{"id":"ea02f7fd-6ca2-4bd3-aed2-c33f7f96d1a8","arxiv_id":"2607.19500","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A deterministic combination of diagonal sampling, diagonal pivoting, an extended admissibility condition, and sustained convergence checks lets ACAPP+ DCC meet prescribed tolerances on all tested acoustic volume and boundary integral operators.","lead":"This paper modifies the Adaptive Cross Approximation (ACA) for compressing dense matrices from acoustic integral equations, adding a diagonal convergence test, diagonal pivots, an extended admissibility rule, and a sustained convergence check. On benchmarks including a transcranial-ultrasound model, the new variant avoids the 'early convergence' failures of standard ACA at the same memory cost.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Admissibility condition (17) uses corner-to-corner box distance, which can overestimate separation and admit near-singular blocks; factor 2 has no proof/sensitivity.","rationale":"The reader's weakest_assumption isolates the extended admissibility condition (17) as the load-bearing heuristic. I agree and sharpen it: the corner-distance definition in Sec. 2.4 is not the true box distance, so condition (17) can certify admissibility for boxes whose actual separation is smaller than 2*max_element_diameter. This is a concrete, technical flaw rather than a mere lack of sensitivity study. It directly threatens the 'always achieves' claim because a near-singular admissible block could produce a large error not sampled by the DCC. The paper's reported benchmarks may still pass; hence the verdict stays CONDITIONAL. The proposed test would settle whether the concern lands. The paper's honesty about the fixed-vpw complexity (Sec. 4.6) and availability of code are positive, but do not address this admissibility gap.","tokens_in":19940,"tokens_out":14815,"duration_ms":145479,"concrete_test":"Create a benchmark with two thin rectangular plates facing each other across a small gap, laterally offset so that the corner-distance between their octree boxes exceeds 2*max_element_diameter while the true box distance is below it. Mesh each plate with elements of diameter d, set epsilon=1e-4, run ACAPP+ DCC, and compute (i) the per-block Frobenius error of each admissible block and (ii) the global relative error (18). If any admissible block's error exceeds epsilon or the global error exceeds tolerance, the condition fails. Alternatively, recompute condition (17) using the exact minimum Euclidean distance between boxes (e.g., via a convex distance algorithm) and count how many admissible blocks switch to inadmissible in the existing benchmarks; a non-zero count demonstrates the current metric under-enforces separation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that ACAPP+ DCC 'always achieves the prescribed error margins' depends on the extended admissibility condition (17), which sends near-singular interactions to dense blocks. The manuscript defines dist(Box_i, Box_j) as the shortest distance between all pairs of box corners (Sec. 2.4). For axis-aligned boxes whose intervals overlap in some coordinates but are offset in others, this corner-based distance is strictly larger than the true Euclidean distance between boxes (which can be attained on faces/edges). Thus a pair of boxes can satisfy dist_corners > 2*max_element_diameter and be certified admissible even when the actual gap between contained elements is below 2*max_element_diameter. In such a block, nearly singular integrands can be compressed by ACA; the DCC samples only the extended diagonal and may miss off-diagonal large errors. Section 3.4 justifies the factor 2 with a single example and gives no sensitivity study or proof, so the universal robustness conclusion is not supported for geometries where box projections are offset (e.g., thin plates, high-aspect-ratio elements).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript addresses early-convergence failures of Adaptive Cross Approximation with Partial Pivoting (ACAPP) when compressing hierarchical (H-)matrix blocks of acoustic volume and boundary integral operators. It proposes four deterministic modifications: a diagonal convergence criterion (DCC) based on an extended diagonal sample of each block, the use of these diagonal entries as additional pivot candidates, an extended admissibility condition intended to keep near-singular interactions in dense blocks, and a sustained convergence check requiring the error estimator to remain below tolerance for α consecutive iterations. Numerical experiments on sphere, cube, ellipsoid, and CT-derived skull models show that ACAPP+ DCC is the only tested variant that meets prescribed relative-vector error tolerances in all benchmarks, while the other variants stagnate or fail in several cases. The paper also reports memory consumption and scaling experiments.","tokens_in":20254,"tokens_out":12993,"duration_ms":136533,"significance":"If the empirical claims hold, the paper makes a practical