--- title: Diagonal-Parallel Picard in Fixed-Point Methods url: https://www.emergentmind.com/topics/diagonal-parallel-picard type: topic --- # Diagonal-Parallel Picard in Fixed-Point Methods Searching arXiv for papers directly relevant to “Diagonal-Parallel Picard” and adjacent fixed-point/parallel-Picard formulations. Diagonal-Parallel Picard is not a standardized method name in the recent arXiv literature. The exact phrase does not appear in the principal papers most closely associated with parallel Picard-style methods. Instead, the phrase is best read as a compact label for several closely related constructions: Picard-style fixed-point iterations made parallel by a diagonal approximation, a diagonal preconditioner, or a coordinatewise decoupling that leaves only a scan-, map-, or component-parallel subproblem. In the literature on parallelizing sequential models, the closest explicit meaning is a diagonal-Jacobian quasi-Newton update inside a unifying linear dynamical systems framework; in spectral deferred corrections and collocation, it corresponds more literally to diagonal-preconditioned Picard iteration across collocation nodes; and in multidimensional reflected \(G\)-BSDEs it denotes Picard iteration under diagonal generators, where each frozen-coordinate update is a one-dimensional reflected equation [2509.21716] [1703.08079] [2108.09012]. ## 1. Terminological status and scope Several recent papers explicitly state that the exact phrase “Diagonal-Parallel Picard” does not appear in their text. In the parallel-sequential-model literature, the nearest named objects are Picard iteration with identity transition, quasi-Newton iteration with diagonal Jacobian approximation, and more generally parallel fixed-point methods obtained by choosing structured approximations of the Jacobian that remain closed under composition. In diffusion and Metropolis sampling, the nearest objects are global Picard updates over an entire trajectory, sometimes with a Jacobi-like time-parallel interpretation or with a moving online frontier rather than a diagonal matrix structure [2509.21716] [2503.19731] [2506.09762]. This suggests that “Diagonal-Parallel Picard” functions less as a canonical algorithm name than as an umbrella description spanning at least three meanings of “diagonal.” In one meaning, “diagonal” refers to a diagonal Jacobian approximation in state space. In another, it refers to a diagonal preconditioner in collocation-node space. In a third, it refers only informally to a diagonal or frontier-like propagation pattern through the time-index/iteration grid. The common invariant across these uses is a Picard-style fixed-point backbone combined with a structural simplification that exposes parallelism. ## 2. Parallel fixed-point iteration for sequential models A central recent formulation studies the sequential nonlinear recursion \[ x_{t+1}=f_{t+1}(x_t), \] recasts rollout as a fixed-point problem over the full trajectory, and shows that Newton, quasi-Newton, Picard, and Jacobi all fit the same linear-dynamical-systems update \[ x_{t+1}^{(i+1)}=f_{t+1}(x_t^{(i)})+A_{t+1}\bigl(x_t^{(i+1)}-x_t^{(i)}\bigr). \] The method is determined by the choice of \(A_{t+1}\): full Newton uses the full Jacobian, quasi-Newton uses the diagonal Jacobian, Picard uses \(I_D\), and Jacobi uses \(0\). The paper states explicitly that “Picard iterations are in fact a type of quasi-Newton iteration, where we approximate the Jacobian of the dynamics function by the identity matrix,” while the diagonal approximation is \[ A_{t+1}=\operatorname{diag}\!\left[\frac{\partial f_{t+1}}{\partial x_t}(x_t^{(i)})\right]. \] Within this framework, the strict paper-grounded interpretation of “Diagonal-Parallel Picard” is therefore not Picard in the paper’s own terminology, but the diagonal quasi-Newton member of the same family [2509.21716]. The same paper makes the computational reason precise. Once \(A_{1:T}\) and the previous iterate are fixed, the update becomes an affine linear time-varying recurrence over \(t\), and affine-map composition is associative. The full trajectory can therefore be evaluated by a parallel scan in \(\mathcal{O}(\log T)\) depth on \(\mathcal{O}(T)\) processors. The diagonal structure matters because diagonal matrices are closed