On variational iterative methods for semilinear problems

Prosper Torsu

REVIEW 4 major objections 4 minor 24 references

On variational iterative methods for semilinear problems

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A variational Adomian-type series method with a log-conductivity expansion reduces semilinear elliptic problems to sequences of fast-solvable Poisson equations.

desk verdict A clean variational Adomian-style iteration with fast Poisson solvers, but the central convergence theorem is conditional on an unproved tail-decay assumption and the Picard-equivalence lemma has an indexing slip. read the letter →

arxiv 1908.00470 v1 pith:ZDETTPZV submitted 2019-08-01 math.NA cs.NA

Prosper Torsu This is my paper · ORCID

classification math.NAcs.NA

keywords problemssemilineariterativemethodnumericalableaccurateapplications

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 deals with equations that mix a linear diffusion term with a nonlinear reaction term, for example the stationary Allen-Cahn or Klein-Gordon equations. Solving such equations numerically usually requires an iterative loop, and each loop typically needs a Jacobian matrix, which is expensive to compute. The author proposes to split the unknown solution into a sum of pieces, and to split the nonlinear term into a sum of so-called Adomian polynomials. Each piece of the solution is then obtained by solving a plain Poisson equation, which can be done very fast with Fourier-based solvers when the domain is a square and the boundary conditions are simple. A second expansion, based on the logarithm of the diffusion coefficient, is used to handle coefficients that vary in space.

The paper proves two basic facts: every piece is stable in the energy norm, and the error between the exact solution and the partial sum is bounded by the size of the remaining terms in the nonlinear series. The proof of this second fact, however, assumes that the remaining terms shrink to zero; the paper does not prove that the Adomian series converges. The numerical experiments use three manufactured solutions on a unit square and compare the series with the classical Picard iteration. The errors are very close to those of Picard, and the paper claims lower cost, but it reports no timing measurements.

Because the core idea is a known series method and the new ingredient is mostly a practical trick, the contribution is a modest extension. The method may be useful in settings where fast Poisson solvers are already available, but the advertised efficiency advantage is not demonstrated.

Extended reading notes

Core claim

The paper's central assertion is that the semilinear problem (1.1) can be solved by the partial sum (3.14), where each mode ψζ is obtained by inverting the Laplacian only, and that Lemma 3.3 guarantees ‖∇(ψ-ψM)‖₂ → 0 as M→∞ so that the series solution is accurate. The abstract adds that the method is 'computationally less expensive' and yields accurate approximations.

Load-bearing premise

The convergence result rests on the unproved premise that the Adomian tail Σ_{ζ>M} Pζ converges to zero in L²(Ω). This is stated as the final step of the proof of Lemma 3.3 ('The theorem follows as Σ_{ζ>M} Pζ → 0 for sufficiently large M') and is the same condition needed to prove that the series (3.10) for r converges. If the tail does not decay, the error bound (3.15) is vacuous and the partial sums need not approach the true solution, so the entire numerical justification collapses.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know two things about this one. First, the method is essentially Adomian decomposition in a variational wrapper, with one genuinely useful twist: the log-conductivity expansion (3.18)-(3.21) decouples the coefficient a so that each subproblem is a Poisson solve, which is a real practical convenience for regular grids. Second, the advertised convergence guarantee in Lemma 3.3 is not proved. The final step just asserts that the Adomian tail tends to zero, and that is exactly the condition needed to make the series converge in the first place. Individual L² membership of the P_ζ does not give you summability, so the error bound (3.15) is conditional on the very thing the lemma claims to establish. This is a genuine gap, not a nitpick.

What the paper does well: the variational framing is clean, the recursive subproblems are explicit and linear, the stability estimate (3.23) is standard Lax-Milgram, and the numerical experiments are honest—errors match Picard and the manufactured solutions across three test problems. The authors also state plainly that convergence theory for these series methods is generally open, and that equivalence with Picard is only shown for linear r. That candor counts for something.

The soft spots beyond the main gap: Lemma 3.4 contains a summation/indexing error in the proof, so the claimed equivalence with Picard for linear reactions is not established as written. The computational-cost claim in the abstract and conclusion is supported by no timing measurements; the tables only give accuracy. And the novelty over existing Adomian/homotopy work is modest, though the log-conductivity expansion is a legitimate addition rather than pure repackaging.

Who gets value from this? Someone implementing a Jacobian-free solver for semilinear elliptic problems on regular grids might find the subproblem structure appealing. A reader looking for a rigorous convergence theorem will be disappointed, because the central result is conditional on an unproved analyticity/summability condition.

