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This work is on a user-friendly reduced basis method for solving a family of parametric PDEs by preconditioned Krylov subspace methods including the conjugate gradient method, generalized minimum residual method, and bi-conjugate gradient method. The proposed methods use a preconditioned Krylov subspace method for a high-fidelity discretization of one parameter instance to generate orthogonal basis vectors of the reduced basis subspace. Then large-scale discrete parameter-dependent problems are approximately solved in the low-dimensional Krylov subspace. As shown in the theory and experiments, only a small number of Krylov subspace iterations are needed to simultaneously generate approximate solutions of a family of high-fidelity and large-scale systems in the reduced basis subspace. This reduces the computational cost dramatically because (1) to construct the reduced basis vectors, we only solve one large-scale problem in the high-fidelity level; and (2) the family of large-scale problems restricted to the reduced basis subspace have much smaller sizes., Comment: 23 pages, 6 figures

In this paper, we study solutions for a weakly coupled system of eikonal equations arising in an optimal path-planning problem with random breakdown. The model considered takes into account two types of breakdown for the vehicle, partial and total, which happen at a known, spatially inhomogeneous rate. In particular, we analyze the complications due to the delicate degenerate coupling condition by using existing results on weakly coupled systems of Hamilton-Jacobi equations. Then we consider finite element method schemes built for convection-diffusion problems to construct approximate solutions for this system and produce some numerical simulations., Comment: 6 pages, two columns

This work is on a fast and accurate reduced basis method for solving discretized fractional elliptic partial differential equations (PDEs) of the form $\mathcal{A}^su=f$ by rational approximation. A direct computation of the action of such an approximation would require solving multiple (20$\sim$30) large-scale sparse linear systems. Our method constructs the reduced basis using the first few directions obtained from the preconditioned conjugate gradient method applied to one of the linear systems. As shown in the theory and experiments, only a small number of directions (5$\sim$10) are needed to approximately solve all large-scale systems on the reduced basis subspace. This reduces the computational cost dramatically because: (1) We only use one of the large-scale problems to construct the basis; and (2) all large-scale problems restricted to the subspace have much smaller sizes. We test our algorithms for fractional PDEs on a 3d Euclidean domain, a 2d surface, and random combinatorial graphs. We also use a novel approach to construct the rational approximation for the fractional power function by the orthogonal greedy algorithm (OGA).

It is well known that via the augmented Lagrangian method, one can solve Stokes' system by solving the nearly incompressible linear elasticity equation. In this paper, we show that the converse holds, and approximate the inverse of the linear elasticity operator with a convex linear combination of parameter-free operators. In such a way, we construct a uniform preconditioner for linear elasticity for all values of the Lam\'{e} parameter $\lambda\in [0,\infty)$. Numerical results confirm that by using inf-sup stable finite-element spaces for the solution of Stokes' equations, the proposed preconditioner is robust in $\lambda$.

Maxwell's equations are a system of partial differential equations that govern the laws of electromagnetic induction. We study a mimetic finite-difference (MFD) discretization of the equations which preserves important underlying physical properties. We show that, after mass-lumping and appropriate scaling, the MFD discretization is equivalent to a structure-preserving finite-element (FE) scheme. This allows for a transparent analysis of the MFD method using the FE framework, and provides an avenue for the construction of efficient and robust linear solvers for the discretized system. In particular, block preconditioners designed for FE formulations can be applied to the MFD system in a straightforward fashion. We present numerical tests which verify the accuracy of the MFD scheme and confirm the robustness of the preconditioners.

We introduce and explain key relations between a posteriori error estimates and subspace correction methods viewed as preconditioners for problems in infinite dimensional Hilbert spaces. We set the stage using the Finite Element Exterior Calculus and Nodal Auxiliary Space Preconditioning. This framework provides a systematic way to derive explicit residual estimators and estimators based on local problems which are upper and lower bounds of the true error. We show the applications to discretizations of $\delta d$, curl-curl, grad-div, Hodge Laplacian problems, and linear elasticity with weak symmetry. We also provide a new regular decomposition for singularly perturbed H(d) norms and parameter-independent error estimators. The only ingredients needed are: well-posedness of the problem and the existence of regular decomposition on continuous level., Comment: 30 pages, 1 figure

In this paper, we study a posteriori error estimators which aid multilevel iterative solvers for linear systems with graph Laplacians. In earlier works such estimates were computed by solving global optimization problems, which could be computationally expensive. We propose a novel strategy to compute these estimates by constructing a Helmholtz decomposition on the graph based on a spanning tree and the corresponding cycle space. To compute the error estimator, we solve efficiently the linear system on the spanning tree, and then we solve approximately a least-squares problem on the cycle space. As we show, such an estimator has a nearly-linear computational complexity for sparse graphs under certain assumptions. Numerical experiments are presented to demonstrate the efficacy of the proposed method.

