{"id":"cb43a253-ff60-4f0f-98e7-807cb8167b54","arxiv_id":"2506.19944","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A new hybrid high-order method for the Gross-Pitaevskii eigenvalue problem is shown to converge at optimal rates and, with a modified quadrature, to deliver guaranteed lower bounds on the ground-state energy without post-processing.","lead":"The paper introduces a hybrid high-order numerical method for the Gross-Pitaevskii eigenvalue problem and proves how fast the computed state, energy, and chemical potential converge. In a modified lowest-order form, the method gives guaranteed lower bounds on the ground-state energy without post-processing, which helps certification of Bose-Einstein condensate simulations.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The abstract's claim of directly obtained asymptotically exact lower energy bounds is not supported for the smooth potentials V1/V2: the numerical protocol fixes a coarse piecewise-constant under-approximation (h=2^-2), leaving a non-vanishing gap to the true energy.","rationale":"The reader's weakest_assumption identifies essentially the same vulnerability: Theorem 3.1 requires V in P0(Th) and condition (3.5), while Section 5.1 uses a fixed coarse under-approximation, so the computed lower bounds target a modified potential. I agree this is the most load-bearing concern because it directly undercuts the abstract's central claim of 'guaranteed and asymptotically exact lower energy bounds ... obtained directly without the need of post-processing' for the problems actually tested. The convergence analysis for the unmodified HHO method (Theorems 4.5-4.6) rests on standard HHO and nonlinear-eigenvalue arguments and is corroborated by the numerical experiments, so I would not reject the paper. The conditional verdict stands: the authors should either restrict the asymptotic-exactness claim to piecewise-constant potentials or prove and implement an under-approximation that is refined with h and satisfies condition (3.5) self-containedly. Secondary gaps, such as the omitted proof of Theorem 4.8, the sketched Remark 4.7, and the unquantified accuracy of the reference solutions, are worth addressing but are less central to the paper's main selling point.","tokens_in":25611,"tokens_out":3439,"duration_ms":36770,"concrete_test":"Run the Section 5.1 harmonic experiment (V1, kappa=1000) with the under-approximating potential V_h defined as the elementwise infimum of V on the actual computational mesh (h=2^-3,...,2^-7) instead of the fixed h=2^-2 grid; choose sigma by rearranging (3.5) with a conforming Q1 upper bound, and record E0_h plus the energy E(V_h) of the under-approximated potential. If E0_h <= E holds and E - E0_h decays like O(h^2), then asymptotically exact lower bounds can be recovered for smooth potentials by refining the potential under-approximation, and the paper's numerical protocol is the obstruction; if the gap remains bounded below, the claim as stated fails for smooth V.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central advertised novelty is that the modified HHO method gives guaranteed and asymptotically exact lower energy bounds directly, without post-processing. The only theorem proving this for the modified scheme (Theorem 3.1 with convergence in Theorem 4.8) is stated under V in P0(Th) and condition (3.5), which involves the unknown discrete energy E0_h. For the smooth potentials in Section 5.1 (V1 harmonic, V2 lattice), the authors enforce V in P0(Th) by replacing V with a piecewise-constant under-approximation on a fixed coarse grid h=2^-2 and prolonging it to all finer meshes. Then E0_h <= E(V_coarse) <= E, where E(V_coarse) is the ground-state energy of the modified potential; the gap E - E(V_coarse) is independent of h. Hence the computed lower bounds are not asymptotically exact for the original problem: as h -> 0 they converge to the energy of a different potential. The abstract and conclusion do not state this limitation. In addition, condition (3.5) is verified in Section 5.1 using the reference energy E from an external conforming Q1 computation, so the guarantee is not self-contained. This is a scope/correctness issue with the headline claim, not an internal inconsistency; the convergence theory for the unmodified HHO method (Theorems 4.5-4.6) is largely independent and appears sound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a hybrid high-order (HHO) discretization for the Gross–Pitaevskii eigenvalue problem (GP-EVP). For the standard HHO method, it proves optimal-order a priori error estimates for the ground