Damped Iterative Water Filling (DIWF)

• DIWF is a damped, iterative power allocation method that blends previous allocations with current water-filling updates to ensure smooth and monotonic convergence.
• The method applies a fixed one-to-one subcarrier assignment in OFDMA-based SAGIN, enforcing minimum user rate constraints and robust QoS under random channel variations.
• Simulation results show DIWF’s rapid stabilization of sum rate and effective constrained power redistribution, offering competitive performance compared to AO and GA methods.

Damped Iterative Water Filling (DIWF) denotes a damped, iterative variant of water-filling-based power allocation in which each update blends the previous allocation with a current water-filling target rather than replacing the iterate outright. In the 2025 Space-Air-Ground Integrated Networks (SAGIN) study that uses the term explicitly, DIWF is a power-allocation method for an OFDMA air-base-station (ABS)-assisted deployment after a fixed one-to-one user–subcarrier mapping has been imposed, with the stated purpose of achieving robust power allocation, guaranteed user rate constraints, smooth convergence, and prevention of oscillations under random channel variations and QoS constraints (Geddam et al., 16 Sep 2025). In the broader water-filling literature, closely related formulations appear under names such as “smoothed asynchronous iterative waterfilling algorithm” and “averaged iterative water-filling,” which provide the principal fixed-point and robustness interpretations of damping (0801.2480).

1. Definition and conceptual role

In the SAGIN resource-allocation framework, DIWF is introduced as “Strategy 2: Subcarrier Assignment followed by DIWF for Power Allocation.” Its scope is therefore narrower than joint assignment-and-power methods: the subcarrier allocation is fixed first, and DIWF acts only on the continuous power-control stage (Geddam et al., 16 Sep 2025). The paper explicitly states that “of the $N$ possible subcarriers, a unique subcarrier is assigned to each of the $K$ users,” so DIWF operates under a one-to-one user–subcarrier mapping rather than over the full mixed-integer search space.

The defining modification relative to conventional water filling is the introduction of a damping factor that combines the previous allocation with the current water-filling update. In that treatment, conventional water filling is described as allocating more power to stronger channels to maximize spectral efficiency, whereas DIWF adds damping to prevent oscillations, enforce monotonic improvement, ensure smooth convergence, and improve stability when random channel variations and QoS constraints are present (Geddam et al., 16 Sep 2025). This makes DIWF a constrained and stabilized water-filling procedure rather than a one-shot allocation rule.

A broader theoretical interpretation appears in the iterative water-filling literature on interference channels. The 2008 asynchronous water-filling paper does not use the name DIWF, but it explicitly defines the relaxed map

$$\mathbf T_q(\mathbf p)\triangleq \alpha_q \mathbf p_q + (1-\alpha_q)\mathsf{WF}_q(\mathbf p_{-q}),$$

and the corresponding “Smoothed Asynchronous Iterative Waterfilling Algorithm,” identifying $\alpha_q\in[0,1)$ as a forgetting factor (0801.2480). This establishes the standard fixed-point meaning of damping: a convex combination of the current iterate and a best-response water-filling solution. A directly related 2025 survey-style analysis likewise writes a relaxed iteration

$$\mathbf p^{(t+1)}=(1-\alpha)\mathbf p^{(t)}+\alpha\,\Phi(\mathbf p^{(t)}),\qquad 0<\alpha\le 1,$$

and treats it as a relaxed or averaged IWF variant, although not as a complete DIWF theory (Wang, 17 Feb 2025).