We consider quasi-polynomial spaces of differential forms defined as weighted (with a positive weight) spaces of differential forms with polynomial coefficients. We show that the unisolvent set of functionals for such spaces on a simplex in any spatial dimension is the same as the set of such functionals used for the polynomial spaces. The analysis in the quasi-polynomial spaces, however, is not standard and requires a novel approach. We are able to prove our results without the use of Stokes' Theorem, which is the standard tool in showing the unisolvence of functionals in polynomial spaces of differential forms. These new results provide tools for studying exponentially-fitted discretizations stable for general convection-diffusion problems in Hilbert differential complexes.

Tutte's spring embedding theorem states that, for a three-connected planar graph, if the outer face of the graph is fixed as the complement of some convex region in the plane, and all other vertices are placed at the mass center of their neighbors, then this results in a unique embedding, and this embedding is planar. It also follows fairly quickly that this embedding minimizes the sum of squared edge lengths, conditional on the embedding of the outer face. However, it is not at all clear how to embed this outer face. We consider the minimization problem of embedding this outer face, up to some normalization, so that the sum of squared edge lengths is minimized. In this work, we show the connection between this optimization problem and the Schur complement of the graph Laplacian with respect to the interior vertices. We prove a number of discrete trace theorems, and, using these new results, show the spectral equivalence of this Schur complement with the boundary Laplacian to the one-half power for a large class of graphs. Using this result, we give theoretical guarantees for this optimization problem, which motivates an algorithm to embed the outer face of a spring embedding.

We present residual-based a posteriori error estimates of mixed finite element methods for the three-field formulation of Biot's consolidation model. The error estimator is an upper and lower bound of the space time discretization error up to data oscillation. As a by-product, we also obtain new a posteriori error estimate of mixed finite element methods for the heat equation., Comment: 25 pages, 7 figures

In this paper, we present block preconditioners for a stabilized discretization of the poroelastic equations developed in [45]. The discretization is proved to be well-posed with respect to the physical and discretization parameters, and thus provides a framework to develop preconditioners that are robust with respect to such parameters as well. We construct both norm-equivalent (diagonal) and field-of-value-equivalent (triangular) preconditioners for both the stabilized discretization and a perturbation of the stabilized discretization that leads to a smaller overall problem after static condensation. Numerical tests for both two- and three-dimensional problems confirm the robustness of the block preconditioners with respect to the physical and discretization parameters.

Invasive species have been recognized as a leading threat to biodiversity. In particular, lakes are especially affected by species invasions because they are closed systems sensitive to disruption. Accurately controlling the spread of invasive species requires solving a complex spatial-dynamic optimization problem. In this work we propose a novel framework for determining the optimal management strategy to maximize the value of a lake system net of damages from invasive species, including an endogenous diffusion mechanism for the spread of invasive species through boaters' trips between lakes. The proposed method includes a combined global iterative process which determines the optimal number of trips to each lake in each season and the spatial-dynamic optimal boat ramp fee., Comment: 10 pages, 4 algorithms

We propose a path cover adaptive algebraic multigrid (PC-$\alpha$AMG) method for solving linear systems of weighted graph Laplacians and can also be applied to discretized second order elliptic partial differential equations. The PC-$\alpha$AMG is based on unsmoothed aggregation AMG (UA-AMG). To preserve the structure of smooth error down to the coarse levels, we approximate the level sets of the smooth error by first forming vertex-disjoint path cover with paths following the level sets. The aggregations are then formed by matching along the paths in the path cover. In such manner, we are able to build a multilevel structure at a low computational cost. The proposed PC-$\alpha$AMG provides a mechanism to efficiently re-build the multilevel hierarchy during the iterations and leads to a fast nonlinear multilevel algorithm. Traditionally, UA-AMG requires more sophisticated cycling techniques, such as AMLI-cycle or K-cycle, but as our numerical results show, the PC-$\alpha$AMG proposed here leads to nearly optimal standard V-cycle algorithm for solving linear systems with weighted graph Laplacians. Numerical experiments for some real world graph problems also demonstrate PC-$\alpha$AMG's effectiveness and robustness, especially for ill-conditioned graphs.