My recommendation: send it to peer review, but the referee should push hard on the tail-decay assumption. The author needs to either prove summability under stated hypotheses or clearly reframe Lemma 3.3 as a conditional result. Fix Lemma 3.4, add timing comparisons, and release code. The underlying idea is plausible and the numerics are consistent, so it is worth referee time despite the gap.

Referee Report

4 major / 4 minor

Summary. The paper proposes a variational iterative method for semilinear elliptic boundary value problems of the form -∇·(a∇ψ)+r(ψ)=f. The nonlinearity is expanded into Adomian polynomials, and each mode is obtained by solving a sequence of linear sub-problems with the (possibly variable) coefficient a handled via a logarithmic Taylor expansion, so that only Poisson solves are required. The authors provide stability estimates for the modes, a comparison with Picard iteration, a fast Poisson solver implementation for the resulting discrete systems, and three numerical tests comparing the series approximation with manufactured exact solutions and with Picard iteration. The central theoretical claim is that the partial sums ψ_M converge to the true solution, with error controlled by the decay of the Adomian tail.

Significance. If the convergence gap were closed, the method would be an attractive Jacobian-free alternative for semilinear problems, because each mode inversion involves only the constant-coefficient Laplacian and the fast solver in Section 4 is clearly described. The explicit stability estimates (3.23)-(3.24), the detailed construction of the fast solver, and the systematic numerical comparison with Picard iteration are useful contributions. However, the main convergence theorem is not actually proved: Lemma 3.3 assumes the decay of the Adomian tail rather than deriving it, so the central claim of the paper is conditional at present. The numerical experiments provide empirical evidence for the three test cases but do not supply the missing general mechanism.

major comments (4)

[Section 3, Lemma 3.3] The proof of Lemma 3.3 ends with "The theorem follows as Σ_{ζ>M}P_ζ → 0 for sufficiently large M." This is not a consequence of the stated hypothesis that P_ζ ∈ L²(Ω) for 0 ≤ ζ ≤ M; membership of each individual term in L² gives no control of the tail. Moreover, the hypothesis does not apply to indices ζ > M, which are precisely those appearing in the tail. Thus the error bound (3.15) is conditional on exactly the statement that Lemma 3.3 is supposed to establish. Please either prove tail decay under explicit assumptions on r (for example, analyticity combined with a smallness condition on the solution), or restate Lemma 3.3 as a conditional result and provide a separate proof of convergence for the Adomian series.
[Section 3.1, Lemma 3.4] For linear r, the Adomian polynomials defined by (3.11) satisfy P_ζ = ψ_ζ, so Σ_{ζ≤M}P_ζ equals ψ_M, not ψ_{M-1} as written in equation (3.16). In addition, equation (3.16) for ψ_M is not the same as the defining equation (3.13) with ζ = M; equation (3.13) gives ∫ a∇ψ_M·∇w = -∫ P_M w for M ≥ 1, while (3.16) sums over all ζ ≤ M. The induction as written is therefore inconsistent, and the base case for M = 1 is not stated. Please correct the indexing and the base case, and then restate the claim precisely.
[Section 5 and abstract] The abstract claims that the method is "computationally less expensive", but the paper reports no timing measurements or operation counts comparing the proposed method with Picard iteration or Newton's method. Since the total cost is (M+1)(N+1) solves of the discrete Laplacian plus the cost of assembling the right-hand sides, the claim needs either a complexity analysis that accounts for the growth of M and N, or quantitative CPU-time experiments. As written, the computational-cost claim is not supported by the evidence in Section 5.
[Section 3.2, Algorithm 2] In Algorithm 2, the loop bounds read "for ζ = 0 to N" and "for ξ = 0 to M", whereas Section 3.1 defines M as the Adomian truncation index and N as the Taylor truncation index for the coefficient a. The loops should run ζ = 0,...,M and ξ = 0,...,N to implement the derived equations (3.20)-(3.22). As written, the pseudo-code does not match the derivation and is inconsistent with the numerical experiments in which M is varied while N is fixed.

minor comments (4)

[Abstract and Introduction] The abstract and introduction refer to "known continuous data f and g", but the model problem (1.1) contains only f; the function g is first introduced in Remark 3.1 as an auxiliary function. Please clarify the notation at the outset.
[Section 5.3, Test 3] In Test 3 the problem is stated with r(ψ) = exp(-ψ²) + tan⁻¹ψ, but the sentence "Moreover, f is chosen accordingly with r(ψ) = ψ" contradicts this. Presumably f is chosen so that the stated exact solution satisfies the equation with the first nonlinearity; please correct the typo.
[Section 3, equation (3.11)] The definition of the Adomian polynomials in (3.11) uses 1/(ζ-1)! and d^{ζ-1}/dλ^{ζ-1}, which is undefined for ζ = 0. Please state P_0 = r(ψ_0) separately and give the formula for ζ ≥ 1.
[Section 3, proof of Lemma 3.3] In the chain of inequalities preceding (3.15), the Poincaré constant is omitted; the step ‖e_M‖₂ ≤ ‖∇e_M‖₂ should read ‖e_M‖₂ ≤ C_P ‖∇e_M‖₂ with a constant depending on Ω. This does not change the conclusion but should be stated for correctness.