2. Optimization setting and mathematical formulation

The explicit DIWF formulation in the SAGIN study is embedded in an OFDMA ABS-assisted scenario in which an ABS substitutes for a dysfunctional terrestrial base station. The original optimization problem maximizes total sum rate over both subcarrier assignment and power allocation: $$\max_{a_{k,n},P_{k,n}} \sum_{k=1}^{K}\sum_{n=1}^{N} a_{k,n}r_{k,n},$$ with decision variables

$a_{k,n}\in\{0,1\}, \qquad P_{k,n}\ge 0,$

channel gains $h_{k,n}$, SINR

$$Yk,n \frac{Pk,nhk,n}{\Sigma j\#kaj,nPj,nhj,n + o2},$$

and achievable rate

$Tk,n = B . \log_2(1+Yk,n).$

The constraints include subcarrier exclusivity,

$K$0

ABS power budget,

$K$1

and minimum-rate QoS,

$K$2

The paper characterizes the resulting problem as a MINLP because of the binary assignment variables and nonlinear SINR (Geddam et al., 16 Sep 2025).

DIWF is not applied to that full MINLP. Instead, the assignment variables are treated as fixed under the one-to-one mapping assumption, and the remaining subproblem is a constrained power-allocation problem over already assigned channels. The procedure first satisfies user minimum-rate constraints, then allocates residual power by water filling, and finally damps each update for stability (Geddam et al., 16 Sep 2025). The paper also states that minimum-rate requirements are converted through Shannon’s formula into minimum SNR and then into minimum required power per user; if the total required minimum power exceeds the ABS budget, the instance is declared infeasible.

This formulation is distinct from the multiuser Gaussian interference-channel setting of classical iterative water-filling. In that setting, each user maximizes

$K$3

subject to total-power and spectral-mask constraints, and the water-filling operator is a best response against interference from other users (0801.2480). The SAGIN DIWF instance is therefore a fixed-assignment constrained power-control problem, whereas the classical theory is usually interference-coupled and game-theoretic. This suggests that the SAGIN use of DIWF is structurally narrower but operationally aligned with the broader relaxed water-filling family.

3. Algorithmic structure and damped update

The SAGIN paper states the DIWF procedure qualitatively rather than as formal pseudocode. Initialization begins from a feasible power allocation after computing the minimum power required for each user to satisfy $K$4 and checking feasibility against the ABS power budget (Geddam et al., 16 Sep 2025). The remaining power is then distributed using water filling, with stronger channels favored in the water-filling stage.

The defining update rule is described verbally as a blend of the previous allocation and the current water-filling solution. The paper gives the damping-factor parameter explicitly as $K$5. A faithful mathematical reconstruction of the stated logic is

$K$6

The paper does not print this exact formula in LaTeX, but it does state that DIWF “introduces a damping factor that combines the previous allocation with the current water-filling update” and “blends the previous allocation with the optimal water-filling solution” (Geddam et al., 16 Sep 2025).

At each iteration, the total achievable sum rate and the rate per user are recalculated. The stopping condition is stated explicitly in prose: the process continues until the change in sum rate between iterations falls below a predefined tolerance, and the reported tolerance is $K$7. The natural reconstructed form is

$K$8

The convergence curve reported for DIWF is Monte Carlo averaged, and the paper states that this averaging ensures reliable performance under random channel variations (Geddam et al., 16 Sep 2025).

Within the broader literature, two related damped constructions are especially important. First, the smoothed asynchronous update

$K$9

provides a distributed asynchronous analogue with stale interference measurements allowed (0801.2480). Second, the averaged iterative water-filling algorithm under noisy interference-plus-noise feedback uses a diminishing-step update

$$\mathbf T_q(\mathbf p)\triangleq \alpha_q \mathbf p_q + (1-\alpha_q)\mathsf{WF}_q(\mathbf p_{-q}),$$0

with $$\mathbf T_q(\mathbf p)\triangleq \alpha_q \mathbf p_q + (1-\alpha_q)\mathsf{WF}_q(\mathbf p_{-q}),$$1 and $$\mathbf T_q(\mathbf p)\triangleq \alpha_q \mathbf p_q + (1-\alpha_q)\mathsf{WF}_q(\mathbf p_{-q}),$$2, and proves almost sure convergence under time-varying IPN estimation error (Hong et al., 2011). These formulations show that “damping” in DIWF can refer either to fixed relaxation or to diminishing-weight averaging, depending on context.

