{"id":"dcc986b9-26f4-4351-953f-910e9615214f","arxiv_id":"2507.15848","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Iterative soft thresholding is combined with collocation-based time stepping to produce low-rank solutions of evolution equations whose ranks stay quasi-optimal for the achieved accuracy.","lead":"The authors design low-rank time-stepping schemes for evolution equations that automatically adjust their approximation rank to hit a user-chosen accuracy, with proof that the rank stays within a constant factor of the best possible rank. A reader interested in high-dimensional Schrödinger or Fokker-Planck simulation would care because this closes a known gap: previous time-stepping methods either controlled error or rank, not both.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 4.3's rank benchmark uses a false equality: min_{∥u−v∥≤η} σmax(v) does not equal the soft-thresholding parameter, so Theorem 27's initialization H_{α0}u(0) is not covered.","rationale":"The paper's central claim is Theorem 27, and the theorem's proof must be internally consistent for the claim to stand. The bounded-potential assumption flagged by the reader is an explicitly stated limitation rather than an internal inconsistency: the authors restrict to uniformly bounded V and defer Coulomb-type potentials to future work via Strichartz estimates. The implementation gap involving intermediate recompressions is also real, but it concerns the discrepancy between the analyzed ideal algorithm and the code used in experiments. The concern I identify is more direct: the definition of the rank benchmark r_n is used in the proof of the main theorem, and the displayed equality between the minimal σmax of an η-approximation and the soft-thresholding parameter is false. A simple two-singular-value example shows the initialization step H_{α0}u(0) can fail both the stated error bound and the stated rank bound. Since the rank bound is the main novelty, this is load-bearing. The issue appears fixable by redefining α_n and α0 via the soft-thresholding condition and initializing with the corresponding hard-thresholded value, so the reader's CONDITIONAL verdict remains appropriate. I recommend UNCHANGED: the verdict should stay conditional, with the additional requirement that the α_n definition and initialization be corrected.","tokens_in":32208,"tokens_out":37775,"duration_ms":392094,"concrete_test":"Take a rank-2 element u with singular values (2,1) and set η=0.5. Compute both sides of the equality in Section 4.3: (i) minimize σmax(v) subject to ∥u−v∥≤0.5 (result: 1.5); (ii) solve ∥u−S_αu∥=0.5 (result: α≈0.354). Then form ũ_0 = H_{1.5}u and check whether ∥ũ_0−u∥ ≤ η and rank(ũ_0) ≤ r_0 = 0.5²/1.5² hold. They fail: the error is 1 and the rank is 1, while r_0≈0.111. Repeating with α=0.354 satisfies both bounds, confirming that the soft-thresholding definition is the correct one and the printed initialization is not.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem's rank bound is stated in terms of r_n = η²/α_n², with α_n defined in Section 4.3 by the claimed equality α_n = inf_t min_{∥u(t)−v∥≤η} σmax(v) = max{α>0 : ∀t, ∥u(t)−S_αu(t)∥ ≤ η}. This equality is false in general. The soft-thresholding parameter solves ∥u−S_αu∥² = Σ_i min(α,σ_i)², whereas the minimal possible largest singular value of an η-approximation v is a different variational quantity, generally much larger. Concretely, if u has singular values (2,1) and η=0.5, the minimal σmax over η-approximations is 1.5 (attained by singular values (1.5,1)), while ∥u−S_αu∥=0.5 is attained at α≈0.354. The theorem proof then initializes ũ_0 = H_{α0}u(0) with α0 given by the left-hand side and asserts ∥ũ_0−u(0)∥ ≤ η and rank(ũ_0) ≤ r_0 = η²/α0². In this example H_{1.5}u keeps only the first singular value, giving error 1 > 0.5 and rank 1 > r_0 ≈ 0.111. Thus the induction in Theorem 27 is not valid as written. The flaw appears repairable by defining α_n and α_0 solely through the soft-thresholding condition and initializing with the corresponding H_{α0}, but the current text does not do so.