state, eigenvalue, and energy under regularity assumptions. It then introduces a lowest-order modified HHO method with a special quadrature for the nonlinearity and proves, under a piecewise-constant potential assumption and a condition involving the discrete energy, that the discrete energy is a guaranteed lower bound for the exact ground-state energy. The authors claim that, unlike previous mixed finite element approaches, these lower bounds are obtained directly without post-processing and are asymptotically exact. Numerical experiments for harmonic, lattice, and disorder potentials compare the modified HHO lower bounds with those of the post-processed Raviart–Thomas method from GHLP25, and convergence experiments illustrate the predicted rates.","tokens_in":25859,"tokens_out":6779,"duration_ms":78119,"significance":"If the main results hold, the paper makes a useful contribution: it extends the HHO methodology to a nonlinear eigenvalue problem, proves sharp-looking convergence rates for both the standard and modified methods, and provides a new avenue toward guaranteed lower energy bounds for the GP-EVP. The paper is careful in separating the convergence analysis for the unmodified HHO method from the lower-bound construction, and it supplies several explicit constants and auxiliary lemmas. The availability of reproducible code is a further strength. However, the headline claim of asymptotically exact lower energy bounds for smooth potentials is not supported by the theorems as stated: the numerical protocol freezes a coarse piecewise-constant under-approximation of the potential, so the computed lower bounds converge to the energy of a different potential. This is a scope issue with the main advertised novelty rather than an internal inconsistency in the convergence proofs.","major_comments":[{"comment":"The claim that the modified HHO method yields 'asymptotically exact lower energy bounds' is not established for the smooth potentials V1 and V2 used in the experiments. Theorem 3.1 requires V ∈ P0(Th). In §5.1 the authors satisfy this by replacing V1 and V2 with a piecewise-constant under-approximation on a fixed Cartesian grid of size h = 2^{-2} and prolonging it to all finer meshes. Since that potential is fixed, the modified method converges as h → 0 to E(V_coarse), the ground-state energy of the modified potential, not to E(V). The gap E(V) - E(V_coarse) is independent of h, so the plotted quantity E - E0_h cannot tend to zero. The lower-bound property is preserved because V_coarse ≤ V, but asymptotic exactness for the original problem is not. The abstract and conclusion should state this limitation explicitly, or the numerical protocol should be changed so that the potential under-approximation is refined with the mesh.","section":"Abstract, Theorem 3.1, Remark 3.2, §5.1"},{"comment":"Condition (3.5) is not a fully computable a priori certificate because it contains the unknown discrete energy E0_h. In §5.1 the authors make it hold by first computing an independent upper bound E with a conforming Q1 method and then choosing σ from (3.5) with E in place of E0_h. This is logically valid only after an external upper bound is available, and it makes the 'guaranteed' lower bound depend on a second computation. The paper should either state this dependence as an explicit hypothesis of Theorem 3.1, or provide a computable upper bound for E0_h that makes the guarantee self-contained. As written, the theorem guarantees E0_h ≤ E conditional on a condition that involves the very quantity to be bounded.","section":"Theorem 3.1, Eq. (3.5), §5.1"},{"comment":"The convergence theorem for the modified HHO method, which underpins the second-order energy and eigenvalue claims, is not actually proved. The proof states that it is 'very similar' to the standard case and that the arguments of Lemmas 4.3-4.6 are repeated using Lemmas A.5-A.7, but the nontrivial adaptations—particularly the treatment of the modified nonlinear form (4.33) and the quadrature error estimates—are not carried out. Since the modified method is a central contribution, the proof should be written out in sufficient detail, or the theorem should be stated with a clear indication of which parts remain conjectural. This is especially important because the modified nonlinearity is non-polynomial in the discrete unknowns and the eigenvalue problem (4.33) requires separate handling.","section":"Theorem 4.8 and its proof"}],"minor_comments":[{"comment":"The caption says 'polynomial degrees k = 1, 2, 3 (from left to right)', but the panels are labeled k = 0, 1, 2 and the text refers to