Weak Convergence in Infinite-Dimensional Analysis

• Weak convergence method is defined by convergence of inner products in Hilbert/Banach spaces, providing a framework for analysis when strong convergence fails.
• It employs tools like Mosco convergence, Lyapunov-type arguments, and demiclosedness principles to guarantee convergence in operator-splitting and variational algorithms.
• The approach is pivotal in applications ranging from infinite-dimensional optimization and PDEs to stochastic numerical schemes and particle system analysis.

The weak convergence method is a foundational analytical approach in the study of infinite-dimensional optimization, partial differential equations, stochastic processes, and variational problems. It refers both to convergence in the topology induced by bounded linear functionals (rather than norm topology) and to a family of methodological tools for proving convergence of algorithms, discretizations, or particle systems toward their continuous or infinite-dimensional limits. The method’s central elements are the analysis of weakly convergent sequences, Mosco-type convergence of functionals/bilinear forms, Lyapunov-type arguments, and the exploitation of Hilbert or Banach space structure to guarantee convergence even in the absence of strong (norm) compactness.

1. Core Principles of Weak Convergence

Weak convergence in a real Hilbert space $H$ is defined by: a sequence $x_k\in H$ converges weakly to $\bar x$ if for every $z\in H$, $\langle x_k,z\rangle\to\langle\bar x,z\rangle$. In optimization and variational analysis, weak convergence often arises where iterates or discretized solutions cannot be proved to converge strongly due to lack of compactness or the noncompactness of infinite-dimensional balls. Instead, arguments focus on monotonicity, convexity, Lyapunov-type functionals, and demiclosedness principles.

Weak convergence is central to the analysis of convex minimization (and saddle-point) problems, operator splitting methods, stochastic processes, and function space-valued equations, providing both convergence guarantees and a platform for further regularity analysis or error estimates (Banert et al., 2023, Svaiter, 2010, Lo et al., 2016).

2. Weak Convergence in Operator-Splitting and Variational Algorithms

Weak convergence is a classical tool in the analysis of optimization algorithms in Hilbert spaces, particularly for monotone inclusions of the form $0\in A(x)+B(x)$. Prototypical instances include the Douglas–Rachford method and the Chambolle–Pock primal–dual method.

Douglas–Rachford Method

Given maximal monotone operators $A,B:H\rightrightarrows H$ and an initial $(x_0,b_0)\in H\times H$, the classical Douglas–Rachford iterates are defined by alternating (proximal) updates:

• $(y_k,a_k)\in A$, with $y_k+\lambda a_k = x_{k-1}-\lambda b_{k-1}$,
• $(x_k,b_k)\in B$, with $x_k+\lambda b_k = y_k+\lambda b_{k-1}$.

Under feasibility and maximal monotonicity, $(x_k,b_k)$ converges weakly to an element of the extended solution set $S(A,B)$: $$p_k=(x_k,b_k)\rightharpoonup(x^*,b^*):\ b^*\in B(x^*),\ -b^*\in A(x^*)$$ The proof relies on quasi-Fejér monotonicity, demiclosedness principles, boundedness, and Opial’s lemma (Svaiter, 2010, Svaiter, 2018).

Chambolle–Pock Method

For the composite convex problem

$\min_{x\in X}\ f(x)+g(Lx)$

with $f,g$ proper, convex, lower semicontinuous, and $L:X\to Y$ bounded, the Chambolle–Pock updates:

• $$x_{k+1} = \operatorname{prox}_{\tau f}(x_k - \tau L^*y_k)$$,
• $\hat y_{k+1} = x_{k+1} + \theta(x_{k+1}-x_k)$,
• $$y_{k+1} = \operatorname{prox}_{\sigma g^*}(y_k + \sigma L\hat y_{k+1})$$,

produce a sequence that converges weakly to a saddle-point $(x_*,y_*)$ under the condition $\theta > 1/2$ and $\tau\sigma\|L\|^2 < 4/(1+2\theta)$. The argument constructs a Lyapunov functional and establishes summability of residuals, then employs the asymptotic regularity and Opial’s lemma (Banert et al., 2023).

3. Bilinear Forms, Mosco Convergence, and Particle Systems

The weak-convergence method is generalized to sequences of Markov semigroups and Dirichlet forms, especially in the analysis of empirical particle systems and mean-field limits. The convergence of $n$-particle processes to deterministic (or sometimes stochastic) infinite-particle limits is established via "Mosco-type" convergence of a family of bilinear forms $S_n$ associated to Dirichlet forms or generators on $L^2$ spaces.

Key steps:

• Define weak (w-) and strong (s-) modes of convergence of $u_n\in L^2(E,v_n)$ to $u\in L^2(E,v)$.
• Show Mosco-type convergence by verifying lower (lim inf) and upper (lim sup) bounds of energy functionals on dense cores.
• Prove resolvent convergence: $G_{n,\beta}g \to G_\beta g$ in $L^2(E,v_n)$ for all suitable $g$.
• Use this to establish convergence of invariant measures and, with additional tightness, weak convergence of stationary measures and process paths.

