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We study two fully discrete evolving surface finite element schemes for the Cahn-Hilliard equation on an evolving surface, given a smooth potential with polynomial growth. In particular we establish optimal order error bounds for a (fully implicit) backward Euler time-discretisation, and an implicit-explicit time-discretisation, with isoparametric surface finite elements discretising space.

We propose an operator learning approach to accelerate geometric Markov chain Monte Carlo (MCMC) for solving infinite-dimensional Bayesian inverse problems (BIPs). While geometric MCMC employs high-quality proposals that adapt to posterior local geometry, it requires repeated computations of gradients and Hessians of the log-likelihood, which becomes prohibitive when the parameter-to-observable (PtO) map is defined through expensive-to-solve parametric partial differential equations (PDEs). We consider a delayed-acceptance geometric MCMC method driven by a neural operator surrogate of the PtO map, where the proposal exploits fast surrogate predictions of the log-likelihood and, simultaneously, its gradient and Hessian. To achieve a substantial speedup, the surrogate must accurately approximate the PtO map and its Jacobian, which often demands a prohibitively large number of PtO map samples via conventional operator learning methods. In this work, we present an extension of derivative-informed operator learning [O'Leary-Roseberry et al., J. Comput. Phys., 496 (2024)] that uses joint samples of the PtO map and its Jacobian. This leads to derivative-informed neural operator (DINO) surrogates that accurately predict the observables and posterior local geometry at a significantly lower training cost than conventional methods. Cost and error analysis for reduced basis DINO surrogates are provided. Numerical studies demonstrate that DINO-driven MCMC generates effective posterior samples 3--9 times faster than geometric MCMC and 60--97 times faster than prior geometry-based MCMC. Furthermore, the training cost of DINO surrogates breaks even compared to geometric MCMC after just 10--25 effective posterior samples.

This work introduces a stabilised finite element formulation for the Stokes flow problem with a nonlinear slip boundary condition of friction type. The boundary condition is enforced with the help of an additional Lagrange multiplier and the stabilised formulation is based on simultaneously stabilising both the pressure and the Lagrange multiplier. We establish the stability and the a priori error analyses, and perform a numerical convergence study in order to verify the theory.

A Peskun ordering between two samplers, implying a dominance of one over the other, is known among the Markov chain Monte Carlo community for being a remarkably strong result. It is however also known for being a result that is notably difficult to establish. Indeed, one has to prove that the probability to reach a state $\mathbf{y}$ from a state $\mathbf{x}$, using a sampler, is greater than or equal to the probability using the other sampler, and this must hold for all pairs $(\mathbf{x}, \mathbf{y})$ such that $\mathbf{x} \neq \mathbf{y}$. We provide in this paper a weaker version that does not require an inequality between the probabilities for all these states: essentially, the dominance holds asymptotically, as a varying parameter grows without bound, as long as the states for which the probabilities are greater than or equal to belong to a mass-concentrating set. The weak ordering turns out to be useful to compare lifted samplers for partially-ordered discrete state-spaces with their Metropolis--Hastings counterparts. An analysis in great generality yields a qualitative conclusion: they asymptotically perform better in certain situations (and we are able to identify them), but not necessarily in others (and the reasons why are made clear). A quantitative study in a specific context of graphical-model simulation is also conducted.

We analyze a bilinear optimal control problem for the Stokes--Brinkman equations: the control variable enters the state equations as a coefficient. In two- and three-dimensional Lipschitz domains, we perform a complete continuous analysis that includes the existence of solutions and first- and second-order optimality conditions. We also develop two finite element methods that differ fundamentally in whether the admissible control set is discretized or not. For each of the proposed methods, we perform a convergence analysis and derive a priori error estimates; the latter under the assumption that the domain is convex. Finally, assuming that the domain is Lipschitz, we develop an a posteriori error estimator for each discretization scheme and obtain a global reliability bound.

We construct a simple and robust finite volume discretization for linearized mechanics, Stokes and poromechanics, based only on co-located, cell-centered variables. The discretization has a minimal stencil, using only the two neighboring cells to a face to calculate numerical stresses and fluxes. We fully justify the method theoretically in terms of stability and convergence, both of which are robust in terms of the material parameters. Numerical experiments support the theoretical results, and shed light on grid families not explicitly treated by the theoretical results.

Maximal regularity is a kind of a priori estimates for parabolic-type equations and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) time-stepping method. We will establish such an estimate without logarithmic factor over a quasi-uniform temporal mesh. To show the main result, we introduce the temporally regularized Green's function and then reduce the discrete maximal regularity to a weighted error estimate for its DG approximation. Our results would be useful for investigation of DG approximation of nonlinear parabolic problems.

Deflation techniques are typically used to shift isolated clusters of small eigenvalues in order to obtain a tighter distribution and a smaller condition number. Such changes induce a positive effect in the convergence behavior of Krylov subspace methods, which are among the most popular iterative solvers for large sparse linear systems. We develop a deflation strategy for symmetric saddle point matrices by taking advantage of their underlying block structure. The vectors used for deflation come from an elliptic singular value decomposition relying on the generalized Golub-Kahan bidiagonalization process. The block targeted by deflation is the off-diagonal one since it features a problematic singular value distribution for certain applications. One example is the Stokes flow in elongated channels, where the off-diagonal block has several small, isolated singular values, depending on the length of the channel. Applying deflation to specific parts of the saddle point system is important when using solvers such as CRAIG, which operates on individual blocks rather than the whole system. The theory is developed by extending the existing framework for deflating square matrices before applying a Krylov subspace method like MINRES. Numerical experiments confirm the merits of our strategy and lead to interesting questions about using approximate vectors for deflation.

In this paper we revisit the discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems in Hilbert spaces and provide some new and improved saturation results under less restrictive conditions, comparing with the existing results in the literature.

We study stochastic approximation procedures for approximately solving a $d$-dimensional linear fixed point equation based on observing a trajectory of length $n$ from an ergodic Markov chain. We first exhibit a non-asymptotic bound of the order $t_{\mathrm{mix}} \tfrac{d}{n}$ on the squared error of the last iterate of a standard scheme, where $t_{\mathrm{mix}}$ is a mixing time. We then prove a non-asymptotic instance-dependent bound on a suitably averaged sequence of iterates, with a leading term that matches the local asymptotic minimax limit, including sharp dependence on the parameters $(d, t_{\mathrm{mix}})$ in the higher order terms. We complement these upper bounds with a non-asymptotic minimax lower bound that establishes the instance-optimality of the averaged SA estimator. We derive corollaries of these results for policy evaluation with Markov noise -- covering the TD($\lambda$) family of algorithms for all $\lambda \in [0, 1)$ -- and linear autoregressive models. Our instance-dependent characterizations open the door to the design of fine-grained model selection procedures for hyperparameter tuning (e.g., choosing the value of $\lambda$ when running the TD($\lambda$) algorithm).