Least-squares Petrov-Galerkin (LSPG) model-reduction techniques such as the Gauss-Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible flow problems where standard Galerkin techniques have failed. However, there has been limited comparative analysis of the two approaches. This is due in part to difficulties arising from the fact that Galerkin techniques perform optimal projection associated with residual minimization at the time-continuous level, while LSPG techniques do so at the time-discrete level.This work provides a detailed theoretical and computational comparison of the two techniques for two common classes of time integrators: linear multistep schemes and Runge-Kutta schemes. We present a number of new findings, including conditions under which the LSPG ROM has a time-continuous representation, conditions under which the two techniques are equivalent, and time-discrete error bounds for the two approaches. Perhaps most surprisingly, we demonstrate both theoretically and computationally that decreasing the time step does not necessarily decrease the error for the LSPG ROM; instead, the time step should be 'matched' to the spectral content of the reduced basis. In numerical experiments carried out on a turbulent compressible-flow problem with over one million unknowns, we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the LSPG reduced-order model by an order of magnitude.

Abstract. We study solution techniques for a linear-quadratic optimal control problem involving fractional powers of elliptic operators. These fractional operators can be realized as the Dirichletto-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. Thus, we consider an equivalent formulation with a nonuniformly elliptic operator as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We discretize the proposed truncated state equation using first degree tensor product finite elements on anisotropic meshes. For the control problem we analyze two approaches: one that is semi-discrete based on the so-called variational approach, where the control is not discretized, and the other one is fully discrete via the discretization of the control by piecewise constant functions. For both approaches, we derive a priori error estimates with respect to the degrees of freedom. Numerical experiments validate the derived error estimates and reveal a competitive performance of anisotropic over quasi-uniform refinement.

Fractional differential operators provide an attractive mathematical tool to model effects with limited regularity properties. Particular examples are image processing and phase field models in which jumps across lower dimensional subsets and sharp transitions across interfaces are of interest. The numerical solution of corresponding model problems via a spectral method is analyzed. Its efficiency and features of the model problems are illustrated by numerical experiments.

In this paper we introduce new characterizations of the spectral fractional Laplacian to incorporate nonhomogeneous Dirichlet and Neumann boundary conditions. The classical cases with homogeneous boundary conditions arise as a special case. We apply our definition to fractional elliptic equations of order s ∈ (0, 1) with nonzero Dirichlet and Neumann boundary condition. Here the domain Ω is assumed to be a bounded, quasi-convex Lipschitz domain. To impose the nonzero boundary conditions, we construct fractional harmonic extensions of the boundary data. It is shown that solving for the fractional harmonic extension is equivalent to solving for the standard harmonic extension in the very-weak form. The latter result is of independent interest as well. The remaining fractional elliptic problem (with homogeneous boundary data) can be realized using the existing techniques. We introduce finite element discretizations and derive discretization error estimates in natural norms, which are confirmed by numerical experiments. We also apply our characterizations to Dirichlet and Neumann boundary optimal control problems with fractional elliptic equation as constraints.For the numerical computation of solutions of (1.5), we rely on well established techniques, see for instance [12,8,7]. It is even possible to apply a standard finite element method especially if the boundary datum g is regular enough. However, the numerical realization of the nonlocal operator (−∆ D,0 ) s in (1.6) is more challenging. Several approaches have been advocated, for instance, computing the eigenvalues and eigenvectors of −∆ D,0 (cf. [39]), Dunford-Taylor integral representation [13], or numerical schemes based on the Caffarelli-Silvestre (or the Stinga-Torrea) extension, just to name a few. In our work, we choose the latter even though the proposed ideas directly apply to other approaches where (−∆ D,0 ) s appears, for instance [13]. Notice that the aforementioned extension of Caffarelli-Silvestre (or the Stinga-Torrea) is only applicable to (−∆ D,0 ) s and not directly to the operator (−∆ D ) s in (1.1).The extension approach was introduced in [17] for R n , see its extensions to bounded domains [19,40]. It states that (−∆ D,0 ) s can be realized as an operator that maps a Dirichlet boundary condition to a Neumann condition via an extension problem on the semi-infinite cylinder C = Ω × (0, ∞), i.e., a Dirichlet-to-Neumann operator. A first finite element method to solve (1.6) based on the extension approach is given in [37]. This was applied to semilinear problems in [4]. In the context of fractional distributed optimal control problems, the extension approach was considered in [3] where related discretization error estimates are established as well.An additional advantage is that our characterization allows for imposing other types of nonhomogeneous boundary conditions such as Neumann boundary conditions (see sections 2.4 and 5) and that it immediately extends to general second order fractional operators (see Section 8).We remark that the diffic...