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops low-rank time integration schemes based on iterative Picard/SDC fixed-point iterations combined with soft thresholding of singular values. The central result, Theorem 27, claims for the Gauss-Legendre-Picard scheme a global error bound of order η + κ_J h^{J+1} + κ_{2J} h^{2J}T (times an exponential in T) and rank bounds proportional, up to constants, to benchmark ranks r_n = η²/α_n². The analysis rests on the contractivity of the Duhamel fixed-point iteration with step size independent of spatial resolution, a constant-factor threshold decrease, and a series of rank and quadrature lemmas. Numerical experiments for a Schrödinger equation and a parabolic equation with anisotropic diffusion illustrate the rank evolution and error behavior.","tokens_in":32523,"tokens_out":17683,"duration_ms":182270,"significance":"Assuming the main theorem can be repaired as indicated below, this is a notable contribution: it appears to be the first time-stepping low-rank integrator that simultaneously provides a guaranteed accuracy bound and a rank bound in terms of natural best-approximation ranks, without requiring a space-time variational formulation. The paper is careful and self-contained, with explicit constants, a detailed induction proof, and clear statements of assumptions. The use of the independent soft-thresholding result [9] is appropriate, and the numerical experiments are informative. The main mathematical idea is sound and the flaws identified below are localized and repairable.","major_comments":[{"comment":"The displayed equality defining α_n, namely α_n = inf_t min_{∥u(t)−v∥≤η} σmax(v) = max{α>0 : ∀t, ∥u(t)−S_αu(t)∥≤η}, is false in general. For u with singular values (2,1) and η=0.5, the left-hand side is 1.5 (attained by v with singular values (1.5,1)), while the right-hand side is approximately 0.354, the threshold at which ∥u−S_αu∥=0.5. The claim 'the minimizer is of the form v=S_{α(t)}u(t)' is also false. Consequently, the initialization ũ_0=H_{α0}u(0) with α0=1.5 produces error 1>η and rank 1>r_0≈0.111, contradicting the asserted properties and invalidating the induction in the proof of Theorem 27. The flaw is repairable by defining α_n as the largest α such that ∥u(t)−S_αu(t)∥≤η for all t in the interval, and by initializing with H_{α0}u(0) (or S_{α0}u(0)) for that α0; then the error and rank bounds in the proof go through. As written, however, the theorem's definition and proof are not valid.","section":"Section 4.3 and Theorem 27"},{"comment":"The proof of Theorem 27 states 'We initialise the scheme with ũ_0 = H_{α0}u(0)', but Algorithm 3.1 contains no such step; line 3 of the algorithm sets α0←σmax(v0) and v0←0, and the initial boundary value is never thresholded. The theorem therefore analyzes a variant of the scheme that differs from the presented algorithm. In addition, the assignment v0←0 in line 3 overwrites the left boundary value that is needed in line 11 to form the next boundary approximation; a separate variable should be used for the fixed-point initial guess. The paper should reconcile the algorithm and the theorem by adding the thresholded initialization to the algorithm and fixing the name conflict.","section":"Theorem 27 proof and Algorithm 3.1"},{"comment":"The practical implementation described after Algorithms 3.1 and 3.2 applies an additional recompression R_δ with tolerance δ_rel·res after every addition of low-rank matrices, and in the SDC case a further smaller recompression for the residual. None of the lemmas in Section 4 account for these intermediate recompressions; the analysis covers only the final recompression at the subinterval boundary (e.g., Lemma 24 and Proposition 25). As a result, the numerical experiments test a heuristic variant rather than the exactly analyzed scheme. The authors should either extend the analysis to include these steps or explicitly state that the experiments demonstrate a practical variant whose analysis is open.","section":"Section 3.2"}],"minor_comments":[{"comment":"There is a typo, 'appoximation', in the sentence on the stability function; it should read 'approximation'.","section":"Section 2.4"},{"comment":"There is a typo, 'mendtioned', in the paragraph on recompression tolerances; it should read 'mentioned'.","section":"Section 6.1"},{"comment":"The sentence 'Here res denotes the current residual error estimator' is vague because res is updated in line 7 as ∥φ_k−v_k∥_J; please give a precise definition at first use.","section":"Section 3.2, after