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On the variational dual formulation of the Nash system and an adaptive convex gradient-flow approach to nonlinear PDEs

T0 review · 4 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Choosing the base state via a matrix-weighted transport equation makes the dual of the Nash system exactly consistent and recovers the solution by an explicit formula.

desk verdict Dual half is solid and worth refereeing; the gradient-flow scheme is an honest conjecture, and the toy model has a reversed case split that needs fixing. read the letter →

arxiv 2512.12878 v3 pith:TUXXDFQ3 submitted 2025-12-14 math.AP cs.NAmath.NAmath.OC

Dmitry Vorotnikov , Amit Acharya This is my paper · ORCID
classification math.APcs.NAmath.NAmath.OC MSC 35D9935L4037K5841A6049Q99
keywords variationaldualformulationNashsystembasestateconsistencygradientflowHamilton-Jacobiequationsconvexdualitygeneralizedoptimaltransport
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

This paper aims to make a variational dual formulation of the noise-free Nash system, a quadratic system of Hamilton-Jacobi type with several interacting players, genuinely useful. It shows that if the base state is not chosen arbitrarily but is generated by a matrix-valued 'density' and a velocity field solving a generalized transport equation, then the dual maximization problem has no duality gap: the optimal dual pair recovers the original weak solution through an explicit inversion formula. The same construction yields a sequence of base states, converging in mean to zero, that are fully consistent in the single-player case, and the paper proves existence of variational dual solutions for any number of players, any base state, and any initial data. The paper also proposes a staged Hilbertian gradient-flow scheme that switches base states along a fake-time variable and is expected to converge to a solution of the original PDE; this convergence is proved only for a two-dimensional algebraic toy model, while the multi-player Nash system is described but not established.

What carries the argument

The central object is the base state v̄, a guess function entering a relative kinetic energy; the identity that carries the argument is the construction (2.19): given a matrix field G≥0 and a velocity u solving the weak transport equation (2.17), the base state is set to G(v−u). The pair (G,u) acts as a weight-and-transport coordinate system that makes the dual functional coincide with the primal one, and the inversion formula v = (I+2B⁺)⁻¹(v̄−E⁺) is the mechanism by which a dual maximizer is turned back into a solution. The paper's second engine is the staged Hilbertian gradient flow, where a fake-time variable drives descent on a convex dual functional and each stage resets the base state

What would settle it

Take a known weak solution v of the multi-player noise-free Nash system (N>1) on a periodic box, choose any pair (G,u) satisfying (2.16)–(2.17), and check whether (E⁺,B⁺) from (2.19) attains the supremum in (2.14); if the supremum is strictly larger, or if formula (2.20) fails on a set of positive measure when G>0, the consistency claim collapses. Alternatively, for the gradient-flow scheme, a concrete counterexample would be a two-player Nash system where the staged flow leaves the region I+2B>0 before equilibration and the generated base states fail to accumulate at any weak solution.

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Extended reading notes

Core claim

The central claim is Theorem 2.4: if v is a weak solution of the Burgers-like formulation (2.5) of the noise-free Nash system, and (G,u) is any pair with G≥0 satisfying the linearized transport constraint (2.17), then for the base state v̄ = G(v−u) the dual variational problem (2.14) has the same value as the primal saddle-point problem (2.8). The explicit maximizer is E⁺ = −Gv+v̄, B⁺ = ½(G−I), and wherever G>0 the solution is recovered by v = (I+2B⁺)⁻¹(v̄−E⁺). The paper also proves, for more than one player, that a maximizer of (2.14) always exists in the L²×L^∞ class for arbitrary base states and initial data, and that in the single-player case a sequence of consistent base states can be m

Load-bearing premise

The load-bearing premise is that a weak solution v of the Burgers-like Nash system actually exists — and for more than one player the paper states that no such existence is known — so Theorem 2.4 is conditional, and the gradient-flow scheme is only formal because it presupposes the dual trajectory never leaves the region where the reconstruction map is invertible and the dual functional is convex, a regularity issue the paper explicitly sets aside.