Concrete applications include Ginzburg–Landau type interacting diffusions, with the method verifying convergence to deterministic (hydrodynamic) limits and the identification of the stationary law in the infinite-dimensional system (Löbus, 2012).

4. Weak Convergence for Stochastic Numerical Schemes

Weak convergence is crucial in the study of stochastic differential equations (SDEs) and stochastic PDEs under numerical approximation. Rather than analyzing strong (pathwise) deviations, weak error analysis quantifies convergence in distribution for the law of relevant functionals of the discretized process.

Key tools:

• Analysis in the Skorohod topology for path-dependent functionals (Song et al., 2013).
• Use of the Itô–Taylor expansion for higher-order weak approximation schemes, including schemes employing non-Gaussian increments for computational efficiency (Wu et al., 2018).
• Use of Malliavin calculus to control weak errors in semilinear stochastic PDEs, which may have unbounded, non-globally Lipschitz nonlinearities. The error is split into spatial and temporal components, each controlled by the regularity of the Kolmogorov equation or via Malliavin integration by parts (Cai et al., 2021).
• For mean-field (McKean–Vlasov) SDEs, analysis of global-in-time uniform moment bounds and decay properties is required to demonstrate improved long-time weak convergence rates for advanced schemes such as the Leimkuhler–Matthews method, especially when the rate is dimension-independent (Chen et al., 2 May 2024).

5. Monotonicity, Bregman Distances, and Fejér-Type Sequences

Weak convergence in optimization and monotone inclusion settings is often proved by exploiting the structure of Fejér or Bregman–Fejér monotone sequences:

• Fejér monotonicity: $\|x_{k+1}-z\|\leq \|x_k-z\|$ for all $z$ in the solution set.
• Forward Bregman monotonicity: $D_f(x_{k+1},z)\le D_f(x_k,z)$ for a Legendre function $f$ and all $z$ in the solution set, with $D_f$ the Bregman distance.

These properties ensure the boundedness and asymptotic regularity of sequences; combined with convexity, demiclosedness of the operators, and Lyapunov-type functions, weak convergence follows. The approach extends to advanced projection and circumcenter schemes (Ouyang, 2021, Cruz et al., 2014, Matsushita, 2016).

6. Connections to Probability and Statistical Mechanics

In probability theory and asymptotic statistics, the weak convergence method underpins much of modern theory:

• The Portmanteau theorem provides multiple equivalent definitions of weak convergence for measures (Lo et al., 2016).
• Prokhorov’s theorem links tightness to the relative compactness of sequences of probability measures, essential for establishing convergence in distribution in metric spaces.
• Weak convergence of functionals is handled via the continuous-mapping theorem, Skorohod–Wichura representations, and functional empirical processes.
• In statistical mechanics, weak convergence underlies hydrodynamic limits, scaling limits of interacting particle systems, and the analysis of central limit-type theorems in infinite dimensions (Löbus, 2012).

7. Tightness, Demiclosedness, and Opial's Lemma

Universal analytic tools in the weak convergence method include:

• Tightness: ensures that sequences of probability laws (or function-valued objects) have convergent subsequences.
• Demiclosedness: for nonexpansive or maximal monotone operators, guarantees that weak cluster points of approximate zeros are zeros of the operator.
• Opial's Lemma: in Hilbert spaces, provides a criterion for weak convergence of Fejér monotone sequences to a point in the nonempty closed convex set of solution points.

These tools are structurally recurring, not only in operator splitting and variational analysis, but also in iterative methods for variational inequalities, projection algorithms, and stochastic approximation (Svaiter, 2010, Karahan et al., 2014, Shehu et al., 2021).

---

References:

• "The Chambolle–Pock method converges weakly with $θ>1/2$ and $τσ\|L\|^2<4/(1+2θ)$" (Banert et al., 2023)
• "Weak convergence on Douglas-Rachford method" (Svaiter, 2010)
• "Weak Convergence of $n$-Particle Systems Using Bilinear Forms" (Löbus, 2012)
• "Weak Convergence Methods for Approximation of Path-dependent Functionals" (Song et al., 2013)
• "Weak convergence of the backward Euler method for stochastic Cahn--Hilliard equation with additive noise" (Cai et al., 2021)
• "Stochastic simulation of collisions using high-order weak convergence algorithms" (Wu et al., 2018)
• "Improved weak convergence for the long time simulation of Mean-field Langevin equations" (Chen et al., 2 May 2024)
• "Convergence analysis of variants of the averaged alternating modified reflections method" (Matsushita, 2016)
• "Bregman Circumcenters: Monotonicity and Forward Weak Convergence" (Ouyang, 2021)
• "On Weak and Strong Convergence of the Projected Gradient Method for Convex Optimization in real Hilbert Spaces" (Cruz et al., 2014)
• "Weak Convergence (IA). Sequences of Random Vectors" (Lo et al., 2016)
• "Weak convergence of adaptive Markov chain Monte Carlo" (Brown et al., 2 Jun 2024)