Optimal Control for Chemotaxis Systems and Adjoint-Based Optimization with Multiple-Relaxation-Time Lattice Boltzmann Models

Aziz Belmiloudi

REVIEW 3 major objections 5 minor 89 references

Optimal Control for Chemotaxis Systems and Adjoint-Based Optimization with Multiple-Relaxation-Time Lattice Boltzmann Models

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that optimal control of coupled anisotropic chemotaxis systems can be solved with a multiple-relaxation-time lattice Boltzmann method and its adjoint, recovering the continuous equations and supplying a discrete gradient…

desk verdict A systematic formal extension of adjoint MRT-LBM to chemotaxis optimal control, but the Chapman-Enskog recovery is not established for the general class and no numerics back the practical claims. read the letter →

arxiv 1908.03876 v1 pith:QLRTHH6A submitted 2019-08-11 math.OC cs.NAmath.NA

Aziz Belmiloudi This is my paper · ORCID

classification math.OCcs.NAmath.NA MSC 92B0549J2035Q9249M2592C5065K1535K5765M99

keywords optimalcontrolchemotaxismultiple-relaxation-timelatticeBoltzmannmethodadjoint-basedoptimizationChapman-Enskogexpansionanisotropicconvection-diffusionsensitivityanalysisgradientdescent

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 tries to establish that a broad class of optimal control problems governed by coupled, anisotropic chemotaxis-type systems can be solved numerically by a multiple-relaxation-time (MRT) lattice Boltzmann method paired with an adjoint MRT model. The payoff would be a gradient-descent workflow in which each optimization iteration needs one forward lattice-Boltzmann solve and one backward adjoint solve, with the same local, parallel-friendly structure as the primal scheme. The paper derives first-order optimality conditions for a general three-equation system, builds an MRT scheme whose Chapman-Enskog expansion is claimed to recover the macroscopic equations, and then derives an adjoint MRT scheme whose discrete gradient is claimed to be exact for the discretized cost functional. Applications are drawn from cell migration, tumor invasion, and crime-pattern models.

What carries the argument

The carrying object is the D2Q9 (two-dimensional, nine-velocity) multiple-relaxation-time lattice Boltzmann scheme with the collision matrix $\Pi_\Theta = \mathrm{diag}(\Lambda_{\Theta,1},\Lambda_{\Theta,2},\Lambda_{\Theta,3})$ in moment space; the block $\Lambda_{\Theta,2}$ containing the parameters $k_{\Theta,ij}$ is what turns the scheme from isotropic into anisotropic diffusion. The adjoint multiple-relaxation-time (AMRT) model is the same type of system run backward in time with the adjoint equilibrium $\psi^\mathrm{eq}_{\Theta,i} = \sum_k ((\hat\Lambda_\Theta^*)^{-1} O_\Theta \hat\Lambda_\Theta^*)_{ik}\psi_{\Theta,k}$. The Chapman-Enskog expansion of the distribution function in the Knudsen number $\varepsilon$ is what links the microscopic streaming-and-collision steps to the macroscopic system (1), and the same expansion supplies the choices $\vec V_\Theta$, $A_{\Theta,k}$, and $K_\Theta$ that produce the target diffusion and cross-diffusion tensors.

What would settle it

Compute the $DV$ operator from (60) for a manufactured problem with $T_\Theta = t\,(x_1,x_2)$ on the D2Q9 lattice and measure $\tau\,\mathrm{div}\bigl((K_\Theta^{-1}-\tfrac12 I)DV\bigr)$ relative to the retained terms; if it exceeds $O(\varepsilon^3)$, the LBM solutions cannot converge to (1). A second check: compare the discrete adjoint gradient (89) with a finite-difference gradient of $J_h$ in (88) on the same test case; any mismatch that does not shrink with the discretization contradicts the asserted exactness.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the discrete MRT lattice Boltzmann equations (30)-(32) approximate the continuous chemotaxis-type system (1) through the Chapman-Enskog expansion with an error of order $O(\varepsilon^3)$, provided the convection tensors are chosen so that the extra term $\tau\,\mathrm{div}\bigl((K_\Theta^{-1}-\tfrac12 I)DV\bigr)$ is null or negligible. Given that recovery, the paper defines an adjoint multiple-relaxation-time lattice Boltzmann model in which adjoint distributions stream backward in time along $-\mathbf{e}_i$, the adjoint equilibrium is built from the transpose of the collision operator, and the gradient of the cost functional is expressed directly in terms of the adjoint distribution functions via (89). This yields a discrete gradient that the paper asserts is exact for the discretized functional, and hence a complete adjoint-based optimization loop for the primal system.