Editorial extensions

If this is right

  • For the single-player Hamilton-Jacobi equation, arbitrarily small base states suffice to eliminate the duality gap and produce smooth dual solutions from which the gradient is exactly recovered, so the measure-valued dual objects of the zero-base-state theory can be avoided by a tiny perturbation.
  • For more than one player, any base state and any initial data lead to a finite-valued dual problem with a maximizer in L²×L^∞, giving a well-defined variational notion of solution where no classical or weak solution of the Nash system is known to exist.
  • Whenever a weak solution exists, the consistency theorem supplies a certificate: check (G,u) against the transport constraint, form (E⁺,B⁺), and the duality gap is zero on the entire interval [0,T], not just on short time intervals.
  • If the staged gradient flow equilibrates, as proved for the two-variable algebraic toy model, the reconstruction map yields a solution of the original PDE, and the switching rule guarantees monotone decrease of the driving dissipation in fake time.

Reading between the lines

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

  • If the consistency theorem extends to other quadratic PDE systems as the authors expect, a practical recipe emerges: choose (G,u) to approximate the solution's own density-weighted evolution, and the dual problem becomes a reliable solver on arbitrarily long time intervals.
  • The single-player sequence of shrinking consistent base states suggests that the zero-base-state dual theory can be recovered as a limit of well-posed consistent problems, potentially providing a selection principle for non-unique weak solutions through the limit of the reconstructed v.
  • A direct test of the scheme on a two-player Nash system with known explicit solutions would clarify whether the expected convergence holds outside the formal setting before deeper existence theory is developed.
  • The paper's own Remark 3.2 warns that maximizers of the dual problem need not be "dual solutions" that recover a primal solution; an editorially important next step is to characterize when the maximizers from Theorem 2.11 actually do recover solutions.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies the variational dual formulation of the noise-free Nash system with a quadratic Hamiltonian and multiple players. It introduces base states into the dual problem and establishes several results. Theorem 2.4 gives a sufficient condition, v̄ = G(v−u) with G satisfying a generalized transport constraint, under which, if v is a weak solution of the Burgers-like formulation, the dual problem has no duality gap and v can be recovered from the dual maximizer. Corollary 2.6 constructs, for N = 1 and a known viscosity solution v, a sequence of consistent base states converging to zero in L¹. Theorem 2.11 proves existence of variational dual solutions for N > 1 for arbitrary base states. Sections 3–5 propose a formal adaptive gradient-flow scheme in a fictitious time variable, illustrated by a two-dimensional algebraic toy model and specialized to the Nash system. The paper explicitly states that convergence of this scheme is not proved.

Significance. If the results are taken as proven, Section 2 provides a conditional consistency theory for base states in the dual formulation and an existence result for variational dual solutions. The proofs in Section 2 appear carefully constructed; the transport construction in Corollary 2.6 is explicit, and the trace-condition argument in Lemma 2.10 is a useful technical contribution. The paper is also commendably honest about its limitations, explicitly marking Section 3 as formal and Section 5's recovered solution as 'allegedly' a solution. However, the consistency theorem is conditional on a known weak solution, which for N > 1 is not available; the existence theorem concerns only variational dual solutions, which need not be dual solutions. The advertised gradient-flow scheme has no convergence proof in any PDE setting. These caveats significantly narrow the scope of the paper's claims relative to its title and abstract.

