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Rob Stevenson

Publications and source records attributed to Rob Stevenson.

At least 19 recordsLinked to original sources
arXiv2026

A Double-Adaptivity Solver for Parabolic PDEs

We study minimal residual space-time finite element discretizations of linear parabolic initial value problems in canonical space-time variational form. To deal with the arising dual norm, we introduce the Riesz lift of the residual as an additional variable. Quasi-optimality of the primal variable of the mixed system follows from a uniform inf-sup condition. This condition is known to be satisfied for finite element spaces w.r.t. prismatic partitions of the space-time cylinder that allow for a decomposition into time-slabs. We prove that this condition cannot be expected to hold otherwise. To recover stability for general partitions and the data at hand, we derive an a posteriori condition on the error between the exact Riesz lift of the residual and its Galerkin approximation -- being the secondary variable of our system -- under which the primal variable is quasi-optimal. We derive a posteriori error estimators for both variables, and use them in a double-adaptive loop that alternates test-space with trial-space enrichment. We illustrate our findings with numerical experiments in $1+1$ and $2+1$ dimensions.

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arXiv2025

Quasi-optimal time-space discretizations for a class of nonlinear parabolic PDEs

We consider parabolic evolution equations with Lipschitz continuous and strongly monotone spatial operators. By introducing an additional variable, we construct an equivalent system where the operator is a Lipschitz continuous mapping from a Hilbert space $Y \times X$ to its dual, with a Lipschitz continuous inverse. Resulting Galerkin discretizations can be solved with an inexact Uzawa type algorithm. Quasi-optimality of the Galerkin approximations is guaranteed under an inf-sup condition on the selected `test' and `trial' subspaces of $Y$ and $X$. To circumvent the restriction imposed by this inf-sup condition, an a posteriori condition for quasi-optimality is developed that is shown to be satisfied whenever the test space is sufficiently large.

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arXiv2024

Quasi-Optimal Least Squares: Inhomogeneous boundary conditions, and application with machine learning

We construct least squares formulations of PDEs with inhomogeneous essential boundary conditions, where boundary residuals are not measured in unpractical fractional Sobolev norms, but which formulations nevertheless are shown to yield a quasi-best approximations from the employed trial spaces. Dual norms do enter the least-squares functional, so that solving the least squares problem amounts to solving a saddle point or minimax problem. For finite element applications we construct uniformly stable finite element pairs, whereas for Machine Learning applications we employ adversarial networks.

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arXiv2024

Ultra-weak least squares discretizations for unique continuation and Cauchy problems

In this paper, conditional stability estimates are derived for unique continuation and Cauchy problems associated to the Poisson equation in ultra-weak variational form. Numerical approximations are obtained as minima of regularized least squares functionals. The arising dual norms are replaced by discretized dual norms, which leads to a mixed formulation in terms of trial- and test-spaces. For stable pairs of such spaces, and a proper choice of the regularization parameter, the $L_2$-error on a subdomain in the obtained numerical approximation can be bounded by the best possible fractional power of the sum of the data error and the error of best approximation. Compared to the use of a standard variational formulation, the latter two errors are measured in weaker norms. To avoid the use of $C^1$-finite element test spaces, nonconforming finite element test spaces can be applied as well. They either lead to the qualitatively same error bound, or in a simplified version, to such an error bound modulo an additional data oscillation term. Numerical results illustrate our theoretical findings.

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arXiv2023

A pollution-free ultra-weak FOSLS discretization of the Helmholtz equation

We consider an ultra-weak first order system discretization of the Helmholtz equation. When employing the optimal test norm, the `ideal' method yields the best approximation to the pair of the Helmholtz solution and its scaled gradient w.r.t.~the norm on $L_2(\Omega)\times L_2(\Omega)^d$ from the selected finite element trial space. On convex polygons, the `practical', implementable method is shown to be pollution-free essentially whenever the order $\tilde{p}$ of the finite element test space grows proportionally with $\max(\log \kappa,p^2)$, with $p$ being the order at trial side. Numerical results also on other domains show a much better accuracy than for the Galerkin method.

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arXiv2023

A convenient inclusion of inhomogeneous boundary conditions in minimal residual methods

Inhomogeneous essential boundary conditions can be appended to a well-posed PDE to lead to a combined variational formulation. The domain of the corresponding operator is a Sobolev space on the domain $\Omega$ on which the PDE is posed, whereas the codomain is a Cartesian product of spaces, among them fractional Sobolev spaces of functions on $\partial\Omega$. In this paper, easily implementable minimal residual discretizations are constructed which yield quasi-optimal approximation from the employed trial space, in which the evaluation of fractional Sobolev norms is fully avoided.

