Homotopy Grid Methods Overview

• Homotopy Grid Method is a family of techniques that combine grid-based discretizations with homotopical embeddings to transform complex problems into tractable forms.
• The approach enhances computational efficiency in applications such as nonlinear differential equations and AC-OPF optimization via spectral and incremental homotopy strategies.
• It enables precise topological analysis in algebraic topology and Floer theory by systematically extracting invariants from grid structures using combinatorial and spectral methods.

The term "Homotopy Grid Method" encompasses a family of methodologies across applied mathematics, computational physics, algebraic topology, and optimization, unified by the use of homotopy-type embeddings, grid-based discretizations, or grid diagrams coupled with homotopical algebraic constructions. Distinct formulations have been developed and applied in numerical PDEs (particularly nonlinear boundary value problems), large-scale power grid optimization, and the homotopy-type analysis of combinatorial or algebraic invariants of grids and grid-like objects. Key developments appear in works on fast nonlinear solvers via spectral homotopy (Cullen et al., 2018), grid-homotopy categories in Floer theory (Manolescu et al., 2021), physics-based power grid optimization (Pandey et al., 2020, McNamara et al., 2021), and simple homotopy types for grid graphs (Okura, 2019).

1. Numerical Homotopy Grid Methods for Nonlinear Differential Equations

The "Gegenbauer Homotopy Analysis Method" (GHAM), introduced by Booker and Viswanath (Cullen et al., 2018), represents the synthesis of the Homotopy Analysis Method (HAM) with sparse, spectrally accurate grid-based spectral discretizations. The core idea is to transform a nonlinear, variable-coefficient boundary value problem,

$N[u](x) = 0,\quad u(a)=A,\,u(b)=B,$

into a sequence of linearized deformation problems via an embedding parameter $p$, an auxiliary linear operator $L$, and a convergence control parameter $\hbar$. The solution $u(x)$ is expanded as

$u(x;p)=u_0(x) + \sum_{m=1}^\infty U_m(x)\,p^m,$

with each $U_m(x)$ computed by solving linear equations derived from the mth deformation:

$L[U_m(x)-\chi_m U_{m-1}(x)] = \hbar R_m(x),$

where $R_m$ encodes nonlinear inhomogeneities at each order.

Crucially, rather than working in collocation space (which leads to dense, poorly conditioned matrices), the GHAM discretizes $L$ via Gegenbauer (ultraspherical) spectral methods, yielding sparse, almost-banded system matrices. This allows a single sparse matrix $K$ to be assembled and factorized once, then efficiently reused at every homotopy step, so that the total computational effort is

$$T(n, M) = O(n^{1.3}) + O(M n) \approx O(n),\;\; (M\ll n),$$

where $n$ is the discretization order and $M$ the number of homotopy steps (Cullen et al., 2018). Empirical results show wall-clock scaling close to linear in $n$, spectral spatial error (exponential decay with grid size), and orders-of-magnitude improvement over Newton iteration and the classical Spectral HAM.

2. Homotopy-Based Algorithms in Power Grid Optimization

The Homotopy Grid Method in power systems refers to homotopy-continuation approaches for solving large-scale, highly nonconvex AC Optimal Power Flow (AC-OPF) problems. Notable instances include the Incremental Model-Building (IMB) Homotopy (Pandey et al., 2020) and the Two-Stage Homotopy incorporating discrete controls (McNamara et al., 2021).

2.1. Incremental Model-Building for AC-OPF

The IMB Homotopy Grid Method embeds the original, "hard" AC-OPF first-order KKT system $F(\mathbf{x})=0$ into a family of progressively-more-physical subproblems,

$$H(\mathbf{x},\lambda) = (1-\lambda) G(\mathbf{x}) + \lambda F(\mathbf{x}) = 0, \qquad \lambda\in [0,1],$$

where $G(\mathbf{x})$ is a trivial, easily-solvable case (e.g., shorted lines, zero loads, wide generator bounds). All intermediate homotopy systems are constructed to be feasible via homotopy-scaled slack current injections at each bus, so the solver traces a continuous solution path from the trivial to the full problem. Physical constraints (series admittances, tap ratios, loads, generator limits, shunt admittances) are smoothly 'turned on' as $\lambda$ increases. The resulting nonlinear systems are solved at each step by a primal–dual interior point (PDIP) Newton solver with feasibility preserving limiting (Pandey et al., 2020). Numerical experiments confirm robust convergence on AC-OPF instances up to 80k buses, far exceeding the reliability of standard solvers.

2.2. Two-Stage Homotopy for Discrete Device AC-OPF

For mixed-integer nonlinear AC-OPF problems including discrete control variables (e.g., switched shunts, transformer taps), the Two-Stage Homotopy Grid Method (McNamara et al., 2021) first computes a relaxed solution where discrete variables are continuous, then 'snaps' to the nearest feasible discrete values via a second homotopy stage. Formally, after continuous relaxation,

• Stage 1: Solve a homotopy-embedded relaxation from trivial settings to the full AC-OPF with all controls continuous.
• Stage 2: Round discrete controls to nearest admissible settings, compute the resulting KKT residual $R$, and solve a second homotopy

$$H_2(\theta,\nu_2) = F_\text{AC-OPF}(\theta) - \nu_2 R = 0$$

as $\nu_2:1\to 0$, gradually correcting feasibility.