contribution: it documents early-convergence failures in volume integral operators as well as boundary operators, and proposes a deterministic, reproducible algorithmic fix that does not increase the final memory footprint. The availability of the GitHub code is a strength, as are the large-scale skull benchmark and the algebraic complexity study. However, the robustness conclusions are heuristic rather than proven: the DCC error estimator and the extended admissibility condition rest on unanalyzed modeling choices, and the error measure uses a single random vector. The paper is therefore a useful engineering contribution to computational acoustics, but it does not provide a guaranteed-robust compression algorithm.","major_comments":[{"comment":"The admissibility safeguard rests on a distance notion that can overestimate geometric separation. Section 2.4 defines dist(Box_i, Box_j) as the shortest distance between all pairs of box corners. For axis-aligned boxes whose intervals overlap in one or two coordinates, the true Euclidean distance between the boxes can be attained on an edge or face and can be strictly smaller than every corner-to-corner distance. A pair of boxes can then satisfy dist_corners > 2·max_element_diameter while containing elements that are actually closer than 2·max_element_diameter. Such a block would be certified admissible and may contain nearly singular interactions; the DCC checks only the extended diagonal and can miss a localized off-diagonal error. The factor 2 is justified in Section 3.4 by a single example and no sensitivity study. Please either (i) replace the corner-based distance by the true Eucl","section":"§3.4, Eq. (17)"},{"comment":"The relative compression error is evaluated for a single random vector v, with a statement that the seed does not matter but without supporting data. This is important because the conclusion is phrased as 'always achieves the prescribed error margins' (Section 4.7) and because several ACAPP+ DCC curves lie close to the tolerance line at strict ε (e.g., Figure 9b at ε=1e-9 and Figure 15 at ε=1e-9). A single random vector cannot establish a universal statement. Please report at least the range over several random vectors or a deterministic worst-case set (e.g., coordinate basis vectors) for the ACAPP+ DCC runs, and clarify whether the error measure is intended as an estimate of the matrix norm or only as a particular vector norm. This does not undermine the contrast with the clearly failing variants, but it is needed to justify the word 'always'.","section":"§4.3, Eq. (18)"}],"minor_comments":[{"comment":"The expression 'mean(|e_ℓ|²)' should be defined explicitly as the average of the squared moduli over the d extended-diagonal entries; the notation is introduced informally and can be misread as a matrix norm.","section":"§3.2, Eq. (15)"},{"comment":"The sustained-convergence parameter α is not subjected to a sensitivity study. The statement that α=2 or 3 is sufficient is supported only by the benchmarks; a small table varying α would help readers apply the method to new problems.","section":"§3.5"},{"comment":"The random vector v is not specified (e.g., distribution, normalization, real or complex). Please state the exact construction used in Equation (18) and whether the reported errors correspond to a fixed realization.","section":"§4.3"},{"comment":"For boundary triangles, 'element diameter' is defined via the circumscribed circle. For high-aspect-ratio triangles this diameter can be much larger than the actual mesh width, so the interpretation of condition (17) should be discussed, especially in relation to the compression rates reported for P1 discretizations.","section":"§2.4/§3.4"},{"comment":"The 'analogous process' note in lines 23–28 is vague. The else branch should be written out explicitly to make the algorithm fully reproducible without referring to an earlier symmetric case.","section":"Algorithm 2"},{"comment":"The percentage labels above the storage bars are dense and overlap visually. Consider reporting the compression rates in a table or with fewer decimal places.","section":"Figs. 12 and 16"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of math.NA and the empirical contribution is useful. The main risk is the unproven extended admissibility condition (Eq. (17)) combined with the corner-based box distance, which can admit near-singular blocks in offset geometries; this directly affects the robustness claim. The single-vector error measure is a second, more easily fixable weakness. If the authors can provide a safe distance computation for (17) (or a convincing sensitivity study) and multi-vector error statistics, I would support publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a good paper, worth refereeing. The package of diagonal convergence criterion, diagonal pivots, extended admissibility condition, and sustained convergence check is genuinely new as a combination, and the experiments show ACAPP+ DCC staying below tolerance on every benchmark, including the transcranial ultrasound application, while the other variants fail in multiple places. Code is on GitHub, so the work is reproducible. That deserves credit.