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arXiv2023

Minimal residual methods in negative or fractional Sobolev norms

For numerical approximation the reformulation of a PDE as a residual minimisation problem has the advantages that the resulting linear system is symmetric positive definite, and that the norm of the residual provides an a posteriori error estimator. Furthermore, it allows for the treatment of general inhomogeneous boundary conditions. In many minimal residual formulations, however, one or more terms of the residual are measured in negative or fractional Sobolev norms. In this work, we provide a general approach to replace those norms by efficiently evaluable expressions without sacrificing quasi-optimality of the resulting numerical solution. We exemplify our approach by verifying the necessary inf-sup conditions for four formulations of a model second order elliptic equation with inhomogeneous Dirichlet and/or Neumann boundary conditions. We report on numerical experiments for the Poisson problem with mixed inhomogeneous Dirichlet and Neumann boundary conditions in an ultra-weak first order system formulation.

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arXiv2022

Interpolation Operators for parabolic Problems

We introduce interpolation operators with approximation and stability properties suited for parabolic problems in primal and mixed formulations. We derive localized error estimates for tensor product meshes (occurring in classical time-marching schemes) as well as locally in space-time refined meshes.

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arXiv2022

Improved rates for a space-time FOSLS of parabolic PDEs

We consider the first-order system space-time formulation of the heat equation introduced in [Bochev, Gunzburger, Springer, New York (2009)], and analyzed in [F\"uhrer, Karkulik, Comput. Math. Appl. 92 (2021)] and [Gantner, Stevenson, ESAIM Math. Model. Numer. Anal.} 55 (2021)], with solution components $(u_1,{\bf u}_2)=(u,-\nabla_{\bf x} u)$. The corresponding operator is boundedly invertible between a Hilbert space $U$ and a Cartesian product of $L_2$-type spaces, which facilitates easy first-order system least-squares (FOSLS) discretizations. Besides $L_2$-norms of $\nabla_{\bf x} u_1$ and ${\bf u}_2$, the (graph) norm of $U$ contains the $L_2$-norm of $\partial_t u_1 +{\rm div}_{\bf x} {\bf u}_2$. When applying standard finite elements w.r.t. simplicial partitions of the space-time cylinder, estimates of the approximation error w.r.t. the latter norm require higher-order smoothness of ${\bf u}_2$. In experiments for both uniform and adaptively refined partitions, this manifested itself in disappointingly low convergence rates for non-smooth solutions $u$. In this paper, we construct finite element spaces w.r.t. prismatic partitions. They come with a quasi-interpolant that satisfies a near commuting diagram in the sense that, apart from some harmless term, the aforementioned error depends exclusively on the smoothness of $\partial_t u_1 +{\rm div}_{\bf x} {\bf u}_2$, i.e., of the forcing term $f=(\partial_t-\Delta_x)u$. Numerical results show significantly improved convergence rates.

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arXiv2022

Applications of a space-time FOSLS formulation for parabolic PDEs

In this work, we show that the space-time first-order system least-squares (FOSLS) formulation [F\"uhrer, Karkulik, Comput. Math. Appl. 92 (2021)] for the heat equation and its recent generalization [Gantner, Stevenson, ESAIM Math. Model. Numer. Anal. 55 (2021)] to arbitrary second-order parabolic PDEs can be used to efficiently solve parameter-dependent problems, optimal control problems, and problems on time-dependent spatial domains.

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arXiv2022

Least squares solvers for ill-posed PDEs that are conditionally stable

This paper is concerned with the design and analysis of least squares solvers for ill-posed PDEs that are conditionally stable. The norms and the regularization term used in the least squares functional are determined by the ingredients of the conditional stability assumption. We are then able to establish a general error bound that, in view of the conditional stability assumption, is qualitatively the best possible, without assuming consistent data. The price for these advantages is to handle dual norms which reduces to verifying suitable inf-sup stability. This, in turn, is done by constructing appropriate Fortin projectors for all sample scenarios. The theoretical findings are illustrated by numerical experiments.