Performance comparisons on large ARPA-E GO Challenge grids indicate superior reliability and competitive or slightly improved objective values compared to state-of-the-art commercial solvers. The method is particularly robust on cases with coarse discrete device steps or no warm-start information (McNamara et al., 2021).

3. Homotopy Grid Methods in Algebraic and Combinatorial Topology

Distinct from the above analytic and numerical frameworks, the "Homotopy Grid Method" also designates tools in algebraic and combinatorial topology, exploiting the homotopy-theoretic properties of grids as combinatorial models.

3.1. Independence Complexes of Grid Graphs

Okura (Okura, 2019) introduces the Homotopy Grid Method as a systematic procedure for deducing the simple homotopy type of the independence complex $\operatorname{Ind}(G)$ of any graph $G$ that contains a grid subgraph as a full subgraph. The key theorem states that specific grid 'replacement' operations—combinations of edge additions, edge deletions, and vertex deletions localized to a small rectangular grid—realize a single simplicial suspension:

$$\operatorname{Ind}(H) \simeq_s \Sigma \operatorname{Ind}(G),$$

after performing the transformation. Recursive application yields closed-form descriptions for the homotopy types of independence complexes of families such as rectangular grids, cylinders, Möbius cylinders, and hexagonal-cut grids, with periodic wedge-of-spheres decompositions. This is algorithmically encoded as a sequence of localized grid replacements, each inducing a suspension on $\operatorname{Ind}(G)$, and can be iterated to reduce to known base cases in closed algebraic forms (Okura, 2019).

4. Stable Homotopy and Grid Diagrams in Knot Theory

The Homotopy Grid Method in the context of knot and link Floer theory refers to the refinement of grid-based combinatorial knot invariants to stable homotopy types (Manolescu et al., 2021). For a grid diagram $G$ representing a link $L\subset S^3$, one constructs a combinatorial grid complex $GC^+(G)$ and then, by inductive models for the moduli spaces of domains (including all bubbling strata), assembles these into a framed flow category. The key technical tool is a chain complex $CDP_*$ of positive domains with partitions, used to resolve the moduli space boundaries and correct for obstructions in extending the flow category to all dimensions. Explicitly, the construction produces a spectrum $X^+(G)$ whose integral homology recovers knot Floer homology:

$H_*(X^+; \mathbb{Z}) \cong HFL^+(L),$

with actions of $U_i$ induced by collapses within the grid complex. The refinement is homotopy-invariant under grid moves (cyclic permutation, commutation, stabilization), and it is conjectured that the resulting stable homotopy type depends only on the isotopy class of the knot, providing a spectrum-level lift of the combinatorially defined Heegaard Floer invariants (Manolescu et al., 2021).

5. Comparative Table: Core Homotopy Grid Methods

Context	Principal Reference	Grid Structure	Homotopy Principle	Key Application
Nonlinear BVPs	(Cullen et al., 2018)	Spectral grid	HAM with spectral, sparse grid	Fast, accurate nonlinear solvers
AC-OPF Optimization	(Pandey et al., 2020, McNamara et al., 2021)	Power grid (network)	Physics-based or device homotopy	Robust, scalable optimization
Independence Complexes	(Okura, 2019)	Graph grid	Grid suspensions, simplicial moves	Topological classification
Floer Theory	(Manolescu et al., 2021)	Grid diagram	Inductive moduli, flow categories	Knot/link stable homotopy types

6. Theoretical and Practical Impact

The unifying feature of Homotopy Grid Methods is the explicit embedding of the original, often nontrivial, mathematical object—be it a nonlinear operator, an optimization system, or a combinatorial complex—within a homotopy connecting it to a tractable starting case, leveraging grid-structured discretizations or combinatorics at each step. The methodology guarantees path-feasibility, robust convergence in high-dimensional settings, or exact topological invariants, depending on the context. In numerical analysis and optimization, the reuse of sparse grids or incremental model-building yields quasi-linear computational scaling and high empirical reliability on problems of previously intractable size.

In algebraic topology and Floer theory, the grid approach provides a combinatorial framework for constructing and analyzing invariants at a finer (spectrum) level, while in graph theory it enables systematic decomposition and computation of complex invariants via suspensions.

The range of domains and technical implementations underscores the versatility of homotopy-grid coupling. Further advances may generalize these grid-homotopy paradigms to higher-dimensional, non-rectangular lattices, or more general operator classes. The explicit analytic and topological control offered by these constructions suggests broad applicability in numerical computation, discrete mathematics, and beyond.