major comments (4)
  1. [Section 4, after Eq. (4.15)] The claimed selection rule is reversed. For the first stage, with d∞ = 1/2 − |c+1/2|, substitution into the DtP map (4.13) yields (c,c) when c > −1/2 and (c+1,c+1) when c < −1/2, the opposite of the text's assertion. The same reversal appears in the rule following (4.22). The induction (4.24) actually supports the corrected rule: for c > −1/2 it proves c+1/2−v_k > 0, i.e., c−v_k > −1/2, which correctly selects (c,c). Since Section 4 is the only demonstration of the switching mechanism, this error must be corrected.
  2. [Section 3.2, Step (15); Section 5] The central algorithmic claim is not proved. Step (15) states 'It remains to prove that one has equilibration after a finite number of steps', and Section 5 says the recovered v 'would allegedly be a solution'. No convergence result is established for any PDE; the only illustration is the 2D algebraic toy model. The dissipation inequality (3.14) alone does not imply convergence of the DtP-generated primal iterates, especially when the dual trajectory may leave the DtP zone (3.15). The paper should state prominently in the abstract and introduction that the gradient-flow scheme is a formal/conjectural construction, or provide a convergence theorem under additional hypotheses.
  3. [Section 2.2, Theorem 2.4 and Corollary 2.6] The sufficient condition (2.18), v̄ = G(v−u), makes the base state depend on the sought solution v. Remark 2.5 notes the simplest admissible pair gives v̄ = v. Thus the consistency result is conditional on a weak solution being already known. For N > 1 the paper itself states that no solvability results are available (Introduction), so Theorem 2.4 does not apply to the multi-player regime unless a solution is supplied. Theorem 2.11 provides only variational dual maximizers, and by Remark 3.2 these need not be dual solutions that recover a primal solution. Consequently, the paper's rigorous results do not establish existence of weak solutions to the Nash system for N > 1. This limitation should be stated more prominently, and the non-trivial content of Theorem 2.4 beyond the trivial case v̄ = v should be clarified.
  4. [Section 3.1, Eqs. (3.4), (3.9), (3.15)] The existence of the DtP map U^(H) and the convexity of S_H on a neighborhood O*_U of D = 0 are asserted at a formal level; footnote 5 says 'we work at a formal level and ignore regularity issues'. The correctness of the gradient-flow scheme, including the envelope-theorem formula (3.11)–(3.12) and the dissipation property (3.14), relies on these assertions. They are load-bearing for the proposed method and should either be proved in a simplified setting or explicitly stated as assumptions rather than derived facts.
minor comments (4)
  1. [Introduction, Section 2] The periodic box is typeset as T N and T N×p; these should be T^N and T^{N×p}.
  2. [Proof of Theorem 2.4, after (2.24)] The algebraic identity (v⊗v,I) − (v,v̄) − 1/2(v,v) = K_{v,v̄} − 1/2(v̄,v̄) is used without comment; adding one line would improve readability.
  3. [Corollary 2.6, Eq. (2.28)] The test function Ψ is only assumed to satisfy Ψ(0)=0, not Ψ(T)=0; it would be helpful to state explicitly that the boundary term at t=T is handled by the terminal condition ρ_m(T)=1.
  4. [Section 4, Eqs. (4.18) and (4.23)] The use of '≈' in the base-state update formulas could be replaced by inequalities; the subsequent argument 'approximately equal' should be made precise or replaced by an explicit perturbation estimate.

Circularity Check

2 steps flagged · score 6.0 of 10

Theorem 2.4's good base states are defined through the sought solution, making the recovery formula an algebraic identity; the gradient-flow convergence is explicitly left open, so the partial circularity is confined to the consistency claims.

  1. self definitional [Theorem 2.4, Eqs. (2.16)-(2.20); Remark 2.5]
    "Assume also that ¯v=G(v−u) a.e. in (0,T)×Ω. ... the pair (E+,B+) defined by E+ := −Gv+ ¯v, B+ := 1/2 (G−I) ... we can retrieve the weak solution v ... by applying the formula v= (I+2B+)−1(¯v−E+) a.e. in O."

    The sufficient condition for a good base state (2.18) is stated in terms of the unknown solution v. The proposed maximizer (2.19) is then built directly from v, and the recovery formula (2.20) is an identity: I+2B+ = G and ¯v−E+ = Gv, so (I+2B+)^−1(¯v−E+) = v. Consistency and retrieval are thus guaranteed by the construction itself, not by any independent property of the dual problem. Remark 2.5 makes this explicit: the simplest admissible pair gives ¯v = v.