Circularity Check

1 steps flagged · score 6.0 of 10

Lemma 3.3's convergence proof is conditional on the unproved decay of the Adomian tail, making the theorem's conclusion an assumption rather than a derivation.

self definitional [Lemma 3.3, Section 3 (proof after Eq. 3.15)]
"Consequently, ‖∇eM‖2 ≤ (Σ_{ζ>M} ‖Pζ‖2) ‖∇eM‖2. ... ‖∇eM‖2 ≤ Σ_{ζ>M} ‖Pζ‖2. (3.15) The theorem follows as Σ_{ζ>M} Pζ → 0 for sufficiently large M."

Lemma 3.3 claims to prove convergence of the partial sum ψM from the hypothesis Pζ ∈ L2(Ω) for ζ ≤ M. In the proof, the error is bounded by the tail sum Σ_{ζ>M} ‖Pζ‖2, and the proof then simply asserts that this tail tends to zero. No argument is given that the Adomian series Σ_{ζ≥0} Pζ converges to r(ψ); indeed, the error equation already uses the identity r(ψ) = Σ_{ζ≥0} Pζ, which is precisely the convergence property needed. The hypothesis Pζ ∈ L2 for finitely many ζ ≤ M gives no control over ζ > M. Thus the theorem reduces to the conditional statement: if the Adomian tail decays, then the error decays. The antecedent is asserted without proof, so the convergence theorem is not established independently; it is a restatement of the required input condition.

full rationale

The paper's main convergence theorem (Lemma 3.3) is not self-contained: its proof ends by assuming the decay of the Adomian tail, the very property needed for the series r(ψ)=Σ Pζ to represent r(ψ). This is a genuine circular step in the theoretical justification. The numerical experiments in Section 5 do provide empirical evidence that the tail decays for the three manufactured problems, and the algorithm itself is well defined, but the general convergence claim is therefore only conditionally established. No load-bearing self-citation circularity appears: the citations to the author's prior work [3] are motivational rather than foundational. The cost claim and the linear-equivalence lemma are outside the circularity finding. Overall, partial circularity in the central proof warrants a score of 6.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No parameters are fitted to data, but the truncation orders M and N and the auxiliary function g are user-chosen and affect the result. The key logical assumptions are the existence of a solution, L²-regularity of the Adomian polynomials, convergence of the Adomian tail (unproved), and validity of the truncated Taylor expansion of a.

free parameters (3)

M = 1, 3, 5, 7 in tests
Truncation order of the Adomian series in (3.14); chosen by hand in each numerical test and controls the accuracy of the approximation.
N = 2, 4 in tests
Truncation order of the Taylor expansion for a in (3.19); chosen by hand and determines the fidelity of the coefficient approximation.
auxiliary function g = not used in tests; example in Remark 3.1
Introduced in Remark 3.1 to avoid the trivial zeroth mode for homogeneous problems; arbitrary user choice with no guidance or numerical experiments.

assumptions (4)

domain assumption A solution to (2.3) exists, not necessarily unique.
Stated in Section 2: 'Under the assumption that a solution (not necessarily unique) exists'. The method produces an approximation only if a solution exists.
ad hoc to paper Pζ ∈ L²(Ω) for all 0 ≤ ζ ≤ M.
Required by Lemma 3.2 and Lemma 3.3 to guarantee existence and uniqueness of each mode and to bound the error. This is stated as a hypothesis, not derived from properties of r.
ad hoc to paper The Adomian series Σ_{ζ} Pζ converges in L² and its tail tends to zero.
Assumed in the final step of Lemma 3.3 to conclude convergence. This is exactly the convergence property the paper claims to establish, making the theoretical justification circular.
domain assumption The diffusion coefficient a admits the truncated Taylor representation a = Σ_{ξ≤N} Y^ξ/ξ! with small truncation error.
Used in (3.19)-(3.21) to convert the variable-coefficient problem into Laplacian subproblems. This holds for smooth a and large N, but the truncation error is not quantified in the convergence analysis.