Load-bearing premise

The recovery proof of Proposition 1 drops the term $\tau\,\mathrm{div}\bigl((K_\Theta^{-1}-\tfrac12 I)DV\bigr)$ as null or negligible although it is generically nonzero for space- and time-dependent convection, and the discrete gradient (89) is derived from the continuous LBE (65) rather than from the discrete scheme (66), which contains an extra $\tau^2/2$ force term.

Editorial extensions

If this is right

Each gradient iteration costs one forward MRT solve and one backward AMRT solve; the gradient formula (89) is assembled locally from distribution functions, preserving the parallel structure of lattice Boltzmann methods.
The framework extends to systems with $N\geq 4$ equations and to mixed or Robin boundary conditions by changing the boundary treatment, so the same code can be adapted across applications.
The discrete adjoint gradient enables gradient and conjugate-gradient descent with line search on the discretized cost functional, including the quadratic case where the step size can be computed explicitly.
For the crime-model, attraction-repulsion, two-species, and tumor-invasion examples, the method gives an implementable route to estimating source terms and parameters from observations.

Reading between the lines

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

A reader could test the recovery claim by evaluating $\tau\,\mathrm{div}\bigl((K_\Theta^{-1}-\tfrac12 I)DV\bigr)$ for a time- and space-dependent convection tensor; the paper does not estimate this term, so the range of validity of Proposition 1 is an open question, not a proven fact.
The $\tau^2/2$ force term in the discrete primal scheme (66) is absent from the continuous LBE (65) from which the adjoint is derived; deriving a fully discrete AMRT directly from (66) would either confirm or correct the exactness of gradient formula (89).
For applications where the convection tensors are independent of space and time, such as leading-order crime-pattern models, the problematic $DV$ term may vanish, so the method's practical range may be cleaner than the general theorem's statement.
Coupling the scheme with proper orthogonal decomposition, which the paper mentions as a possibility, is a natural next step that could reduce the memory and CPU cost of the backward-in-time adjoint solves.

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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

Dear [colleague],

The reader's report is largely right: the central consistency claim in Proposition 1 has a real hole, and the adjoint derivation stops short of a true discrete adjoint. But this is not a throwaway paper. There is solid formal work here.

What is actually new: the paper extends the author's earlier AMRT framework and existing MRT convection-diffusion models to a three-equation anisotropic cross-diffusion chemotaxis system, with controls in the source terms. The continuous adjoint analysis (Theorem 2) is careful and complete, and the construction of the equilibrium distribution with the integrability condition (37) for anisotropic convection is a systematic piece of calculus. If you want a worked example of how to derive gradient formulas for LBM discretizations of coupled PDE systems, the derivation is instructive.

The soft spots are exactly where the reader puts them. In Proposition 1 the proof defines DV at equation (60) and then says: if this term is null... or is negligible, (63) becomes (64). For TΘ depending on x or t, which the examples allow (the flow velocity in (7)-(9)), DV is generically nonzero and sits at the same order as the target diffusion term. The stress-test example with TΘ=(1+βt)Θe1 gives DV=βΘe1, so the recovered PDE picks up a spurious diffusion-like term. No estimate, no scaling relation, no subclass restriction is given. Proposition 1 as stated—O(ε^3) recovery for the general class—is not established, and this is load-bearing because the MRT primal solver's whole purpose is to reproduce (1).