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arXiv2022

A well-posed First Order System Least Squares formulation of the instationary Stokes equations

In this paper, a well-posed simultaneous space-time First Order System Least Squares formulation is constructed of the instationary incompressible Stokes equations with slip boundary conditions. As a consequence of this well-posedness, the minimization over any conforming triple of finite element spaces for velocities, pressure and stress tensor gives a quasi-best approximation from that triple. The formulation is practical in the sense that all norms in the least squares functional can be efficiently evaluated. Being of least squares type, the formulation comes with an efficient and reliable a posteriori error estimator. In addition, a priori error estimates are derived, and numerical results are presented.

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arXiv2021

Minimal residual space-time discretizations of parabolic equations: Asymmetric spatial operators

We consider a minimal residual discretization of a simultaneous space-time variational formulation of parabolic evolution equations. Under the usual `LBB' stability condition on pairs of trial- and test spaces we show quasi-optimality of the numerical approximations without assuming symmetry of the spatial part of the differential operator. Under a stronger LBB condition we show error estimates in an energy-norm which are independent of this spatial differential operator.

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arXiv2021

Accuracy controlled data assimilation for parabolic problems

This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain under consideration. Unlike previous approaches to this and related problems our starting point is a regularized least squares formulation in a continuous infinite-dimensional setting that is based on stable variational time-space formulations of the parabolic PDE. This allows us to derive a priori as well as a posteriori error bounds for the recovered states with respect to a certain reference solution. In these bounds the regularization parameter is disentangled from the underlying discretization. An important ingredient for the derivation of a posteriori bounds is the construction of suitable Fortin operators which allow us to control oscillation errors stemming from the discretization of dual norms. Moreover, the variational framework allows us to contrive preconditioners for the discrete problems whose application can be performed in linear time, and for which the condition numbers of the preconditioned systems are uniformly proportional to that of the regularized continuous problem. In particular, we provide suitable stopping criteria for the iterative solvers based on the a posteriori error bounds. The presented numerical experiments quantify the theoretical findings and demonstrate the performance of the numerical scheme in relation with the underlying discretization and regularization.

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arXiv2021

Efficient numerical approximation of a non-regular Fokker--Planck equation associated with first-passage time distributions

In neuroscience, the distribution of a decision time is modelled by means of a one-dimensional Fokker--Planck equation with time-dependent boundaries and space-time-dependent drift. Efficient approximation of the solution to this equation is required, e.g., for model evaluation and parameter fitting. However, the prescribed boundary conditions lead to a strong singularity and thus to slow convergence of numerical approximations. In this article we demonstrate that the solution can be related to the solution of a parabolic PDE on a rectangular space-time domain with homogeneous initial and boundary conditions by transformation and subtraction of a known function. We verify that the solution of the new PDE is indeed more regular than the solution of the original PDE and proceed to discretize the new PDE using a space-time minimal residual method. We also demonstrate that the solution depends analytically on the parameters determining the boundaries as well as the drift. This justifies the use of a sparse tensor product interpolation method to approximate the PDE solution for various parameter ranges. The predicted convergence rates of the minimal residual method and that of the interpolation method are supported by numerical simulations.

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arXiv2021

A weighted POD-reduction approach for parametrized PDE-constrained Optimal Control Problems with random inputs and applications to environmental sciences

Reduced basis approximations of Optimal Control Problems (OCPs) governed by steady partial differential equations (PDEs) with random parametric inputs are analyzed and constructed. Such approximations are based on a Reduced Order Model, which in this work is constructed using the method of weighted Proper Orthogonal Decomposition. This Reduced Order Model then is used to efficiently compute the reduced basis approximation for any outcome of the random parameter. We demonstrate that such OCPs are well-posed by applying the adjoint approach, which also works in the presence of admissibility constraints and in the case of non linear-quadratic OCPs, and thus is more general than the conventional Lagrangian approach. We also show that a step in the construction of these Reduced Order Models, known as the aggregation step, is not fundamental and can in principle be skipped for noncoercive problems, leading to a cheaper online phase. Numerical applications in three scenarios from environmental science are considered, in which the governing PDE is steady and the control is distributed. Various parameter distributions are taken, and several implementations of the weighted Proper Orthogonal Decomposition are compared by choosing different quadrature rules.

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arXiv2021

Operator preconditioning: the simplest case

Using the framework of operator or Calder\'on preconditioning, uniform preconditioners are constructed for elliptic operators discretized with continuous finite (or boundary) elements. The preconditioners are constructed as the composition of an opposite order operator, discretized on the same ansatz space, and two diagonal scaling operators.

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