Algorithm 3.1"},{"comment":"In the proof, the final equality '2 Re(−i Σ ∥e^{it_j∆}p(t_j)∥²)=0' is correct but could be made clearer by explicitly noting that the sum is real and the real part of a purely imaginary number vanishes.","section":"Lemma 15"}],"recommendation":"major_revision","confidential_remarks":"The paper is a good fit for the journal. The main theorem's proof contains a repairable but currently invalid step; after the definition of α_n is corrected and the initialization is aligned with the algorithm, the contribution is solid. No concerns about novelty or attribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper delivers something genuinely new: a time-stepping low-rank integrator where the rank of the iterates is shown to stay within a constant factor of the best approximation rank at the same accuracy, with explicit error bounds. The combination of the Lawson transform, Picard iteration, and soft thresholding with a decreasing threshold is, as far as I know, absent from the prior literature. The authors also deserve credit for laying out the limitations: the parabolic and SDC variants are marked as unanalyzed, and the numerical experiments are honest.\n\nThe main soft spot is a definitional error in Section 4.3. The paper defines α_n as both the infimum of minimal largest singular values over η-approximations and the soft-threshold parameter that gives error η. These are not equal. For a matrix with singular values (2,1) and η=0.5, the first quantity is 1.5, the second is about 0.354. The proof of Theorem 27 then uses H_{α0}u(0) with the first quantity, which can produce an initial error larger than η and a rank that violates the stated bound r_0. That breaks the induction. The repair is straightforward: define α_n from the soft-threshold condition alone, and use H_{α_n} with that same parameter. I believe the theorem can be made correct this way, but the text as written is not.\n\nTwo smaller issues. The implementation uses intermediate recompressions with a relative tolerance that are not part of the analyzed algorithm; the authors should either extend the analysis or explicitly say those are outside the guarantees. And the parabolic and SDC sections are explicitly preliminary, so the proven region is only the Picard-Schrödinger case with bounded, finite-rank potential. The paper is open about this, which I respect.\n\nThis is a good paper that needs revision before it can be trusted as stated. I would send it to a serious referee.","headline":"First time-stepping low-rank integrator with quasi-optimal rank bounds, but a false equality in Section 4.3 breaks the proof as written.","tokens_in":33081,"tokens_out":6146,"would_cite":true,"duration_ms":63804,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65F55","65M12","65Y20","65L70"],"pacs":[],"model":"deepseek-v4-flash","headline":"This integrator guarantees accuracy with near-optimal ranks.","keywords":["low-rank approximation","time integration","soft thresholding","Schrödinger equation","parabolic PDE","quasi-optimal ranks","Gauss-Legendre collocation","spectral deferred correction"],"falsifier":"A decisive test is to run Algorithm 3.1 on the two-dimensional Schrödinger equation with a bounded finite-rank potential, initial data of full rank, and the paper's parameters (for instance $h=0.1$, $J=11$, $\\eta=10^{-6}$), and check on every interval whether $\\operatorname{rank}(\\tilde u_n)$ is bounded by the explicit constant of Theorem 27 times $\\max_{1\\le i\\le n} \\eta^2/\\alpha_i^2$; a single violation would refute the quantitative claim. The same experiment with an unbounded potential such as $V(x)=1/|x|$ falls outside the theorem's assumptions, so observing divergence or rank blow-up there would show the bounded-potential premise is essential.","tokens_in":31979,"feed_emoji":"🧮","tokens_out":13650,"duration_ms":135070,"temperature":0.7,"pith_summary":"This paper claims that time integration in low-rank matrix representation can be made both provably accurate and provably rank-efficient: for a prescribed accuracy $\\eta$, the method produces approximate solutions whose representation rank stays within a constant factor of the best-approximation rank at that accuracy. The model problem is the linear time-dependent Schrödinger equation on a product