  2. self definitional [Corollary 2.6, Eqs. (2.26)-(2.25)]
    "Set ¯vm := ρm(v−um) = Gm(v−um). ... the gradient v of the solution to the Hamilton-Jacobi equation can be fully retrieved from the variational dual solutions according to the formula v = (I+2Bm)^−1(¯vm −Em)."

    The sequence of good base states is constructed using the solution v and smooth approximations um to v via the transport equation for ρm. Hence the base states are fitted to v; the subsequent full retrieval of v from the dual solutions (Em,Bm) = (−ρmum, 1/2(ρm−1)I) is just the same algebraic identity as in Theorem 2.4. The convergence ¯vm → 0 is real, but it does not convert the construction into a prediction of v.

full rationale

The circular component is localized. Theorem 2.4 is a consistency statement whose class of admissible base states is defined through the sought solution v (Eq. (2.18)); the dual maximizer is explicitly built from v (Eq. (2.19)), and the recovery formula (2.20) reduces to I+2B+ = G and ¯v−E+ = Gv. Corollary 2.6 repeats this for N=1 by choosing base states ¯v_m = ρ_m(v−u_m). These are genuine self-definitional steps: the good base state encodes v, so consistency and retrieval are by construction. The paper's other major advertised result, the staged Hilbertian gradient-flow scheme (Sec. 3.2), is not circular: the manuscript explicitly says 'It remains to prove that one has equilibration after a finite number of steps' (step (15)), and Section 5 says the generated v 'would allegedly be a solution'; a formal conjecture is a gap, not a circularity. Theorem 2.11 (existence of variational dual solutions for N>1) is independent: its proof is self-contained via the trace condition Lemma 2.10. The Introduction also admits that no solvability results are currently available for N>1, so the multi-player consistency theorem is conditional on an open hypothesis; this is a limitation, not a circular step. Self-citations are used for background and framework rather than as a load-bearing uniqueness or existence theorem. A separate correctness concern, not circularity, is that the toy-model case split in Section 4 appears reversed relative to the stated selection rule. Overall, the central consistency claim partially reduces to its input, but the paper still contains independent existence and framework contributions, so the score is 6 rather than higher.

Assumptions & free parameters 5 free parameters · 4 assumptions · 1 invented entities

The ledger is small and honest: the paper introduces no fitted constants, and the Section-2 theorems are parameter-free. The degrees of freedom that matter are the hand-chosen scheme parameters (a,b,p; ν,µ,τ; D*; initial base state) and the free transport pair (G,u) that parameterizes the family of consistent base states. The axiomatic load is concentrated in (i) the assumed existence of primal weak solutions for N>1 (open, flagged by the authors), and (ii) the formal DtP-zone/convexity assumptions of Section 3, which the authors explicitly decline to justify rigorously. The staged gradient flow is the one genuinely new construct; it lacks an independent falsifiable handle within the paper.

