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We present a loosely coupled, non-iterative time-splitting scheme based on Robin-Robin coupling conditions. We apply a novel unified analysis for this scheme applied to both a Parabolic/Parabolic coupled system and a Parabolic/Hyperbolic coupled system. We show for both systems that the scheme is stable, and the error converges as $\mathcal{O}\big(\Delta t \sqrt{T +\log{\frac{1}{\Delta t}}}\big)$, where $\Delta t$ is the time step

In this paper, we develop a new type of accelerated algorithms to solve some classes of maximally monotone equations as well as monotone inclusions. Instead of using Nesterov's accelerating approach, our methods rely on a so-called Halpern-type fixed-point iteration in [32], and recently exploited by a number of researchers, including [24, 70]. Firstly, we derive a new variant of the anchored extra-gradient scheme in [70] based on Popov's past extra-gradient method to solve a maximally monotone equation $G(x) = 0$. We show that our method achieves the same $\mathcal{O}(1/k)$ convergence rate (up to a constant factor) as in the anchored extra-gradient algorithm on the operator norm $\Vert G(x_k)\Vert$, , but requires only one evaluation of $G$ at each iteration, where $k$ is the iteration counter. Next, we develop two splitting algorithms to approximate a zero point of the sum of two maximally monotone operators. The first algorithm originates from the anchored extra-gradient method combining with a splitting technique, while the second one is its Popov's variant which can reduce the per-iteration complexity. Both algorithms appear to be new and can be viewed as accelerated variants of the Douglas-Rachford (DR) splitting method. They both achieve $\mathcal{O}(1/k)$ rates on the norm $\Vert G_{\gamma}(x_k)\Vert$ of the forward-backward residual operator $G_{\gamma}(\cdot)$ associated with the problem. We also propose a new accelerated Douglas-Rachford splitting scheme for solving this problem which achieves $\mathcal{O}(1/k)$ convergence rate on $\Vert G_{\gamma}(x_k)\Vert$ under only maximally monotone assumptions. Finally, we specify our first algorithm to solve convex-concave minimax problems and apply our accelerated DR scheme to derive a new variant of the alternating direction method of multipliers (ADMM).

The discretization of robust quadratic optimal control problems under uncertainty using the finite element method and the stochastic collocation method leads to large saddle-point systems, which are fully coupled across the random realizations. Despite its relevance for numerous engineering problems, the solution of such systems is notoriusly challenging. In this manuscript, we study efficient preconditioners for all-at-once approaches using both an algebraic and an operator preconditioning framework. We show in particular that for values of the regularization parameter not too small, the saddle-point system can be efficiently solved by preconditioning in parallel all the state and adjoint equations. For small values of the regularization parameter, robustness can be recovered by the additional solution of a small linear system, which however couples all realizations. A mean approximation and a Chebyshev semi-iterative method are investigated to solve this reduced system. Our analysis considers a random elliptic partial differential equation whose diffusion coefficient $\kappa(x,\omega)$ is modeled as an almost surely continuous and positive random field, though not necessarily uniformly bounded and coercive. We further provide estimates on the dependence of the preconditioned system on the variance of the random field. Such estimates involve either the first or second moment of the random variables $1/\min_{x\in \overline{D}} \kappa(x,\omega)$ and $\max_{x\in \overline{D}}\kappa(x,\omega)$, where $D$ is the spatial domain. The theoretical results are confirmed by numerical experiments, and implementation details are further addressed.

In this work we consider a class of non-linear eigenvalue problems that admit a spectrum similar to that of a Hamiltonian matrix, in the sense that the spectrum is symmetric with respect to both the real and imaginary axis. More precisely, we present a method to iteratively approximate the eigenvalues of such non-linear eigenvalue problems closest to a given purely real or imaginary shift, while preserving the symmetries of the spectrum. To this end the presented method exploits the equivalence between the considered non-linear eigenvalue problem and the eigenvalue problem associated with a linear but infinite-dimensional operator. To compute the eigenvalues closest to the given shift, we apply a specifically chosen shift-invert transformation to this linear operator and compute the eigenvalues with the largest modulus of the new shifted and inverted operator using an (infinite) Arnoldi procedure. The advantage of the chosen shift-invert transformation is that the spectrum of the transformed operator has a "real skew-Hamiltonian"-like structure. Furthermore, it is proven that the Krylov space constructed by applying this operator, satisfies an orthogonality property in terms of a specifically chosen bilinear form. By taking this property into account in the orthogonalization process, it is ensured that even in the presence of rounding errors, the obtained approximation for, e.g., a simple, purely imaginary eigenvalue is simple and purely imaginary. The presented work can thus be seen as an extension of [V. Mehrmann and D. Watkins, "Structure-Preserving Methods for Computing Eigenpairs of Large Sparse Skew-Hamiltonian/Hamiltonian Pencils", SIAM J. Sci. Comput. (22.6), 2001], to the considered class of non-linear eigenvalue problems. Although the presented method is initially defined on function spaces, it can be implemented using finite dimensional linear algebra operations.