domain, treated through Duhamel's formula in the twisted variable $v(t)=e^{-it\\Delta}u(t)$, collocation at Gauss-Legendre points, and a fixed-point iteration (Picard, or spectral deferred correction) in which every update is soft-thresholded with a geometrically decaying threshold. The main theorem bounds the global error by a term of order $h^{J+1}$ (with $h^{2J}$ accuracy at subinterval boundaries) times an exponential in $T$, and bounds every computed rank by an explicit constant times the ranks of the exact solution's best low-rank approximations at the target accuracy. To the authors' knowledge, this is the first time-stepping scheme, outside of space-time variational formulations, that provably achieves both a guaranteed error and quasi-optimal ranks; the paper also constructs and tests numerically a parabolic variant.","feed_headline":"Time stepper achieves guaranteed accuracy at near-optimal rank","feed_subtitle":"Soft thresholding inside each fixed-point sweep keeps computed ranks within a constant of the best possible.","key_machinery":"The load-bearing mechanism is a contractive fixed-point iteration whose every sweep is followed by soft thresholding of singular values. The twisted variable $v(t)=e^{-it\\Delta}u(t)$ converts the stiff Schrödinger equation into $v'=F_t v$ with $F_t=-ie^{-it\\Delta}Ve^{it\\Delta}$, so the Picard map $\\Phi(v)_j = v_0 + \\sum_m \\omega_{j,m}F_{t_m}v_m$ has contraction constant $\\rho = h\\Lambda_J C_V$ depending only on the bounded potential norm, not on the spatial mesh. Soft thresholding $S_\\alpha$ shrinks every singular value by $\\alpha$ and is non-expansive, and it satisfies $\\|v-S_\\alpha v\\|^2 = \\min_w (\\|v-w\\|^2 + \\alpha^2\\operatorname{rank}(w))$, which makes the soft-thresholding error the natural benchmark for quasi-optimal ranks. A comparison lemma, $\\operatorname{rank}(S_{\\alpha+\\beta}u) \\le \\operatorname{rank}(S_\\alpha v) + \\|u-v\\|^2/\\beta^2$, turns the convergence of the iterates into rank bounds. The Gauss-Legendre nodes (roots of the Legendre polynomial on each subinterval) make the fixed point an implicit Gauss Runge-Kutta method of order $2J$ that preserves the $L^2$ norm at interval boundaries, so the local error accumulates only linearly in the number of intervals.","core_discovery":"The paper's central claim is that rank adaptation in low-rank time integration can be certified. For the linear Schrödinger equation $i\\partial_t u = -\\Delta u + V u$ on a product domain, with $V$ a bounded potential of finite tensor rank, Algorithm 3.1 is shown to satisfy the global error bound\n$$\\sup_{1\\le n\\le N} \\|\\tilde u_n - u(nh)\\| \\le \\left(\\eta + \\kappa_J $h^{{J+1}}$ + \\kappa_{2J} $h^{{2J}}$ T\\right) \\exp\\!\\left(8\\Lambda_J C_V \\frac{1+\\rho}{(1-\\rho)^3} T\\right)$$\nand the rank bound\n$$\\operatorname{rank}(\\tilde u_n) \\le \\frac{16}{\\$rho^{3}$}\\left(\\frac{1+\\rho}{1-\\rho}\\right)^2 \\left(1 + J\\,\\operatorname{rank}(V) \\max_{1\\le i\\le n} r_i\\right), \\qquad r_i = \\frac{\\$eta^{2}$}{\\$alpha_i^{2}$},$$\nwhere $\\alpha_i$ is the soft-thresholding level at which the exact solution has soft-thresholding error $\\eta$ on the $i$-th subinterval. In words, the computed ranks stay within a constant factor, depending on $h$, $J$, and the rank of the potential, of the ranks of the best low-rank approximations at the target accuracy. A parabolic version, using Radau-Legendre nodes and a dyadic secondary quadrature for the Duhamel integrals, is constructed and tested numerically with quasi-optimal rank behavior, but its analysis is explicitly left open.","pith_inferences":["Editorial inference: the same soft-thresholding-plus-contractive-fixed-point recipe should transfer to tensor-train and hierarchical tensor formats, since rank reduction there reduces to SVD truncation of matricizations; the paper says the scheme generalizes directly, but the dimension-dependence of the constants is not analyzed.","Editorial inference: the $h^{-3}$ factor in the rank bound suggests the method should be used with $h$ fixed and $J$ increased, and that taking $h$ too small actively worsens the rank guarantee; a natural testable extension is an adaptive strategy that raises $J$ per interval rather than shrinking $h$.","Editorial inference: the assumption that $V$ is uniformly