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Finite element simulations have been used to solve various partial differential equations (PDEs) that model physical, chemical, and biological phenomena. The resulting discretized solutions to PDEs often do not satisfy requisite physical properties, such as positivity or monotonicity. Such invalid solutions pose both modeling challenges, since the physical interpretation of simulation results is not possible, and computational challenges, since such properties may be required to advance the scheme. We, therefore, consider the problem of computing solutions that preserve these structural solution properties, which we enforce as additional constraints on the solution. We consider in particular the class of convex constraints, which includes positivity and monotonicity. By embedding such constraints as a postprocessing convex optimization procedure, we can compute solutions that satisfy general types of convex constraints. For certain types of constraints (including positivity and monotonicity), the optimization is a filter, i.e., a norm-decreasing operation. We provide a variety of tests on one-dimensional time-dependent PDEs that demonstrate the method's efficacy, and we empirically show that rates of convergence are unaffected by the inclusion of the constraints.

We study a class of spatial discretizations for the Vlasov-Poisson system written as an hyperbolic system using Hermite polynomials. In particular, we focus on spectral methods and discontinuous Galerkin approximations. To obtain L 2 stability properties, we introduce a new L 2 weighted space, with a time dependent weight. For the Hermite spectral form of the Vlasov-Poisson system, we prove conservation of mass, momentum and total energy, as well as global stability for the weighted L 2 norm. These properties are then discussed for several spatial discretizations. Finally, numerical simulations are performed with the proposed DG/Hermite spectral method to highlight its stability and conservation features.

In this paper, we propose a numerical method to solve the mass-conserved Ohta-Kawasaki equation with finite element discretization. An unconditional stable convex split-ting scheme is applied to time approximation. The Newton method and its variant are used to address the implicitly nonlinear term. We rigorously analyze the convergence of the Newton iteration methods. Theoretical results demonstrate that two Newton iteration methods have the same convergence rate, and the Newton method has a smaller convergent factor than the variant one. To reduce the condition number of discretized linear system, we design two efficient block preconditioners and analyze their spectral distribution. Finally, we offer numerical examples to support the theoretical analysis and indicate the efficiency of the proposed numerical methods for the mass-conserved Ohta-Kawasaki equation.

For the general class of residual distribution (RD) schemes, including many finite element (such as continuous/discontinuous Galerkin) and flux reconstruction methods, an approach to construct entropy conservative/ dissipative semidiscretizations by adding suitable correction terms has been proposed by Abgrall (J.~Comp.~Phys. 372: pp. 640--666, 2018). In this work, the correction terms are characterized as solutions of certain optimization problems and are adapted to the SBP-SAT framework, focusing on discontinuous Galerkin methods. Novel generalizations to entropy inequalities, multiple constraints, and kinetic energy preservation for the Euler equations are developed and tested in numerical experiments. For all of these optimization problems, explicit solutions are provided. Additionally, the correction approach is applied for the first time to obtain a fully discrete entropy conservative/dissipative RD scheme. Here, the application of the deferred correction (DeC) method for the time integration is essential. This paper can be seen as describing a systematic method to construct structure preserving discretization, at least for the considered example.

An energy conservative discontinuous Galerkin scheme for a generalised third order KdV type equation is designed. Based on the conservation principle, we propose techniques that allow for the derivation of optimal a priori bounds for the linear KdV equation and a posteriori bounds for the linear and modified KdV equation. Extensive numerical experiments showcasing the good long time behaviour of the scheme are summarised which are in agreement with the analysis proposed.

In this work, we consider the linear inverse problem $y=Ax+\epsilon$, where $A\colon X\to Y$ is a known linear operator between the separable Hilbert spaces $X$ and $Y$, $x$ is a random variable in $X$ and $\epsilon$ is a zero-mean random process in $Y$. This setting covers several inverse problems in imaging including denoising, deblurring, and X-ray tomography. Within the classical framework of regularization, we focus on the case where the regularization functional is not given a priori but learned from data. Our first result is a characterization of the optimal generalized Tikhonov regularizer, with respect to the mean squared error. We find that it is completely independent of the forward operator $A$ and depends only on the mean and covariance of $x$. Then, we consider the problem of learning the regularizer from a finite training set in two different frameworks: one supervised, based on samples of both $x$ and $y$, and one unsupervised, based only on samples of $x$. In both cases, we prove generalization bounds, under some weak assumptions on the distribution of $x$ and $\epsilon$, including the case of sub-Gaussian variables. Our bounds hold in infinite-dimensional spaces, thereby showing that finer and finer discretizations do not make this learning problem harder. The results are validated through numerical simulations.