free parameters (5)
  • Auxiliary potential coefficients (a, b, p) of H_k
    Chosen by hand with only constraints a>0, b≥0, p>2 (Sec. 3.1–3.2(1)); the DtP map, the DtP zone, and the gradient flow all depend on this choice; no guidance is given for setting them for the Nash system.
  • Stage-switch and stopping parameters (ν, µ, τ)
    User-defined tolerances/caps that determine when a stage ends and when the scheme stops (Sec. 3.2 steps 10 and 15); the toy model's finite-stage proof is independent of them, but the PDE scheme's practical termination depends on these choices.
  • Initial base state v̄₁ (Ū₁)
    Arbitrary; the paper says the closer it is to a solution the better the convergence chances (Sec. 3.2 step 6) and Remark 4.1 shows that certain base states (e.g., zero in the symmetric c=−1/2 toy) stall the scheme indefinitely — the convergence claim is conditional on a good free choice.
  • Dual Dirichlet boundary data D*
    Set to 0 'for simplicity' (Sec. 3.1, Eq. (3.5)); Remark 3.1 documents that this choice creates boundary layers and suggests a data-dependent fix — a free design element affecting performance.
  • Transport pair (G,u) in Theorem 2.4
    Any (G,u) solving (2.16)–(2.17) is admissible; G≥0 PSD matrix-valued, u∈L². The family of consistent base states is indexed by this free choice, and the retrieval formula (2.20) only works where G>0 — the size of the retrieval set O is controlled by the choice of (G,u).
assumptions (4)
  • domain assumption For N>1, weak solutions v of the noise-free Nash system (2.5) are assumed to exist wherever consistency is claimed (Theorem 2.4).
    The paper states no solvability result for N>1 exists (Introduction, Sec. 1). Theorem 2.4 is an implication 'if v is a weak solution...'; for N>1 the hypothesis is currently unverifiable, so the consistency result has no known instance in the multi-player case.
  • ad hoc to paper In Section 3, the DtP map U^(H)(D, Ū) exists and the dual functional S_H is convex on a neighborhood O*_{Ū} of D=0, with the envelope-theorem gradient formula (3.11)–(3.12) valid.
    Justified only formally: 'we work at a formal level and ignore regularity issues' (Sec. 3.1, fn. 5) and via the implicit function theorem given strict convexity of H at Ū. The scheme's correctness (steps 8–13 of Sec. 3.2) presupposes the trajectory stays inside this DtP zone (3.15).
  • domain assumption For N=1, the gradient v=∇ψ of the viscosity solution of the H-J equation (1.6) is a weak solution in the sense of Definition 2.1, and continuity (Cor. 2.6) or essential boundedness (Remark 2.8) of v holds.
    Used to build the consistent sequence of base states and to claim full consistency for shocks; the continuity assumption is flagged by the authors as restrictive (Remark 2.8).
  • standard math Standard functional-analytic facts used without proof: weak-* compactness arguments and upper semicontinuity of concave infima in Theorem 2.11; classical linear transport theory in Corollary 2.6; gradient-flow dissipation monotonicity (3.14) after [13].
    Background tools; no formal verification is shipped.
invented entities (1)
  • Fictitious gradient-flow time s with staged base-state switching
    purpose: Algorithmic device (Sec. 3.2): gradient descent in H on the dual functional S_{H_k} in a fake time s; at the end of each stage the current DtP reconstruction is frozen as the next base state, producing a piecewise-smooth U-trajectory claimed to converge to a primal solution.
    No convergence proof for any PDE; the paper explicitly leaves equilibration open (Sec. 3.2 step 15) and Section 5's Nash-system version is hedged ('would allegedly be a solution'). The toy model (Sec. 4) is the only in-paper evidence; prior computational evidence [23,24,25] concerns Newton-type base-state schemes rather than this gradient flow.

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Pith. "Pith review of On the variational dual formulation of the Nash system and an adaptive convex gradient-flow approach to nonlinear PDEs." pith.science (2026). https://pith.science/paper/TUXXDFQ3

@misc{pith2026251212878,
  author       = {Pith},
  title        = {Pith review of: On the variational dual formulation of the Nash system and an adaptive convex gradient-flow approach to nonlinear PDEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TUXXDFQ3}},
  note         = {Machine review of arXiv:2512.12878}
}
read the original abstract

We investigate the influence of base states on the consistency of the dual variational formulation for quadratic systems of PDEs, which are not necessarily conservative (typical examples include the noise-free Nash system with a quadratic Hamiltonian and multiple players). We identify a sufficient condition under which consistency holds over large time intervals. In particular, in the single-player case, there exists a sequence of base states (each exhibiting full consistency) that converges in mean to zero. We also prove existence of variational dual solutions to the noise-free Nash system for arbitrary base states. Furthermore, we propose a scheme based on Hilbertian gradient flows that, starting from an arbitrary base state, generates a sequence of new base states that is expected to converge to a solution of the original PDE.