We extend and analyze the energy-based discontinuous Galerkin method for second order wave equations on staggered and structured meshes. By combining spatial staggering with local time-stepping near boundaries, the method overcomes the typical numerical stiffness associated with high order piecewise polynomial approximations. In one space dimension with periodic boundary conditions and suitably chosen numerical fluxes, we prove bounds on the spatial operators that establish stability for CFL numbers $c \frac {\Delta t}{h} < C$ independent of order when stability-enhanced explicit time-stepping schemes of matching order are used. For problems on bounded domains and in higher dimensions we demonstrate numerically that one can march explicitly with large time steps at high order temporal and spatial accuracy.

We consider a mesh-based approach for training a neural network to produce field predictions of solutions to parametric partial differential equations (PDEs). This approach contrasts current approaches for "neural PDE solvers" that employ collocation-based methods to make point-wise predictions of solutions to PDEs. This approach has the advantage of naturally enforcing different boundary conditions as well as ease of invoking well-developed PDE theory -- including analysis of numerical stability and convergence -- to obtain capacity bounds for our proposed neural networks in discretized domains. We explore our mesh-based strategy, called NeuFENet, using a weighted Galerkin loss function based on the Finite Element Method (FEM) on a parametric elliptic PDE. The weighted Galerkin loss (FEM loss) is similar to an energy functional that produces improved solutions, satisfies a priori mesh convergence, and can model Dirichlet and Neumann boundary conditions. We prove theoretically, and illustrate with experiments, convergence results analogous to mesh convergence analysis deployed in finite element solutions to PDEs. These results suggest that a mesh-based neural network approach serves as a promising approach for solving parametric PDEs with theoretical bounds.

Much recent interest has focused on the design of optimization algorithms from the discretization of an associated optimization flow, i.e., a system of differential equations (ODEs) whose trajectories solve an associated optimization problem. Such a design approach poses an important problem: how to find a principled methodology to design and discretize appropriate ODEs. This paper aims to provide a solution to this problem through the use of contraction theory. We first introduce general mathematical results that explain how contraction theory guarantees the stability of the implicit and explicit Euler integration methods. Then, we propose a novel system of ODEs, namely the Accelerated-Contracting-Nesterov flow, and use contraction theory to establish it is an optimization flow with exponential convergence rate, from which the linear convergence rate of its associated optimization algorithm is immediately established. Remarkably, a simple explicit Euler discretization of this flow corresponds to the Nesterov acceleration method. Finally, we present how our approach leads to performance guarantees in the design of optimization algorithms for time-varying optimization problems.

We analyse an energy minimisation problem recently proposed for modelling smectic-A liquid crystals. The optimality conditions give a coupled nonlinear system of partial differential equations, with a second-order equation for the tensor-valued nematic order parameter $\mathbf{Q}$ and a fourth-order equation for the scalar-valued smectic density variation $u$. Our two main results are a proof of the existence of solutions to the minimisation problem, and the derivation of a priori error estimates for its discretisation using the $\mathcal{C}^0$ interior penalty method. More specifically, optimal rates in the $H^1$ and $L^2$ norms are obtained for $\mathbf{Q}$, while optimal rates in a mesh-dependent norm and $L^2$ norm are obtained for $u$. Numerical experiments confirm the rates of convergence.

This paper studies the convergence of a spatial semi-discretization for a backward semilinear stochastic parabolic equation. The filtration is general, and the spatial semi-discretization uses the standard continuous piecewise linear element method. Firstly, higher regularity of the solution to the continuous equation is derived. Secondly, the first-order spatial accuracy is derived for the spatial semi-discretization. Thirdly, an application of the theoretical result to a stochastic linear quadratic control problem is presented.

Interpretation of Deep Neural Networks (DNNs) training as an optimal control problem with nonlinear dynamical systems has received considerable attention recently, yet the algorithmic development remains relatively limited. In this work, we make an attempt along this line by reformulating the training procedure from the trajectory optimization perspective. We first show that most widely-used algorithms for training DNNs can be linked to the Differential Dynamic Programming (DDP), a celebrated second-order trajectory optimization algorithm rooted in the Approximate Dynamic Programming. In this vein, we propose a new variant of DDP that can accept batch optimization for training feedforward networks, while integrating naturally with the recent progress in curvature approximation. The resulting algorithm features layer-wise feedback policies which improve convergence rate and reduce sensitivity to hyper-parameter over existing methods. We show that the algorithm is competitive against state-ofthe-art first and second order methods. Our work opens up new avenues for principled algorithmic design built upon the optimal control theory.