k = 0, 1, 2; the caption should be corrected.","section":"Figure 5.3"},{"comment":"The disorder potential V3 is described as piecewise constant on a Cartesian grid with mesh size h = 20; within Ω = (-8,8)^2 this value is ambiguous and is likely intended to be 2^0 or a similar small value. Please clarify.","section":"§5.1, disorder potential"},{"comment":"Remark 3.2 suggests approximating V by its elementwise minimum, while §5.1 says the potentials are 'projected' onto piecewise constants. These are different procedures, and only an under-approximation preserves the lower-bound property. Please clarify which construction was actually used in the experiments.","section":"Remark 3.2 and §5.1"}],"recommendation":"major_revision","confidential_remarks":"The convergence analysis for the unmodified HHO method appears sound and is a meaningful contribution independent of the lower-bound claim. The main problem is the overclaim in the abstract and conclusion regarding asymptotically exact lower bounds for smooth potentials: the numerical protocol fixes a coarse potential under-approximation, so the advertised exactness is not delivered by the theorems. This is fixable by narrowing the claims and/or refining the potential with the mesh, but the current text presents the limitation only implicitly. I recommend major revision rather than rejection because the core HHO convergence results are likely correct and useful."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the quick take: this is a genuine extension of HHO methods to the nonlinear Gross–Pitaevskii eigenvalue problem, with a real lower-bound result and a careful convergence analysis for the standard method. But the abstract sells more than the theorems deliver. The guaranteed lower bound (Theorem 3.1) is proved only for piecewise-constant V, and condition (3.5) involves the unknown discrete energy, so the guarantee is not self-contained. More importantly, for the smooth potentials in Section 5.1 the authors replace V by a fixed coarse piecewise-constant under-approximation (h = 2^-2) and prolong it to all finer meshes. Then the computed E0_h is a lower bound on the energy of the modified potential, not the original one, and the gap E - E(V_coarse) does not vanish as h -> 0. So the abstract's \"asymptotically exact\" claim for the original problem is not supported. The stress-test note holds up. The paper should either restrict the claim to the piecewise-constant setting or provide an analysis of the potential-under-approximation error.\n\nWhat is actually new and good: the HHO discretization of the GP-EVP with optimal-order convergence proofs (Theorems 4.5–4.6) for ground state, eigenvalue, and energy, including the improved L2 rate h^{r+2}; the modified quadrature in (3.1) that makes Jensen applicable for the nonlinear term; and the proof that the modified lowest-order method gives guaranteed lower bounds directly without post-processing, with O(h^2) energy convergence (Theorem 4.8). The numerical experiments support the convergence rates and show the lower bounds are much tighter than the post-processed mixed method from GHLP25, particularly for smooth potentials. The code is available on GitHub, which makes the numerical claims checkable.\n\nSoft spots: Theorem 4.8's proof is mostly a sketch (\"repeating the arguments...\") and Remark 4.7 omits details of the improved eigenvalue rate. The reference solution for the numerical experiments is a Q_{k+1} approximation on a finer grid, but its accuracy is not quantified, so the observed rates should be read with that caveat. Condition (3.5) is verified using an external conforming upper bound, which is fine in practice but means the lower-bound guarantee is not fully a priori. These are fixable in revision.\n\nBottom line: this deserves a serious referee. The central convergence analysis for the standard HHO method is careful and mostly self-contained against established HHO results. The lower-bound contribution is real but scoped more narrowly than the abstract claims. I'd send it to review and ask for a rewrite of the claims and the missing details.","headline":"Solid HHO convergence analysis for the GP-EVP, but the headline lower-bound claim overreaches: for smooth potentials the numerics freeze a coarse piecewise-constant potential, so the asymptotic exactness advertised in the abstract is not established.","tokens_in":26431,"tokens_out":2194,"would_cite":true,"duration_ms":22066,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N12","65N15","65N25","65N30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces an HHO discretisation