under uncertainty using adaptive model reduction and sparse grids.SIAM/ASA J. Uncertain. Quantif., 7(3):877–912, 2019. ISSN 2166-2525. URLhttps://doi.org/10. 1137/18M1220996

    2019

  60. [60]

    J. C. Ziems. Adaptive multilevel inexact SQP-methods for PDE-constrained optimization with control constraints.SIAM J. Optim., 23(2):1257–1283, 2013. ISSN 1052-6234,1095-7189. URL https://doi.org/10.1137/110848645. 23

    work page doi:10.1137/110848645 2013

  61. [61]

    J. C. Ziems and S. Ulbrich. Adaptive multilevel inexact SQP methods for PDE-constrained optimization.SIAM J. Optim., 21(1):1–40, 2011. ISSN 1052-6234,1095-7189. doi: 10.1137/ 080743160

    2011

  62. [62]

    O. C. Zienkiewicz and J. Z. Zhu. A simple error estimator and adaptive procedure for practical engineering analysis.Int. J. Numer. Methods Eng., 24:337–357, 1987. doi: 10.1002/nme. 16202040206

    work page doi:10.1002/nme 1987

  63. [63]

    Z. Zou, D. P. Kouri, and W. Aquino. A locally adapted reduced-basis method for solving risk-averse PDE-constrained optimization problems.SIAM/ASA J. Uncertain. Quantif., 10 (4):1629–1651, 2022. 24

    2022