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Reference graph

Works this paper leans on

34 extracted references · cited by 2 Pith papers

  1. [1]

    A. Acharya. A dual variational principle for nonlinear dislocation dynamics.Journal of Elasticity, 154(1):383–395, 2023

    2023

  2. [2]

    A. Acharya. Coupled dislocations and fracture dynamics at finite deformation: model derivation, and physical questions.Journal of Materials Science: Materials Theory, 8(1):6, 2024

    2024

  3. [3]

    A. Acharya. A hidden convexity in continuum mechanics, with application to classical, continuous-time, rate-(in)dependent plasticity.Mathematics and Mechanics of Solids, 30(3):701–719, 2025

    2025

  4. [4]

    Acharya, I

    A. Acharya, I. Fonseca, L. Ganedi, and K. Stinson. Vector field models for nematic disclinations. Journal of Nonlinear Science, 33(5):85, 2023

    2023

  5. [5]

    Acharya and J

    A. Acharya and J. Ginster. A convex variational principle for the necessary conditions of classical optimal control.arXiv e-prints, Feb. 2025

    2025

  6. [6]

    Acharya, J

    A. Acharya, J. Ginster, and A. N. Sengupta. Variational Dual Solutions of Chern-Simons Theory. arXiv e-prints, Nov. 2024

    2024

  7. [7]

    Acharya and A

    A. Acharya and A. N. Sengupta. Action principles for dissipative, non-holonomic Newtonian mechanics. Proc. A., 480(2293):Paper No. 20240113, 21, 2024

    2024

  8. [8]

    Acharya and M

    A. Acharya and M. Slemrod. Existence, uniqueness, and long-time behavior of linearized field disloca- tion dynamics.Quarterly of Applied Mathematics, LXXXI:247–258, 2023

    2023

Show all 34 references
  1. [9]

    Acharya, B

    A. Acharya, B. Stroffolini, and A. Zarnescu. Variational Dual Solutions for Incompressible Fluids. arXiv e-prints, Sept. 2024

    2024

  2. [10]

    Acharya and L

    A. Acharya and L. Tartar. On an equation from the theory of field dislocation mechanics.Bollettino dell’Unione Matematica Italiana, 9:409–444, 2011

    2011

  3. [11]

    Alvarez, P

    O. Alvarez, P. Hoch, Y. L. Bouar, and R. Monneau. Dislocation dynamics: short-time existence and uniqueness of the solution.Archive for Rational Mechanics and Analysis, 181(3):449–504, 2006

    2006

  4. [12]

    Arora and A

    A. Arora and A. Acharya. Emergent fault friction and supershear in a continuum model of geophysical rupture.Journal of the Mechanics and Physics of Solids, page 105827, 2024

    2024

  5. [13]

    Attouch, G

    H. Attouch, G. Buttazzo, and G. Michaille.Variational analysis in Sobolev and BV spaces, volume 17 of MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadel- phia, PA; Mathematical Optimization Society, Philadelphia, PA, second edition...

    2014

  6. [14]

    Y. Brenier. The initial value problem for the Euler equations of incompressible fluids viewed as a concave maximization problem.Comm. Math. Phys., 364(2):579–605, 2018

    2018

  7. [15]

    Y. Brenier. Examples of hidden convexity in nonlinear pdes.Preprint, 2020

    2020

  8. [16]

    Brenier and I

    Y. Brenier and I. Moyano. Relaxed solutions for incompressible inviscid flows: a variational and grav- itational approximation to the initial value problem.Philos. Trans. Roy. Soc. A, 380(2219):Paper No. 20210078, 12, 2022

    2022

  9. [17]

    Cardaliaguet, F

    P. Cardaliaguet, F. c. Delarue, J.-M. Lasry, and P.-L. Lions.The master equation and the convergence problem in mean field games, volume 201 ofAnnals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2019

    2019

  10. [18]

    Ghoussoub.Self-dual partial differential systems and their variational principles