bounded and of finite tensor rank excludes Coulomb-type many-body potentials, the physically central case; the paper only mentions Strichartz estimates as a future direction, so extending the theorem to singular potentials is the key open step.","Editorial inference: in the parabolic case, the secondary dyadic quadrature breaks the equivalence to a collocation/Runge-Kutta method, and the paper leaves the analysis open; the numerical evidence suggests a quasi-optimal rank bound may hold, but proving it likely requires tracking the additional quadrature error inside the fixed-point contraction."],"forward_implications":["For solutions whose best-approximation ranks grow like $\\varepsilon^{-1/s}$ with accuracy $\\varepsilon$, the theorem gives computed ranks bounded by $h^{-3}(1+J\\operatorname{rank}(V))\\eta^{-1/s}$ up to a constant, so algebraic singular-value decay is preserved in the computed ranks.","For exponentially decaying singular values, where the optimal rank behaves like $(1+|\\log\\varepsilon|)^\\beta$, the same argument gives computed ranks of order $h^{-3}(1+J\\operatorname{rank}(V))(1+|\\log\\eta|)^\\beta$, so high-accuracy solutions remain cheap to represent.","The time step $h$ is limited only by the contraction condition $\\rho = h\\Lambda_J C_V < 1$ and by the $h^{-3}$ factor in the rank constant, not by the spatial mesh; accuracy is obtained by raising the number $J$ of collocation points.","The Gauss-Legendre scheme preserves the $L^2$ norm at subinterval boundaries, matching the unitary character of the Schrödinger flow, while the parabolic Radau-Legendre variant reproduces the damping of high frequencies; the numerical tests show the error accumulating essentially linearly in time.","The approach sidesteps the tangent-space projection step of dynamical low-rank approximation, so the error analysis does not require the projection error to be small; the paper presents this as the first such guarantee for a time-stepping method outside of space-time variational formulations."],"supporting_citations":[{"why":"Supplies the soft-thresholding iteration, the non-expansiveness property, and the rank-vs-error comparison Lemma 20 used for the quasi-optimal rank bounds.","marker":"[9]"},{"why":"Introduces the Lawson twist $v=e^{-it\\Delta}u$ that removes the stiff Laplacian from the fixed-point contraction, making $h$ independent of spatial resolution.","marker":"[34]"},{"why":"Identifies the collocation fixed point with an implicit Runge-Kutta (Gauss) method and supplies the order and stability facts behind the $h^{2J}$ accuracy.","marker":"[26]"},{"why":"Provides the Gauss-Legendre quadrature error bounds used in Lemma 16 and Lemma 32 for the local accuracy $h^{2J+1}$.","marker":"[41]"},{"why":"Introduces spectral deferred correction and the spectral integration matrix used in Algorithm 3.2 and in the remarks on interval transitions.","marker":"[20]"},{"why":"Establishes the prior space-time variational method with near-optimal ranks for parabolic problems, the benchmark against which the present method's novelty is measured.","marker":"[8]"},{"why":"Dynamical low-rank approximation, the main contrasting method that controls ranks without a convergence guarantee.","marker":"[31]"},{"why":"Provides the Mirsky-type singular-value inequality used in Lemma 34, which is load-bearing for Proposition 11's characterization of soft-thresholding error.","marker":"[38]"},{"why":"Shows that Picard iteration and spectral deferred correction pick up at least one order of accuracy per sweep, used for the SDC order claims in Remark 7.","marker":"[13]"},{"why":"Gives the transition to the subinterval boundary and the fact that iterating to convergence reaches order $2J$ for Gauss-Legendre nodes.","marker":"[25]"}],"fun_headline_variants":["Low-rank time stepper certifies near-optimal ranks","Iterative thresholding locks ranks near optimal for time integration","Adaptive low-rank ODE solver provably matches best rank","Soft thresholding yields provably near-best low-rank time stepping","Time integrator adapts rank with guaranteed accuracy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the potential $V$ is uniformly bounded, giving a contraction constant $\\rho = h\\Lambda_J C_V < 1$; the rest of the argument assumes $V$ has finite tensor rank and