M. Croci, M. B. Giles, M. E. Rognes, and P. E. Farrell, Efficient white noise sampling and coupling for multilevel Monte Carlo with non-nested meshes, SIAM/ASA Journal on Uncertainty Quantification 6-4 (2018), pp. 1630-1655. Article. Talk.

When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. Here, we present a new sampling technique that can be used to efficiently compute white noise samples in a finite element method (FEM) and multilevel Monte Carlo (MLMC) setting. The key idea is to exploit the finite element matrix assembly procedure and factorize each local mass matrix independently, hence avoiding the factorization of a large matrix. Moreover, in an MLMC framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of nonnested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments. We observe optimal convergence rates for the finite element solution of the elliptic SPDEs of interest in 2D and 3D and we show convergence of the sampled field covariances. In an MLMC setting, a good coupling is enforced and the telescoping sum is respected.

P. E. Farrell, M. Croci, T. M. Surowiec, Deflation for semismooth equations, Optimization Methods and Software (2020) - Charles Broyden prize for best OMS paper in 2020. Article.

Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, with- out varying the initial guess supplied to the solver. The central idea is the combi- nation of a semismooth Newton method with a deflation operator that eliminates known solutions from consideration. Given one root of a semismooth residual, defla- tion constructs a new problem for which a semismooth Newton method will not con- verge to the known root, even from the same initial guess. This enables the discovery of other roots. We prove the effectiveness of the deflation technique under the same assumptions that guarantee locally superlinear convergence of a semismooth New- ton method. We demonstrate its utility on various finite- and infinite-dimensional examples drawn from constrained optimization, game theory, economics and solid mechanics.

M. Croci, J. Munoz-Matute, Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of phi-functions of matrices via quadrature, Journal of Computational Science (2023) Article, Preprint.

In this article, we propose an algorithm for approximating the action of phi-functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the phi-functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art.

J. Enderlein, D. Sakhapov, I. Gregor, M. Croci, N. Karedla, Modeling charge separation in charged nanochannels for single-molecule electrometry, The Journal of Chemical Physics (2022) Article, Preprint.

We model the transport of electrically charged solute molecules by a laminar flow within a nanoslit microfluidic channel with electrostatic surface potential. We derive the governing convection–diffusion equation, solve it numerically, and compare it with a Taylor–Aris-like approximation, which gives excellent results for small Péclet numbers. We discuss our results in light of designing an assay that can measure simultaneously the hydrodynamic size and electric charge of single molecules by tracking their motion in such nanoslit channels with electrostatic surface potential.