    N. Ghoussoub.Self-dual partial differential systems and their variational principles. Springer, 2009

    2009

  11. [19]

    Gimperlein, M

    H. Gimperlein, M. Grinfeld, R. J. Knops, and M. Slemrod. The least action admissibility principle. Archive for Rational Mechanics and Analysis, 249(2):22, 2025

    2025

  12. [20]

    Gimperlein, M

    H. Gimperlein, M. Grinfeld, R. J. Knops, and M. Slemrod. On action rate admissibility criteria.arXiv e-prints, 2025

    2025

  13. [21]

    R. Hynd. The Hamilton-Jacobi equation, then and now.Notices Amer. Math. Soc., 68(9):1457–1467, 2021

    2021

  14. [22]

    Kouskiya.Computational Analysis of a duality based approach for solving ordinary and partial differential equations

    U. Kouskiya.Computational Analysis of a duality based approach for solving ordinary and partial differential equations. PhD thesis, Carnegie Mellon University, 2024

    2024

  15. [23]

    Kouskiya and A

    U. Kouskiya and A. Acharya. Hidden convexity in the heat, linear transport, and Euler’s rigid body equations: A computational approach.Quarterly of Applied Mathematics, LXXXII:673–703, 2024. NASH SYSTEM 27

    2024

  16. [24]

    Kouskiya and A

    U. Kouskiya and A. Acharya. Inviscid Burgers as a degenerate elliptic problem.Quarterly of Applied Mathematics, LXXXIII:315–360, 2025

    2025

  17. [25]

    Kouskiya, R

    U. Kouskiya, R. L. Pego, and A. Acharya. Traveling wave profiles for a semi-discrete Burgers equation. Physica D, page 134961, 2025

    2025

  18. [26]

    Mirebeau and E

    J.-M. Mirebeau and E. Stampfli. Discretization and convergence of the ballistic Benamou-Brenier formulation of the porous medium and Burgers equations.arXiv e-prints, Nov. 2025

    2025

  19. [27]

    Morin and A

    L. Morin and A. Acharya. Analysis of a model of field crack mechanics for brittle materials.Computer Methods in Applied Mechanics and Engineering, 386:114061, 2021

    2021

  20. [28]

    F. Rindler. Energetic solutions to rate-independent large-strain elasto-plastic evolutions driven by discrete dislocation flow.Journal of the European Mathematical Society, 27(7):2795–2864, 2025

    2025

  21. [29]

    Singh, J

    S. Singh, J. Ginster, and A. Acharya. A hidden convexity of nonlinear elasticity.Journal of Elasticity, 156(3):975–1014, 2024

    2024

  22. [30]

    Sukumar and A

    N. Sukumar and A. Acharya. Variational formulation based on duality to solve partial differential equa- tions: Use of B-splines and machine learning approximants.Computer Methods in Applied Mechanics and Engineering, 441:117909, 2025

    2025

  23. [31]

    Vorotnikov

    D. Vorotnikov. Partial differential equations with quadratic nonlinearities viewed as matrix-valued optimal ballistic transport problems.Arch. Ration. Mech. Anal., 243(3):1653–1698, 2022

    2022

  24. [32]

    Vorotnikov

    D. Vorotnikov. Dafermos’ principle and Brenier’s duality scheme for defocusing dispersive equations. arXiv e-prints, Jan. 2025

    2025

  25. [33]

    Zhang, X

    C. Zhang, X. Zhang, A. Acharya, D. Golovaty, and N. Walkington. A non-traditional view on the modeling of nematic disclination dynamics.Quarterly of Applied Mathematics, 75(2):309–357, 2017

    2017

  26. [34]

    Zhang, A

    X. Zhang, A. Acharya, N. J. Walkington, and J. Bielak. A single theory for some quasi-static, super- sonic, atomic, and tectonic scale applications of dislocations.Journal of the Mechanics and Physics of Solids, 84:145–195, 2015. (D. Vorotnikov)University of Coimbra, CMUC, Dep...

    2015

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.