The adjoint issue is real but less severe: the gradient (70) comes from the continuous LBE (65), while the actual primal solve uses the discretization (66) with the τ²/2 force term. So (89) is an approximate discrete gradient, not the exact derivative of the discrete cost (88). The paper never quantifies the mismatch, but uses the phrase 'discrete adjoint' more strongly than the derivation supports.

There are no numerical experiments. 'Reliable, efficient, practical' in the abstract is a promise, not a demonstration.

Net: this deserves a serious referee—the machinery is coherent and the gaps are specific and fixable. As submitted I would not accept it: the Chapman-Enskog result needs a controlled estimate on DV or an explicit restriction to TΘ independent of x,t, the adjoint needs a genuine discrete-adjoint consistency check, and a 2D test case would show whether the method actually works. That is a major-revision agenda, not a rejection of the idea.

Referee Report

3 major / 5 minor

Summary. This paper develops a continuous and discrete adjoint-based optimal control framework for a class of nonlinear coupled anisotropic convection-diffusion chemotaxis-type systems (NCACDCS). The author derives first-order necessary optimality conditions using sensitivity and adjoint calculus, then proposes a multiple-relaxation-time lattice Boltzmann method (MRT) for the primal problem and an adjoint MRT model (AMRT) for the adjoint problem. The central claims are that, through a Chapman-Enskog expansion, the MRT scheme recovers the target macroscopic system to order O(ε^3), and that the AMRT formulation provides a discrete gradient of the cost functional suitable for gradient-descent optimization. The paper also sketches several application examples from biology, criminology, and tumor modeling.

Significance. If the derivations were sound, the paper would supply a useful unified MRT-LBM/AMRT workflow for optimal control of a broad class of chemotaxis-type systems, extending the author's earlier work in [7] and the MRT literature. The paper is self-contained in setting up the continuous optimality system and gives explicit formulas for the equilibrium distributions and adjoint equilibrium distributions; these are useful and appear formally coherent. The application examples are relevant and cover a wide range of intended uses. However, the central recovery statement in Proposition 1 and the discrete adjoint gradient formula in Theorem 3 are not established as written, and the manuscript contains no numerical verification of the proposed algorithms. The intended contribution is therefore not realized in the present form.

major comments (3)

[Section 3.1, Proposition 1 (Eqs. (60), (63), (64))] The proof of Proposition 1 is incomplete at the step from Eq. (63) to Eq. (64). The term -τ div_1((K_Θ^{-1}-1/2 I)DV(x,t;Θ)) is discarded based only on the assertion that it is 'null or negligible', but no hypothesis in the proposition and no estimate in the proof implies this. For T_Θ depending explicitly on x and t, as in Examples 2.2.2-2.2.4 where the flow velocity ω enters the convection terms, DV in Eq. (60) is generically nonzero. For instance, in a D2Q9 setting with T_Θ = a(x,t)Θ e_1 and a = 1 + βt, Eq. (37) gives C_Θ = a^2 Θ e_1 e_1^T and DV = βΘ e_1, so the dropped term is a spurious τ-order term of the same formal size as the target diffusion term. Since no scaling relation between τ and ε is provided, this residual cannot be absorbed into the claimed O(ε^3) error, and recovery of the target system (1) from Eqs. (30)-(32) is not established for the general class stated.
[Section 3.2, Theorem 3 and Eq. (89)] The discrete gradient formula (89) is presented as the gradient of the discretized cost functional Jh in Eq. (88), but the adjoint system in Theorem 3 is derived by varying the continuous LBE (65) and integrating by parts in time and space; it is not obtained by differentiating the fully discrete scheme (66). In particular, the discrete primal (66) contains the force correction (τ^2/2)(∂/∂t + κ_x e_i·∇)F_{Θ,i}, which is absent from the continuous LBE (65) on which the adjoint derivation is based. The backward discrete equation described after Eq. (86) is therefore not the true discrete adjoint of (66), and no consistency or error analysis is supplied for the mismatch. The paper must either derive the exact adjoint of the discrete scheme (66) or prove that the continuous adjoint gradient agrees with the discrete gradient to a controlled order; as written, the 'discrete adjoint' claim is unsupported.
[Sections 3.1-3.2 and Section 4] The manuscript contains no numerical experiments for either the MRT primal solver or the AMRT optimization loop, despite the abstract and conclusion asserting that the method is 'reliable, efficient, practical to implement' and despite the algorithmic description in Section 3.2.1. In a numerical-methods paper, such claims require at least a validation test on a model problem, a convergence study, or a comparison with a standard discretization; without one, the practical value of the proposed workflow is unsubstantiated.