\n\nThe strongest part is the empirical demonstration of early convergence on volume operators and on the cube double-layer operator, with block-level visualizations making the failure mechanism clear. The memory-efficiency argument is also honest: the added machinery does not increase storage, and compression rates stay in line with the alternatives.\n\nSoft spots, roughly in order of importance.\n\n1. The admissibility condition (17) uses corner-to-corner distance between boxes, which overestimates true separation when box projections overlap in some coordinates. The stress-test note is right: a pair of boxes can pass the 2·max_element_diameter test even when actual element gaps are smaller. The factor 2 is justified by a single diagram and a one-line argument. For the tested meshes, mostly regular voxel grids, this probably caused no trouble, but the universal \"always\" in the abstract and conclusions is a step beyond the evidence. A sensitivity study on the factor 2 and on η, or a more careful distance definition, would close this gap.\n\n2. The main existing baselines are absent. RSCC from [27] and the diagonal-pivot variant from [28] are described but never run. The paper would be stronger if it showed ACAPP+ DCC beating those methods on the same benchmarks, not just vanilla ACAPP and ACAPP+.\n\n3. The error metric is a single random vector with no variance, and α, η, and the factor 2 are all empirical. These are minor if the central claim is the practical one, but they limit how strongly one can conclude \"robustness.\"\n\nNone of this undermines the core finding: for the operators and meshes considered, the deterministic DCC reliably fixes early convergence without memory overhead. The paper is honest about the frequency-dependent complexity results, and the transcranial simulation is a useful real-world capstone.\n\nWho should read it: anyone implementing ACA or H-matrix compression for integral equations, and people working on robust fast solvers. It is not a mathematical breakthrough; it is a careful engineering contribution with a clear practical payoff. I would engage with it, and I would want the revision to address the missing baselines and the admissibility sensitivity before accepting.","headline":"Solid engineering paper with reproducible code and a genuinely more robust ACA variant, but the 'always' claim rests on an admissibility heuristic with no sensitivity analysis.","tokens_in":20694,"tokens_out":3684,"would_cite":true,"duration_ms":39237,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N38","65R20","65F30","65F55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper argues that four deterministic extensions—a diagonal convergence criterion, diagonal pivots, a tightened admissibility condition, and a sustained convergence check—make ACAPP compression of acoustic volume and boundary operators","keywords":["hierarchical matrix compression","adaptive cross approximation","ACAPP","Helmholtz equation","volume integral operators","boundary integral operators","early convergence","transcranial ultrasound"],"falsifier":"Construct a mesh with thin, elongated elements where two octree boxes satisfy condition (17) yet contain nearly singular interacting points closer than the effective element size, compress the single-layer operator with ACAPP+ DCC at a strict tolerance such as ε=10⁻⁶, and compare the measured relative error (18) with ε. If the measured error exceeds ε, the claimed all-benchmark robustness fails.","tokens_in":19847,"feed_emoji":"🔊","tokens_out":6150,"duration_ms":64236,"temperature":0.7,"pith_summary":"Standard adaptive cross approximation with partial pivoting (ACAPP) can declare convergence early, because its error estimate is built from only the rows and columns it happens to visit; the actual low-rank error then stays above the requested tolerance. This paper proposes four deterministic fixes—checking an extra extended diagonal of each block (DCC), offering those diagonal entries as pivots, tightening admissibility so nearly singular interactions go to dense blocks, and requiring the tolerance to hold over several consecutive iterations. The central claim is that with these fixes, the ACAPP+ variant compresses every benchmark volume and boundary acoustic operator to the prescribed tolerance at the same memory cost as the standard algorithms. If true, large dense integral-equation simulations—including a 972 GB VSIE system for transcranial ultrasound—can be run reliably with controlled error and log-linear memory at constant frequency.","feed_headline":"Modified ACA compression passes every acoustic error test","feed_subtitle":"Standard ACAPP can stop early and miss its tolerance; the DCC variant does not—at the same memory cost.","key_machinery":"The load-bearing mechanism is the Diagonal Convergence Criterion (DCC): for each m×n admissible block the