Very recently Warma [42] has shown that for nonlocal PDEs associated with the fractional Laplacian, the classical notion of controllability from the boundary does not make sense and therefore it must be replaced by a control that is localized outside the open set where the PDE is solved. Having learned from the above mentioned result, in this paper we introduce a new class of source identification and optimal control problems where the source/control is located outside the observation domain where the PDE is satisfied. The classical diffusion models lack this flexibility as they assume that the source/control is located either inside or on the boundary. This is essentially due to the locality property of the underlying operators. We use the nonlocality of the fractional operator to create a framework that now allows placing a source/control outside the observation domain. We consider the Dirichlet, Robin and Neumann source identification or optimal control problems. These problems require dealing with the nonlocal normal derivative (that we shall call interaction operator). We create a functional analytic framework and show well-posedness and derive the first order optimality conditions for these problems. We introduce a new approach to approximate, with convergence rate, the Dirichlet problem with nonzero exterior condition. The numerical examples confirm our theoretical findings and illustrate the practicality of our approach.where the source is either f (force or load) or z (boundary control) see [6,29,36]. In (1.1) there is no provision to place the source in Ω (cf. Figure 1). The issue is that the operator ∆ has "lesser 2010 Mathematics Subject Classification. 49J20, 49K20, 35S15, 65R20, 65N30.

We present an algorithm to generate application-specific, global reduced order quadratures (ROQ) for multiple fast evaluations of weighted inner products between parameterized functions. If a reduced basis (RB) or any other projection-based model reduction technique is applied, the dimensionality of integrands is reduced dramatically; however, the cost of approximating the integrands by projection still scales as the size of the original problem. In contrast, using discrete empirical interpolation (DEIM) points as ROQ nodes leads to a computational cost which depends linearly on the dimension of the reduced space. Generation of a reduced basis via a greedy procedure requires a training set, which for products of functions can be very large. Since this direct approach can be impractical in many applications, we propose instead a two-step greedy targeted towards approximation of such products. We present numerical experiments demonstrating the accuracy and the efficiency of the two-step approach. The presented ROQ are expected to display very fast convergence whenever there is regularity with respect to parameter variation. We find that for the particular application here considered, one driven by gravitational wave physics, the two-step approach speeds up the offline computations to build the ROQ by more than two orders of magnitude. Furthermore, the resulting ROQ rule is found to converge exponentially with the number of nodes, and a factor of ∼ 50 savings, without loss of accuracy, is observed in evaluations of inner products when ROQ are used as a downsampling strategy for equidistant samples using the trapezoidal rule. While the primary focus of this paper is on quadrature rules for inner products of parameterized functions, our method can be easily adapted to integrations of single parameterized functions, and some examples of this type are considered.

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In this work we consider a generalized bilevel optimization framework for solving inverse problems. We introduce fractional Laplacian as a regularizer to improve the reconstruction quality, and compare it with the total variation regularization. We emphasize that the key advantage of using fractional Laplacian as a regularizer is that it leads to a linear operator, as opposed to the total variation regularization which results in a nonlinear degenerate operator. Inspired by residual neural networks, to learn the optimal strength of regularization and the exponent of fractional Laplacian, we develop a dedicated bilevel optimization neural network with a variable depth for a general regularized inverse problem. We also draw some parallels between an activation function in a neural network and regularization. We illustrate how to incorporate various regularizer choices into our proposed network. As an example, we consider tomographic reconstruction as a model problem and show an improvement in reconstruction quality, especially for limited data, via fractional Laplacian regularization. We successfully learn the regularization strength and the fractional exponent via our proposed bilevel optimization neural network. We observe that the fractional Laplacian regularization outperforms total variation regularization. This is specially encouraging, and important, in the case of limited and noisy data.