Linear discriminant analysis (LDA) is a classical method for dimensionality reduction, where discriminant vectors are sought to project data to a lower dimensional space for optimal separability of classes. Several recent papers have outlined strategies for exploiting sparsity for using LDA with high-dimensional data. However, many lack scalable methods for solution of the underlying optimization problems. We propose three new numerical optimization schemes for solving the sparse optimal scoring formulation of LDA based on block coordinate descent, the proximal gradient method, and the alternating direction method of multipliers. We show that the per-iteration cost of these methods scales linearly in the dimension of the data provided restricted regularization terms are employed, and cubically in the dimension of the data in the worst case. Furthermore, we establish that if our block coordinate descent framework generates convergent subsequences of iterates, then these subsequences converge to the stationary points of the sparse optimal scoring problem. We demonstrate the effectiveness of our new methods with empirical results for classification of Gaussian data and data sets drawn from benchmarking repositories, including time-series and multispectral X-ray data, and provide Matlab and R implementations of our optimization schemes.

We introduce a general framework for enforcing local or global maximum principles in high-order space-time discretizations of a scalar hyperbolic conservation law. We begin with sufficient conditions for a space discretization to be bound preserving (BP) and satisfy a semi-discrete maximum principle. Next, we propose a global monolithic convex (GMC) flux limiter which has the structure of a flux-corrected transport (FCT) algorithm but is applicable to spatial semi-discretizations and ensures the BP property of the fully discrete scheme for strong stability preserving (SSP) Runge-Kutta time discretizations. To circumvent the order barrier for SSP time integrators, we constrain the intermediate stages and/or the final stage of a general high-order RK method using GMC-type limiters. In this work, our theoretical and numerical studies are restricted to explicit schemes which are provably BP for sufficiently small time steps. The new GMC limiting framework offers the possibility of relaxing the bounds of inequality constraints to achieve higher accuracy at the cost of more stringent time step restrictions. The ability of the presented limiters to preserve global bounds and recognize well-resolved smooth solutions is verified numerically for three representative RK methods combined with weighted essentially nonoscillatory (WENO) finite volume space discretizations of linear and nonlinear test problems in 1D.

A new scheme for communication between overset grids using subcells and Weighted Essentially Non Oscillatory (WENO) reconstruction for two-dimensional problems has been proposed. The effectiveness of this procedure is demonstrated using the discontinuous Galerkin method (DGM). This scheme uses WENO reconstruction using cell averages by dividing the immediate neighbors into subcells to find the degrees of freedom in cells near the overset interface. This also has the added advantage that it also works as a limiter if a discontinuity passes through the overset interface. Accuracy tests to demonstrate the maintenance of higher order are provided. Results containing shocks are also provided to demonstrate the limiter aspect of the data communication procedure.

We explore the connections between Green's functions for certain differential equations, covariance functions for Gaussian processes, and the smoothing splines problem. Conventionally, the smoothing spline problem is considered in a setting of reproducing kernel Hilbert spaces, but here we present a more direct approach. With this approach, some choices that are implicit in the reproducing kernel Hilbert space setting stand out, one example being choice of boundary conditions and more elaborate shape restrictions. The paper first explores the Laplace operator and the Poisson equation and studies the corresponding Green's functions under various boundary conditions and constraints. Explicit functional forms are derived in a range of examples. These examples include several novel forms of the Green's function that, to the author's knowledge, have not previously been presented. Next we present a smoothing spline problem where we penalize the integrated squared derivative of the function to be estimated. We then show how the solution can be explicitly computed using the Green's function for the Laplace operator. In the last part of the paper, we explore the connection between Gaussian processes and differential equations, and show how the Laplace operator is related to Brownian processes and how processes that arise due to boundary conditions and shape constraints can be viewed as conditional Gaussian processes. The presented connection between Green's functions for the Laplace operator and covariance functions for Brownian processes allows us to introduce several new novel Brownian processes with specific behaviors. Finally, we consider the connection between Gaussian process priors and smoothing splines.