under multiplication, giving a cheap scan representation. The paper records the cost gap explicitly: full Newton requires \(\mathcal{O}(T D^2)\) memory and \(\mathcal{O}(T D^3)\) work, whereas diagonal quasi-Newton requires \(\mathcal{O}(T D)\) space and \(\mathcal{O}(T D)\) work [2509.21716]. A later thesis places the same family inside a broader parallel Newton framework. There the generic update is \[ s_t^{(i+1)}=f_t(s_{t-1}^{(i)})+\tilde A_t\bigl(s_{t-1}^{(i+1)}-s_{t-1}^{(i)}\bigr), \] with \(\tilde A_t=A_t\) for Newton, \(\tilde A_t=\operatorname{diag}(A_t)\) for diagonal quasi-DEER, \(\tilde A_t=I_D\) for Picard, and \(\tilde A_t=0\) for Jacobi. The thesis is explicit that the phrase “Diagonal-Parallel Picard” is not defined there; the closest explicit method is diagonal quasi-DEER, while Picard itself remains the identity approximation [2603.16850]. ## 3. Diagonal-preconditioned Picard in collocation and spectral deferred correction In collocation-based time integration, Picard iteration arises from the collocation system \[ \left(\mathbf{I}-\Delta t\,Q\mathbf{F}\right)(\mathbf{u})=\mathbf{u}_0, \] and spectral deferred correction is written as a preconditioned Picard iteration \[ \left(\mathbf{I}-\Delta t\,Q_\Delta \mathbf{F}\right)(\mathbf{u}^{k+1}) = \mathbf{u}_0+\Delta t\,(Q-Q_\Delta)\mathbf{F}(\mathbf{u}^k). \] Classical SDC uses a lower-triangular \(Q_\Delta\), which makes the sweep Gauss-Seidel-like and serial across nodes. The diagonal-parallel variant replaces that by a diagonal \(Q_\Delta\), so that all node updates decouple and can be computed simultaneously. One paper tests three such diagonal preconditioners: \[ Q_\Delta^{\mathrm{Qpar}}=\operatorname{diag}(q_{11},\dots,q_{MM}), \] \[ Q_\Delta^{\mathrm{IEpar}}=\operatorname{diag}(\tau_1-t_0,\dots,\tau_M-t_0), \] and \[ Q_\Delta^{\mathrm{MIN}}=\operatorname{diag}(\hat{\mathbf q}),\qquad \hat{\mathbf q}=\operatorname*{argmin}_{\mathbf q\in\mathbb{R}^M}\rho\!\left(\mathbf I-\operatorname{diag}(\mathbf q)Q\right). \] This is the most literal sense in which the literature contains a diagonal-parallel Picard construction: a Picard or SDC sweep whose preconditioner is diagonal in node space [1703.08079]. A later paper makes the relationship sharper by stating that plain Picard iteration is recovered by choosing \(Q_\Delta=0\), while diagonal SDC is a diagonal-preconditioned Picard iteration. It proposes analytically designed diagonal sweepers, including \[ Q_{\Delta,M}=\operatorname{diag}\!\left(\frac{\tau_1}{M},\dots,\frac{\tau_M}{M}\right), \] for which \[ (Q-Q_{\Delta,M})^M=0, \] and a nonstationary family \[ Q_\Delta^{(k)}=\operatorname{diag}\!\left(\frac{\tau_1}{k},\dots,\frac{\tau_M}{k}\right), \] for which the stiff-limit propagator product vanishes after \(M\) sweeps. In that paper’s formulation, diagonal-parallel SDC is “Picard iteration for collocation” plus diagonal implicit preconditioning, rather than Picard in the strict unpreconditioned sense [2403.18641]. ## 4. Time-parallel trajectory Picard in diffusion and Metropolis sampling A separate line of work applies Picard iteration not to state-space Jacobians or collocation matrices, but to an entire denoising or Markov trajectory. For diffusion sampling, the trajectory-form Picard update is \[ X^{k+1}\leftarrow \Phi(X^k),\qquad x_t^{k+1}\leftarrow x_0^k+\frac{1}{T}\sum_{i=0}^{t-1}s_\theta(x_i^k,i/T). \] This is parallel across denoising time because all timestep states are updated from the previous global iterate. The paper explicitly says that it does not introduce a new diagonal block decomposition; if mapped to “diagonal-parallel” language, the closest analogy is a Jacobi-like global update over the time dimension. Its main contribution is the Picard Consistency Model, trained to predict the fixed-point trajectory more directly, together with model switching to recover exact convergence to the base sampler. The reported headline gains are “up to a 2.71x speedup over sequential sampling” and “a 1.77x speedup over Picard iteration” [2503.19731]. For zeroth-order Metropolis chains, the relevant object is the Online Picard algorithm. The chain \[ X_{i+1}=X_i+f(X_i,W_i) \] is rewritten via the Picard map \[ X_i'=\Phi_i(X,W)= \begin{cases} X_0,& i=0,\\ X_0+\sum_{\ell=0}^{i-1}f(X_\ell,W_\ell),&0