of the Gross–Pitaevskii ground state that converges at optimal order and, in a modified lowest-order variant, yields guaranteed lower energy bounds directly, without post-processing.","keywords":["Gross-Pitaevskii eigenvalue problem","hybrid high-order method","guaranteed lower energy bounds","a priori error analysis","Bose-Einstein condensates","nonlinear eigenvalue problem","lower eigenvalue bounds","Jensen's inequality"],"falsifier":"For a smooth potential, run the modified HHO method on refined meshes while keeping the piecewise-constant under-approximation of the potential on a fixed coarse grid, as in Section 5.1; if the gap $E-E^0_h$ does not shrink to zero as $h \\to 0$ but levels off at a positive constant, then the claim of asymptotically exact lower bounds for the original smooth problem fails.","tokens_in":25348,"feed_emoji":"⚛️","tokens_out":8583,"duration_ms":82437,"temperature":0.7,"pith_summary":"The paper sets out to prove that a hybrid high-order (HHO) discretisation—using cell and face polynomials linked by a reconstruction operator—approximates the ground state of the Gross–Pitaevskii eigenvalue problem at optimal order in the state, the chemical potential, and the energy. Its second goal is to construct a modified lowest-order version whose computed energy is a guaranteed lower bound on the true ground-state energy, obtained directly from the discrete solution without any post-processing. This matters because conforming finite element methods can only overestimate the ground-state energy; a provable and accurate lower bound turns an energy computation into a two-sided certificate. The paper proves L2 convergence of order $h^{r+2}$, H1 convergence of order $h^{r+1}$, energy convergence of order $h^{2r+2}$, and eigenvalue convergence of order $h^{r+2}$ under regularity, together with $E^0_h \\le E$ and $O(h^2)$ convergence for the modified method.","feed_headline":"Guaranteed lower energy bounds, no post-processing, at optimal rate","feed_subtitle":"Proved: a lowest-order HHO variant gives lower energy bounds with error O(h²), no post-processing.","key_machinery":"The load-bearing object is the pair (reconstruction, stabilisation): given cell unknowns in $P_{k+1}(T)$ and face unknowns in $P_k(F)$, the operator $R_h$ from (2.3) reconstructs a broken polynomial that matches the cell unknown's average and whose gradient against test polynomials equals the boundary and cell data, while $s_h$ penalises the mismatch between cell and face components with explicit weights. This defines the discrete energy $E_h$. For the lower bound, the modified energy $E^0_h$ changes only the quartic term to $\\frac{\\kappa}{4}(|\\Pi^0_{T_h} v_{T_h}|^2 v_{T_h}, v_{T_h})$, and the proof chains a Poincaré inequality with constant $1/\\pi^2$, a trace inequality with constant $C_{\\mathrm{tr}}$, and Jensen's inequality to dominate each discrete term by the continuous energy. A moment-preserving smoother $J_h$ is used in the error analysis to convert nonconforming discrete errors into conforming test functions.","core_discovery":"The central discovery is that the HHO energy functional, minimised over the space $\\hat{V}_h = P_{k+1}(T_h) \\times P_k(F_h)$, reproduces for the nonlinear Gross–Pitaevskii problem the optimal convergence behaviour known for linear elliptic eigenvalue problems: the reconstructed ground state converges like $h^{r+1}$ in the H1-seminorm, the cell component like $h^{r+2}$ in L2, and the energy like $h^{2r+2}$; the eigenvalue error is $h^{r+2}$ under an additional regularity assumption. The lower-bound variant modifies only the nonlinear term, replacing $|v_{T_h}|^2 v_{T_h}$ by $|\\Pi^0_{T_h} v_{T_h}|^2 v_{T_h}$, which makes Jensen's inequality applicable and yields $E^0_h \\le E$ for piecewise-constant potentials under the explicit mesh condition (3.5). The same modified method is shown to converge at first order in H1 and second order in L2, energy, and eigenvalue, so the guaranteed lower bound is asymptotically exact without any post-processing step.","pith_inferences":["Inference: for smooth potentials the experiments under-approximate $V$ on a fixed coarse grid and never refine it, so the asymptotic exactness is proved for the modified potential; I would expect a fixed, non-vanishing gap $E-E^0_h$ as $h \\to 0$ for the original smooth problem.","Inference: condition (3.5) contains the unknown discrete energy, so a self-contained method would need either an a-priori choice of $\\sigma$ small enough or an upper bound computed by the same HHO scheme rather than an external conforming method.","Inference: because the