the exact solution is smooth with decaying singular values, so none of the conclusions of Theorem 27 are claimed for Coulomb-type singular potentials.","fun_headline_variants_meta":{"raw":{"variants":["Low-rank time stepper certifies near-optimal ranks","Iterative thresholding locks ranks near optimal for time integration","Adaptive low-rank ODE solver provably matches best rank","Soft thresholding yields provably near-best low-rank time stepping","Time integrator adapts rank with guaranteed accuracy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000221,"raw_usage":{"total_tokens":1455,"prompt_tokens":956,"completion_tokens":499,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":572,"completion_tokens_details":{"reasoning_tokens":418}},"tokens_in":572,"tokens_out":499,"duration_ms":5178,"temperature":1.0,"reasoning_tokens":418,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:23:52.269186+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive test is to run Algorithm 3.1 on the two-dimensional Schrödinger equation with a bounded finite-rank potential, initial data of full rank, and the paper's parameters (for instance $h=0.1$, $J=11$, $\\eta=10^{-6}$), and check on every interval whether $\\operatorname{rank}(\\tilde u_n)$ is bounded by the explicit constant of Theorem 27 times $\\max_{1\\le i\\le n} \\eta^2/\\alpha_i^2$; a single violation would refute the quantitative claim. The same experiment with an unbounded potential such as $V(x)=1/|x|$ falls outside the theorem's assumptions, so observing divergence or rank blow-up there would show the bounded-potential premise is essential.","supporting_citations":[{"cited_title":"Iterative methods based on soft thresholding of hierarchical tensors","cited_arxiv_id":null,"evidence_quote":"Supplies the soft-thresholding iteration, the non-expansiveness property, and the rank-vs-error comparison Lemma 20 used for the quasi-optimal rank bounds."},{"cited_title":"Generalized Runge-Kutta processes for stable systems with large Lipschitz con- stants","cited_arxiv_id":null,"evidence_quote":"Introduces the Lawson twist $v=e^{-it\\Delta}u$ that removes the stiff Laplacian from the fixed-point contraction, making $h$ independent of spatial resolution."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies the collocation fixed point with an implicit Runge-Kutta (Gauss) method and supplies the order and stability facts behind the $h^{2J}$ accuracy."},{"cited_title":"Introduction to Numerical Analysis","cited_arxiv_id":null,"evidence_quote":"Provides the Gauss-Legendre quadrature error bounds used in Lemma 16 and Lemma 32 for the local accuracy $h^{2J+1}$."},{"cited_title":"Spectral deferred correction methods for ordinary differential equations","cited_arxiv_id":null,"evidence_quote":"Introduces spectral deferred correction and the spectral integration matrix used in Algorithm 3.2 and in the remarks on interval transitions."},{"cited_title":"A space-time adaptive low-rank method for high-dimensional parabolic partial differential equations","cited_arxiv_id":null,"evidence_quote":"Establishes the prior space-time variational method with near-optimal ranks for parabolic problems, the benchmark against which the present method's novelty is measured."},{"cited_title":"Dynamical low-rank approximation","cited_arxiv_id":null,"evidence_quote":"Dynamical low-rank approximation, the main contrasting method that controls ranks without a convergence guarantee."},{"cited_title":"The eigen- and singular values of the sum and product of linear operators","cited_arxiv_id":null,"evidence_quote":"Provides the Mirsky-type singular-value inequality used in Lemma 34, which is load-bearing for Proposition 11's characterization of soft-thresholding error."},{"cited_title":"On the convergence of spectral deferred correction methods","cited_arxiv_id":null,"evidence_quote":"Shows that Picard iteration and spectral deferred correction pick up at least one order of accuracy per sweep, used for the SDC order claims in Remark 7."},{"cited_title":"On the spectral deferred correction of splitting methods for initial value problems","cited_arxiv_id":null,"evidence_quote":"Gives the transition to the subinterval boundary and the fact that iterating to convergence reaches order $2J$ for Gauss-Legendre nodes."}],"review_version":1}