4. Convergence theory and robustness claims

The SAGIN paper’s convergence claims are empirical and qualitative: DIWF steadily improves the system sum rate, rises rapidly in the first few iterations, then stabilizes, and its monotonic increase is said to confirm that each update moves the allocation closer to the optimal solution without overshooting (Geddam et al., 16 Sep 2025). The paper does not provide a formal computational-complexity expression or a contraction-based theorem.

Formal theory comes from adjacent iterative water-filling work. In the asynchronous Gaussian interference-channel model, the main sufficient condition for uniqueness and global convergence is

$$\mathbf T_q(\mathbf p)\triangleq \alpha_q \mathbf p_q + (1-\alpha_q)\mathsf{WF}_q(\mathbf p_{-q}),$$3

where $$\mathbf T_q(\mathbf p)\triangleq \alpha_q \mathbf p_q + (1-\alpha_q)\mathsf{WF}_q(\mathbf p_{-q}),$$4 is the interference-coupling matrix constructed from worst-case normalized cross-link gains over relevant carriers (0801.2480). Under that condition, the asynchronous IWFA converges to the unique Nash equilibrium for any feasible initialization and any totally asynchronous update schedule satisfying the standard causality and infinitely-often update assumptions. For the smoothed relaxed mapping

$$\mathbf T_q(\mathbf p)\triangleq \alpha_q \mathbf p_q + (1-\alpha_q)\mathsf{WF}_q(\mathbf p_{-q}),$$5

the paper proves convergence under the same condition for any $$\mathbf T_q(\mathbf p)\triangleq \alpha_q \mathbf p_q + (1-\alpha_q)\mathsf{WF}_q(\mathbf p_{-q}),$$6 (0801.2480).

A subtle but important limitation is stated explicitly there: the damping or smoothing parameters do not enlarge the proven convergence region. The contraction analysis is built around

$$\mathbf T_q(\mathbf p)\triangleq \alpha_q \mathbf p_q + (1-\alpha_q)\mathsf{WF}_q(\mathbf p_{-q}),$$7

and the result shows that convergence of the smoothed algorithm is guaranteed whenever the undamped map is already contractive (0801.2480). This directly counters a common overstatement that damping alone is a proven cure for noncontractive interference coupling.

The 2025 comprehensive analysis of sequential, simultaneous, and asynchronous IWF makes a similar distinction. It explicitly writes the relaxed update

$$\mathbf T_q(\mathbf p)\triangleq \alpha_q \mathbf p_q + (1-\alpha_q)\mathsf{WF}_q(\mathbf p_{-q}),$$8

and discusses damping as a way to mitigate oscillations and improve robustness when $$\mathbf T_q(\mathbf p)\triangleq \alpha_q \mathbf p_q + (1-\alpha_q)\mathsf{WF}_q(\mathbf p_{-q}),$$9 is close to $\alpha_q\in[0,1)$0, but it does not provide a full damping-dependent convergence theorem (Wang, 17 Feb 2025). By contrast, the 2011 averaged-IWF paper does provide rigorous robustness results for diminishing-step damping under time-varying IPN estimation error, showing convergence to the unique fixed point under the standard contraction condition together with summability or stochastic-approximation assumptions on the noise (Hong et al., 2011). Taken together, these results support DIWF as a stabilization device, but they do not justify the stronger claim that fixed damping universally rescues convergence in strongly coupled settings.

5. Simulation behavior and comparison with related methods

In the ABS-assisted SAGIN simulations, all algorithms were implemented in MATLAB and evaluated over 10,000 Monte Carlo runs with stochastic channel gains. The common reported parameters include subcarrier bandwidth $\alpha_q\in[0,1)$1 and ABS maximum power $\alpha_q\in[0,1)$2. For DIWF specifically, the listed parameters are minimum rate $\alpha_q\in[0,1)$3, damping factor $\alpha_q\in[0,1)$4, and tolerance $\alpha_q\in[0,1)$5 (Geddam et al., 16 Sep 2025).