minor comments (5)

[Section 2.3, Eq. (20)] In Eq. (20), the second and third integrals on the right-hand side multiply ∂Φ_v/∂f_2 and ∂Φ_w/∂f_3 by ilde u; they should multiply by ilde v and ilde w, respectively.
[Section 3.1] The text refers to 'Fig. 1' for the collision-streaming process, but no figure appears in the manuscript; the figure should be included or the reference removed.
[Section 2.4, Remark 7] In Remark 7, 'bellows' should read 'below'.
[Section 3.1, Remark 12] The CFL-type condition τ ≤ C h^2 is stated without derivation or numerical evidence; as written it is an assertion rather than a justified stability condition.
[Section 2.3, Theorem 2] The notation is confusing because ilde u^* denotes both the full adjoint vector ( ilde u^*, ilde v^*, ilde w^*) and its first scalar component in the boundary conditions of the displayed adjoint system; a different symbol for the vector would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central Chapman-Enskog and adjoint-gradient claims are consistency derivations with correctness gaps, not reductions of outputs to inputs.

full rationale

The paper does not fit parameters to data and then rename them as predictions; there are no numerical experiments or external benchmarks. Proposition 1 is a formal multiscale consistency check: the LBE (30)-(32) is constructed with equilibrium/source tensors TTheta, CTheta, ATheta,k, VTheta and relaxation KTheta, and then (62)-(63) show these choices reproduce (1) when the extra term -tau div((K^{-1}-1/2)DV) vanishes. The quote 'If this term is null ... or is negligible compared to the other terms, then (63) becomes (64)' is a gap in the proof (DV is not estimated and is generically nonzero for x,t-dependent TTheta), but this is a correctness/completeness limitation, not equivalence of the derivation to its input. The same holds for the discrete adjoint gradient: Theorem 3 derives the costate equation from the continuous LBE (65), while the algorithm evaluates (89) for the discrete scheme (66) that includes an additional tau^2/2 force term; this is a discrete-adjoint consistency issue, not a circular self-reference. Self-citations to [7] and [13] occur ('Based on work presented in [7,81]', 'using similar approach as in [7]', 'see [13]'), but the load-bearing derivations are reproduced in the text (e.g., Theorem 2 proof, Theorem 3 proof), so the citations are not the sole support and do not force the results. No uniqueness theorem or fitted input is invoked. Accordingly, no specific circular step can be exhibited.

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

The central derivation rests on the standard formal machinery of LBM plus several choices made by hand: the relaxation matrix, the diffusion scale parameter, and the force-correction coefficients. It also depends on the unverified negligibility of the DV term and on an implicit adjoint-consistency assumption for the discrete scheme. No new physical entities are introduced.

free parameters (3)

relaxation matrix KΘ (or ΠΘ) = unspecified; chosen so DΘ = dΘ C_s^2 τ(K^{-1}-1/2)
The macroscopic diffusion tensors are imposed by selecting these relaxation rates; different K give different effective diffusion, so the model does not derive the physical diffusion from first principles.
dΘ = unspecified scale factor
A parameter linked to the target diffusion tensor through the equilibrium function (36); it is set to match the desired macroscopic diffusivity rather than derived from the underlying physics.
κ_t and κ_x = κ_t set to 1; κ_x left free
κ_t is chosen by hand to remove a source time derivative in Eq. (62), and κ_x enters the force correction and the convection velocity definition; neither is pinned by data or by an independent principle.