algorithm evaluates d=max(m,n) extra entries along the extended main diagonal, updates their residuals e_ℓ[s]=e_{ℓ-1}[s]-u_ℓ[i_s]v_ℓ[j_s] each iteration, and adds the block-normalized mean of their squared residuals to the stopping criterion (eq. 16). This gives deterministic coverage of all rows and columns, feeds the extra entries back into pivot selection when the standard criteria pass but DCC does not, and is paired with an admissibility guard (distance > 2×maximum element diameter) and a requirement that convergence hold for α consecutive iterations.","core_discovery":"On its own terms, the paper's discovery is that early convergence in ACAPP is not an irreducible flaw but a failure of the error estimator and pivot selection to sample the right parts of a block, and that a deterministic diagonal-based sampling scheme fixes it. Specifically, the paper shows that matrix blocks with two separated nonzero sub-blocks (cube double-layer operator) or with repeating internal sub-structures (ellipsoid volume operator) defeat the standard estimator, yet the ACAPP+ DCC algorithm—with the combined criterion (16), diagonal pivots, condition (17), and sustained convergence check—achieves the target tolerance in every test, including a CT-derived skull model.","pith_inferences":["Not tested in the paper: applying DCC to the hypersingular operator or to H2/directional compression could reveal different failure modes; the diagonal sweep's coverage of all rows and columns may transfer, but this is an extrapolation.","The factor 2 in condition (17) is heuristic; a sensitivity sweep on meshes with thin or stretched elements would show whether the robustness holds beyond the tested geometries or whether a per-element distance check is needed.","One could turn DCC into an adaptive sampler: if the mean diagonal residual concentrates on a few entries, add rows and columns through those entries as pivots, generalizing the diagonal fix to blocks whose hidden structure is not diagonal-aligned.","Because criterion (16) is the maximum of three normalized residuals, monitoring which term dominates at termination could serve as a diagnostic for early-convergence risk in user codes."],"forward_implications":["On all reported benchmarks, ACAPP+ DCC meets the prescribed tolerance; standard ACAPP and ACAPP+ stall far above tolerance in cube, ellipsoid, and skull cases, so users of these operators can switch to the new variant and get bounded errors.","Compression rates stay essentially unchanged—e.g., 22.5% storage versus 22.4–23.0% for other variants on the cube at ε=10⁻¹⁰—so the reliability gain costs no memory.","At constant frequency, memory scales O(n log n) with mesh refinement (tested up to about 16.8 million voxels), enabling larger high-resolution acoustic simulations than dense storage allows.","For the 972 GB VSIE skull system, ACAPP+ DCC compresses to about 335 GB at ε=10⁻¹⁰ and to roughly 18.4% of dense storage at ε=10⁻⁵, and the resulting acoustic field displays expected reflection, absorption, and focal-spot behavior.","The DCC replaces random restart and random reference-column choices with deterministic choices, improving reproducibility of the compression."],"fun_headline_variants":["New pivot rule stops ACA early-convergence failures","Robust ACA variant meets all acoustic error tolerances","Diagonal-based sampling fixes acoustic matrix compression","Modified ACA passes every acoustic error test on Helmholtz"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Everything rests on condition (17): the heuristic distance > 2 × maximum element diameter, with no proof or sensitivity study, must be enough to keep every nearly singular interaction out of admissible low-rank blocks; if a thin or high-aspect-ratio element slips a near singularity into a compressed block, the error estimator may not catch it.","fun_headline_variants_meta":{"raw":{"variants":["New pivot rule stops ACA early-convergence failures","Robust ACA variant meets all acoustic error tolerances","Diagonal-based sampling fixes acoustic matrix compression","Modified ACA passes every acoustic error test on Helmholtz"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000435,"raw_usage":{"total_tokens":2070,"prompt_tokens":781,"completion_tokens":1289,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":525,"completion_tokens_details":{"reasoning_tokens":1229}},"tokens_in":525,"tokens_out":1289,"duration_ms":10622,"temperature":1.0,"reasoning_tokens":1229,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T12:32:38.986378+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a mesh with thin, elongated elements where two octree boxes satisfy condition (17) yet contain nearly singular interacting points closer than the effective element size, compress the single-layer operator with ACAPP+ DCC at a strict tolerance such as ε=10⁻⁶, and compare the measured relative error (18) with ε. If the measured error exceeds ε, the claimed all-benchmark robustness fails.","supporting_citations":[],"review_version":1}