The main quantitative DIWF result is the convergence figure of Monte Carlo average sum rate versus iteration. From the figure labels, the average sum rate lies approximately in the range of about $\alpha_q\in[0,1)$6 to $\alpha_q\in[0,1)$7, with rapid early increase followed by flattening and stable convergence (Geddam et al., 16 Sep 2025). The paper attributes its robustness and fairness behavior to reserving minimum power for weaker users before allocating additional power to stronger channels.

The paper compares DIWF with Alternating Optimization (AO) and Genetic Algorithm (GA). AO jointly optimizes subcarrier assignment and power allocation, alternates between nonlinear power optimization and greedy reassignment, converges in about 2 to 11 iterations, and is summarized as achieving rapid convergence and balanced throughput. GA jointly optimizes assignment and power allocation by population-based search, has noisier convergence due to crossover and mutation, and reports a best fitness of $\alpha_q\in[0,1)$8 and a Monte Carlo average of $\alpha_q\in[0,1)$9. DIWF, by contrast, assumes fixed one-to-one assignment, optimizes only power, and is characterized by smooth and stable convergence, monotonic increase in sum rate, and explicit QoS-feasibility handling (Geddam et al., 16 Sep 2025).

The resulting comparison is not a general ranking of water-filling methods. It is specific to this SAGIN setup. GA appears to achieve the highest reported throughput among the explicitly visible values, while DIWF achieves slightly lower throughput but is tied more directly to feasibility, minimum-rate guarantees, and non-oscillatory behavior (Geddam et al., 16 Sep 2025). A plausible implication is that DIWF is most attractive when a mapping is already available and the operational priority is robust constrained power redistribution rather than combinatorial search over assignments.

6. Scope, misconceptions, and research directions

A recurrent misconception is to treat DIWF as synonymous with full resource allocation. In the SAGIN study, that is incorrect: DIWF does not solve the full joint MINLP but only the reduced power-allocation problem after fixed assignment (Geddam et al., 16 Sep 2025). A second misconception is to assume that every paper with “iterative water-filling” is about DIWF. Several adjacent works are only partially related: the reconfigurable distributed MIMO-interference algorithm combines MMSE-based subspace updates with a water-filling step but does not use damped or averaged best-response updates (Alexandropoulos et al., 2012); the “Edge Waterfilling” work derives a waterfilling-type policy for a risk-aware interference-free AWGN setting, but its actual algorithm is an outer projected dual subgradient method with no damping mechanism (Yaylali et al., 14 Jul 2025).

Another important clarification concerns terminology. Some of the mathematically closest antecedents to DIWF do not use the name. “Smoothed asynchronous iterative waterfilling algorithm” in the frequency-selective Gaussian interference-channel literature is effectively a relaxed water-filling iteration, and “averaged iterative water-filling algorithm” is effectively a diminishing-step damped water-filling method under noisy feedback (0801.2480). This suggests that DIWF is best understood as a family resemblance centered on relaxed best-response water-filling, rather than as a single universally standardized algorithmic specification.

The SAGIN paper also explicitly notes a modeling nuance: classical water filling is fundamentally a closed-form one-step solution when the problem is convex and involves a single user with known channel gains, but realistic multi-user and networked settings become more complex because of interference coupling, fairness constraints, and joint optimization across heterogeneous resources (Geddam et al., 16 Sep 2025). That observation aligns with the broader literature in which damping, asynchronous updates, active-set logic, or dual iterations appear as adaptations of the water-filling principle rather than as the principle itself.

Future directions named in the SAGIN study include satellite handover, multi-domain orchestration, multi-satellite constellations, cross-domain orchestration, UAV trajectory optimization, real-time traffic dynamics, mobility-aware resource scheduling, and security/privacy considerations (Geddam et al., 16 Sep 2025). Relative to DIWF specifically, this suggests several natural extensions: joint subcarrier assignment plus damped power allocation, interference-coupled multiuser DIWF beyond one-to-one mapping, trajectory-aware DIWF for UAV-supported links, and mobility-aware real-time variants. These are plausible implications of the paper’s stated scope rather than completed results.