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Pith. "Pith review of On variational iterative methods for semilinear problems." pith.science (2026). https://pith.science/paper/ZDETTPZV

@misc{pith2026190800470,
  author       = {Pith},
  title        = {Pith review of: On variational iterative methods for semilinear problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZDETTPZV}},
  note         = {Machine review of arXiv:1908.00470}
}

read the original abstract

This paper presents an iterative method suitable for inverting semilinear problems which are important kernels in many numerical applications. The primary idea is to employ a parametrization that is able to reduce semilinear problems into linear systems which are solvable using fast Poisson solvers. Theoretical justifications are provided and supported by several experiments. Numerical results show that the method is not only computationally less expensive, but also yields accurate approximations.

Figures

Figures reproduced from arXiv: 1908.00470 by the authors.

**Figure 1.** Approximation of various modes compared with the analytical solution. Number of elements 2 3 × 2 3 2 4 × 2 4 2 5 × 2 5 2 6 × 2 6 2 7 × 2 7 M = 1 3.80095(−01) 9.56161(−02) 2.99741(−02) 1.41233(−02) 1.02027(−02) M = 3 3.88708(−01) 9.61027(−02) 2.98165(−02) 1.38540(−02) 9.90701(−03) M = 5 3.89504(−01) 9.61117(−02) 2.98152(−02) 1.38520(−02) 9.90491(−03) M = 7 3.89612(−01) 9.61119(−02) 2.98152(−02) 1.38520(−02) 9.90491(−… view at source ↗

**Figure 2.** Approximation of various modes compared with the analytical solution. N = 2 Number of elements 2 3 × 2 3 2 4 × 2 4 2 5 × 2 5 2 6 × 2 6 2 7 × 2 7 M = 1 1.77952(−01) 1.75391(−01) 1.72629(−01) 1.72591(−01) 1.72622(−01) M = 3 8.85117(−02) 3.72049(−02) 1.08392(−02) 5.66948(−03) 5.07559(−03) M = 5 8.90338(−02) 3.66827(−02) 9.31148(−03) 2.35028(−03) 6.04344(−04) M = 7 8.90409(−02) 3.66799(−02) 9.30586(−03) 2.34311(−03) 5.9… view at source ↗

**Figure 3.** Approximation of various modes compared with the analytical solution. N = 2 Number of elements 2 3 × 2 3 2 4 × 2 4 2 5 × 2 5 2 6 × 2 6 2 7 × 2 7 M = 1 3.67061(−01) 3.25203(−01) 3.18984(−01) 3.17578(−01) 3.17235(−01) M = 3 1.02834(−02) 3.62913(−02) 2.20172(−02) 1.93606(−02) 1.88047(−02) M = 5 9.31325(−02) 2.41869(−02) 6.25023(−03) 1.74825(−03) 6.98383(−04) M = 7 9.29634(−02) 2.40161(−02) 6.06422(−03) 1.52422(−03) 3.8… view at source ↗

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Adomian, A review of the decomposition method and some recent results for nonlinear equations, Mathematical and Computer Modelling, 13 (1990), pp

G. Adomian, A review of the decomposition method and some recent results for nonlinear equations, Mathematical and Computer Modelling, 13 (1990), pp. 17–43

work page 1990

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Amrein and T

M. Amrein and T. P. Wihler, Fully adaptive newton–galerkin methods for semi- linear elliptic partial diﬀerential equations, SIAM Journal on Scientiﬁc Computing, 37 (2015), pp. A1637–A1657

work page 2015

[3] [3]

Aryana, F

S. Aryana, F. Furtado, V. Ginting, and P. Torsu , On series solution for second order semilinear parabolic ibvps , Journal of Computational and Applied Mathematics, 330 (2018), pp. 499–518

work page 2018

[4] [4]

Badiale and E

M. Badiale and E. Serra , Semilinear Elliptic Equations for Beginners: Exis- tence Results via the Variational Approach , Springer Science & Business Media, 2010

work page 2010

[5] [5]

N. A. Baker, D. Sept, S. Joseph, M. J. Holst, and J. A. McCammon , Electrostatics of nanosystems: application to microtubules and the ribosome , Pro- ceedings of the National Academy of Sciences, 98 (2001), pp. 10037–10041

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[6] [6]

Boccardo and L

L. Boccardo and L. Orsina , Semilinear elliptic equations with singular non- linearities, Calculus of Variations and Partial Diﬀerential Equations, 37 (2010), pp. 363–380

work page 2010

[7] [7]

Bochev and R

P. Bochev and R. B. Lehoucq , On the ﬁnite element solution of the pure neumann problem, SIAM review, 47 (2005), pp. 50–66. 15

work page 2005

[8] [8]

Colton and R

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, vol. 93, Springer Science & Business Media, 2012

work page 2012