lower bound is obtained before any post-processing, the same discrete solution could feed an adaptive loop that refines where the local gap between upper and lower energies is largest; the paper does not explore adaptivity.","Inference: the reliance on Jensen's inequality suggests that the proof technique itself limits guaranteed lower bounds to lowest order; a high-order guaranteed lower bound would need a different treatment of the nonlinearity."],"forward_implications":["A $k$-th order HHO discretisation computes a ground-state approximation with L2 error $O(h^{k+2})$ and H1 error $O(h^{k+1})$, and an energy error $O(h^{2k+2})$; under extra regularity the eigenvalue error is $O(h^{k+2})$.","The modified $k=0$ method produces an energy $E^0_h$ that is guaranteed to satisfy $E^0_h \\le E$ and converges to $E$ at $O(h^2)$, so it can be combined with any upper-bound method to bracket the true ground-state energy.","The lower bound needs no post-processing, which numerical tests show makes it one to two orders of magnitude tighter than the post-processed lowest-order mixed method for smooth potentials.","For piecewise-constant potentials, such as the disorder-potential experiments, the guaranteed lower bound holds with explicit constants, and the convergence statements apply to the discrete solution produced by the method.","The analysis covers convex Lipschitz domains in two and three space dimensions with homogeneous Dirichlet conditions, the standard truncation model for confined condensates."],"supporting_citations":[{"why":"Supplies the conforming finite-element analysis of the Gross–Pitaevskii eigenvalue problem, including the auxiliary problem and error estimates that the HHO proof adapts.","marker":"[CCM10]"},{"why":"The mixed finite-element method with post-processed guaranteed lower energy bounds that the modified HHO method improves upon and is compared with in experiments.","marker":"[GHLP25]"},{"why":"Provides the stabilisation and the lower-eigenvalue-bound framework for linear HHO methods, plus the implementation basis for the code.","marker":"[Tra24]"},{"why":"Supplies the classical HHO convergence theory for Poisson problems used in Theorem 4.2 and the approximation lemmas.","marker":"[DPD20]"},{"why":"Gives the finite-dimensional approximation theory for the GP ground state that motivates the a priori error analysis.","marker":"[Zho04]"},{"why":"Establishes lower eigenvalue bounds for mixed methods and supplies the explicit trace constant $C_{\\mathrm{tr}}$ used in the lower-bound proof.","marker":"[Gal23]"},{"why":"Constructs the moment-preserving smoothing operator $J_h$ that converts discrete errors into conforming test functions.","marker":"[EZ20]"},{"why":"Provides the explicit Poincaré constant used to control L2 errors in the lower-bound argument.","marker":"[Beb03]"}],"fun_headline_variants":["Guaranteed lower energy bounds without post-processing","Optimal-rate HHO method yields lower bounds directly","HHO method: direct lower energy bounds at optimal rate","Lower energy bounds proven directly, no post-processing","Direct lower bounds for GP eigenvalue problem, no post-processing"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The lower-bound theorem assumes the potential is constant on each mesh element and requires condition (3.5), which involves the unknown discrete energy, so in practice the guarantee depends on a separate upper-bound computation.","fun_headline_variants_meta":{"raw":{"variants":["Guaranteed lower energy bounds without post-processing","Optimal-rate HHO method yields lower bounds directly","HHO method: direct lower energy bounds at optimal rate","Lower energy bounds proven directly, no post-processing","Direct lower bounds for GP eigenvalue problem, no post-processing"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000967,"raw_usage":{"total_tokens":4080,"prompt_tokens":879,"completion_tokens":3201,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":3125}},"tokens_in":495,"tokens_out":3201,"duration_ms":19444,"temperature":1.0,"reasoning_tokens":3125,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:23:20.058273+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a smooth potential, run the modified HHO method on refined meshes while keeping the piecewise-constant under-approximation of the potential on a fixed coarse grid, as in Section 5.1; if the gap $E-E^0_h$ does not shrink to zero as $h \\to 0$ but levels off at a positive constant, then the claim of asymptotically exact lower bounds for the original smooth problem fails.","supporting_citations":[],"review_version":2}