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In many situations across computational science and engineering, multiple computational models are available that describe a system of interest. These different models have varying evaluation costs and varying fidelities. Typically, a computationally expensive highfidelity model describes the system with the accuracy required by the current application at hand, while lower-fidelity models are less accurate but computationally cheaper than the high-fidelity model. Outer-loop applications, such as optimization, inference, and uncertainty quantification, require multiple model evaluations at many different inputs, which often leads to computational demands that exceed available resources if only the high-fidelity model is used. This work surveys multifidelity methods that accelerate the solution of outer-loop applications by combining high-fidelity and low-fidelity model evaluations, where the low-fidelity evaluations arise from an explicit low-fidelity model (e.g., a simplified physics approximation, a reduced model, a data-fit surrogate) that approximates the same output quantity as the high-fidelity model. The overall premise of these multifidelity methods is that low-fidelity models are leveraged for speedup while the highfidelity model is kept in the loop to establish accuracy and/or convergence guarantees. We categorize multifidelity methods according to three classes of strategies: adaptation, fusion, and filtering. The paper reviews multifidelity methods in the outer-loop contexts of uncertainty propagation, inference, and optimization.

Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling.

The quantification of probabilistic uncertainties in the outputs of physical, biological, and social systems governed by partial differential equations with random inputs require, in practice, the discretization of those equations. Stochastic finite element methods refer to an extensive class of algorithms for the approximate solution of partial differential equations having random input data, for which spatial discretization is effected by a finite element method. Fully discrete approximations require further discretization with respect to solution dependences on the random variables. For this purpose several approaches have been developed, including intrusive approaches such as stochastic Galerkin methods, for which the physical and probabilistic degrees of freedom are coupled, and non-intrusive approaches such as stochastic sampling and interpolatory-type stochastic collocation methods, for which the physical and probabilistic degrees of freedom are uncoupled. All these method classes are surveyed in this article, including some novel recent developments. Details about the construction of the various algorithms and about theoretical error estimates and complexity analyses of the algorithms are provided. Throughout, numerical examples are used to illustrate the theoretical results and to provide further insights into the methodologies.

We propose a new parallel Robin—Robin domain decomposition method for the coupled Stokes—Darcy system with Beavers—Joseph—Saffman—Jones interface boundary condition. In particular, we prove that, with an appropriate choice of parameters, the scheme converges geometrically independent of the mesh size.

We present a new family of stabilized methods for the Stokes problem. The focus of the paper is on the lowest order velocity-pressure pairs. While not LBB compliant, their simplicity and attractive computational properties make these pairs a popular choice in engineering practice. Our stabilization approach is motivated by terms that characterize the LBB \"deficiency\" of the unstable spaces. The stabilized methods are defined by using these terms to modify the saddle-point Lagrangian associated with the Stokes equations. The new stabilized methods offer a number of attractive computational properties. In contrast to other stabilization procedures, they are parameter free, do not require calculation of higher order derivatives or edge-based data structures, and always lead to symmetric linear systems. Furthermore, the new methods are unconditionally stable, achieve optimal accuracy with respect to solution regularity, and have simple and straightforward implementations. We present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods.

The mathematically correct specification of a fractional differential equation on a bounded domain requires specification of appropriate boundary conditions, or their fractional analogue. This paper discusses the application of nonlocal diffusion theory to specify well-posed fractional diffusion equations on bounded domains.

A multi-phase-field model for simulating the adhesion between two vesicles is constructed. Two phase field functions are introduced to simulate each of the two vesicles. An energy model is defined which accounts for the elastic bending energy of each vesicle and the contact potential energy between the two vesicles; the vesicle volume and surface area constraints are imposed using a penalty method. Numerical results are provided to verify the efficacy of our model and to provide visual illustrations of the different types of contact. The method can be adjusted to solve endocytosis problems by modifying the bending rigidity coefficients of the two elastic bending energies. The method can also be extended to simulate multi-cell adhesions, one example of which is erythrocyte rouleaux. A comparison with laboratory observations demonstrates the effectiveness of the multi-phase field approach.

A sparse-grid method for solving multi-dimensional backward stochastic differential equations （BSDEs） based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathe- matical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and （adaptive） hierarchical sparse-grid interpolation. Error estimates are proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Finally, several numerical examples are provided to illustrate the accuracy and efficiency of our scheme.

The numerical modeling of glacier and ice sheet evolution is a subject of growing interest, in part because of the potential for models to inform estimates of global sea level change. This paper focuses on the development of a numerical model that determines the velocity and pressure fields within an ice sheet. Our numerical model features a high‐fidelity mathematical model involving the nonlinear Stokes system and combinations of no‐sliding and sliding basal boundary conditions, high‐order accurate finite element discretizations based on variable resolution grids, and highly scalable parallel solution strategies, all of which contribute to a numerical model that can achieve accurate velocity and pressure approximations in a highly efficient manner. We demonstrate the accuracy and efficiency of our model by analytical solution tests, established ice sheet benchmark experiments, and comparisons with other well‐established ice sheet models. Key Points A parallel high‐order accurate finite element Stokes ice sheet model Both no‐sliding and sliding boundary conditions are studied Effective and efficient solvers using preconditioning techniques

We study mathematically and computationally optimal control problems for stochastic partial differential equations with Neumann boundary conditions. The control objective is to minimize the expectation of a cost functional, and the control is of the deterministic, boundary-value type. Mathematically, we prove the existence of an optimal solution and of a Lagrange multiplier; we represent the input data in terms of their Karhunen—Loève expansions and deduce the deterministic optimality system of equations. Computationally, we approximate the finite element solution of the optimality system and estimate its error through the discretizations with respect to both spatial and random parameter spaces.