assumptions (4)

standard math Chapman-Enskog expansion and Taylor truncation require smooth distribution functions and small Knudsen number ε (Eqs. 42-46).
The recovery proof is formal and assumes the usual LBM multiscale hierarchy without giving convergence conditions for the specific nonlinear coupled system.
domain assumption The control-to-state map F is well-posed and Fréchet differentiable on the admissible set, with the state living in a suitable Banach space (Section 2.1).
The paper supposes Hadamard well-posedness and Fréchet differentiability as hypotheses rather than proving them for the general NCACDCS class.
ad hoc to paper The deviation term DV in Eq. (60) is null or negligible, allowing Eq. (63) to reduce to Eq. (64).
Proposition 1's recovery of the target macroscopic system depends on discarding this term without an error estimate; for TΘ depending on x,t the term is generically nonzero.
ad hoc to paper The adjoint LBE derived from the continuous LBE (65) is an adequate adjoint for the discrete LBE (66) that includes the τ²/2 force term.
The gradient formula (89) is presented as the discrete gradient, but the adjoint equations are not derived from the fully discretized scheme, leaving an unexamined consistency gap.

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Cite this review

Pith. "Pith review of Optimal Control for Chemotaxis Systems and Adjoint-Based Optimization with Multiple-Relaxation-Time Lattice Boltzmann Models." pith.science (2026). https://pith.science/paper/QLRTHH6A

@misc{pith2026190803876,
  author       = {Pith},
  title        = {Pith review of: Optimal Control for Chemotaxis Systems and Adjoint-Based Optimization with Multiple-Relaxation-Time Lattice Boltzmann Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QLRTHH6A}},
  note         = {Machine review of arXiv:1908.03876}
}

read the original abstract

This paper is devoted to continuous and discrete adjoint-based optimization approaches for optimal control problems governed by an important class of Nonlinear Coupled Anisotropic Convection-Diffusion Chemotaxis-type System (NCACDCS). This study is motivated by the fact that the considered complex systems (with complex geometries) appear in diverse biochemical, biological and biosocial criminology problems. To solve numerically the corresponding nonlinear optimization problems, the primal problem NCACDCS is discretised by a coupled Lattice Boltzmann Method with a general Multiple-Relaxation-Time collision operators (MRT) while for the adjoint problem, an Adjoint Multiple-Relaxation-Time lattice Boltzmann model (AMRT) is proposed and investigated. First, the optimal control problems are formulated and first-order necessary optimality conditions are established by using sensitivity and adjoint calculus. The resulting problems are discretised by the coupled MRT and AMRT models and solved via gradient descent methods. First of all, an efficient and stable modified MRT model for NCACDCS is developed, and through the Chapman-Enskog analysis we show that NCACDCS can be correctly recovered from the proposed MRT model. For the adjoint problem, the discretisation strategy is based on AMRT model, which is found to be as simple as MRT model with also highly-efficient parallel nature. The derivation of AMRT model and the discrete cost functional gradient are derived mathematically in detail using the developed MRT model. The obtained method is reliable, efficient, practical to implement and can be easily incorporated into any existing MRT code.

Figures

Figures reproduced from arXiv: 1908.03876 by the authors.

**Figure 1.** Collision-Streaming process the expression of microscopic velocities ei , we prove easily that X 8 i=0 ei = X 8 i=0 ωiei = 0 0 , X 8 i=0 ωiei ⊗ ei = C 2 s Id and X 8 i=0 ωifi = 0. The transportation matrix M can be written as M = E0M0 with M0 given by (see e.g. [48]) M0 =   1 1 1 1 1 1 1 1 1 −4 −1 −1 −1 −1 2 2 2 2 4 −2 −2 −2 −2 1 1 1 1 0 1 0 −1 0 1 −1 −1 1 0 −2 0 2 0 1 −1 −1 1 0 0 1 0 −1 1 1 −1 −1 0… view at source ↗

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