Local Fourier analysis (LFA) is a useful tool in predicting the convergence factors of geometric multigrid methods (GMG). As is well known, on rectangular domains with periodic boundary conditions this analysis gives the exact convergence factors of such methods. In this work, using the Fourier method, we extend these results by proving that such analysis yields the exact convergence factors for a wider class of problems.

We introduce an efficient method for computing the Stekloff eigenvalues associated with the Helmholtz equation. In general, this eigenvalue problem requires solving the Helmholtz equation with Dirichlet and/or Neumann boundary condition repeatedly. We propose solving the related constant coefficient Helmholtz equation with Fast Fourier Transform (FFT) based on carefully designed extensions and restrictions of the equation. The proposed Fourier method, combined with proper eigensolver, results in an efficient and clear approach for computing the Stekloff eigenvalues., Comment: 12 pages, 4 figures

In this paper, we design robust and efficient block preconditioners for the two-field formulation of Biot's consolidation model, where stabilized finite-element discretizations are used. The proposed block preconditioners are based on the well-posedness of the discrete linear systems. Block diagonal (norm-equivalent) and block triangular preconditioners are developed, and we prove that these methods are robust with respect to both physical and discretization parameters. Numerical results are presented to support the theoretical results., Comment: submitted

We generalize some of the functional (hyper-circle) a posteriori estimates from finite element settings to general graphs or Hilbert space settings. We provide several theoretical results in regard to the generalized a posteriori error estimators. We use these estimates to construct aggregation based coarse spaces for graph Laplacians. The estimator is used to assess the quality of an aggregation adaptively. Furthermore, a reshaping algorithm based is tested on several numerical examples., Comment: 17 pages

This paper is to give an overview of AMG methods for solving large scale systems of equations such as those from the discretization of partial differential equations. AMG is often understood as the acronym of "Algebraic Multi-Grid", but it can also be understood as "Abstract Muti-Grid". Indeed, as it demonstrates in this paper, how and why an algebraic multigrid method can be better understood in a more abstract level. In the literature, there are a variety of different algebraic multigrid methods that have been developed from different perspectives. In this paper, we try to develop a unified framework and theory that can be used to derive and analyze different algebraic multigrid methods in a coherent manner. Given a smoother $R$ for a matrix $A$, such as Gauss-Seidel or Jacobi, we prove that the optimal coarse space of dimension $n_c$ is the span of the eigen-vectors corresponding to the first $n_c$ eigenvalues of $\bar RA$ (with $\bar R=R+R^T-R^TAR$). We also prove that this optimal coarse space can be obtained by a constrained trace-minimization problem for a matrix associated with $\bar RA$ and demonstrate that coarse spaces of most of existing AMG methods can be viewed some approximate solution of this trace-minimization problem. Furthermore, we provide a general approach to the construction of a quasi-optimal coarse space and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation, and the anisotropy. Our theory applies to most existing multigrid methods, including the standard geometric multigrid method, the classic AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG, and spectral AMGe., Comment: 128 Pages, 29 figures

Analysis of big data has become an increasingly relevant area of research, with data often represented on discrete networks both constructed and organic. While for structured domains, there exist intuitive definitions of signals and frequencies, the definitions are much less obvious for data sets associated with a given network. Often, the eigenvectors of an induced graph Laplacian are used to construct an orthogonal set of low-frequency vectors. For larger graphs, however, the computational cost of creating such structures becomes untenable, and the quality of the approximation is adequate only for signals near the span of the set. We propose a construction of a full basis of frequencies with computational complexity that is near-linear in time and linear in storage. Using this frequency domain, we can compress data sets on unstructured graphs more robustly and accurately than spectral-based constructions., Comment: 15 pages approx 20 figures

Graph disaggregation is a technique used to address the high cost of computation for power law graphs on parallel processors. The few high-degree vertices are broken into multiple small-degree vertices, in order to allow for more efficient computation in parallel. In particular, we consider computations involving the graph Laplacian, which has significant applications, including diffusion mapping and graph partitioning, among others. We prove results regarding the spectral approximation of the Laplacian of the original graph by the Laplacian of the disaggregated graph. In addition, we construct an alternate disaggregation operator whose eigenvalues interlace those of the original Laplacian. Using this alternate operator, we construct a uniform preconditioner for the original graph Laplacian., Comment: 19 pages

In this paper, we design robust and efficient linear solvers for the numerical approximation of solutions to Maxwell's equations with dissipative boundary conditions. We consider a structure-preserving finite-element approximation with standard Nedelec--Raviart--Thomas elements in space and a Crank--Nicolson scheme in time to approximate the electric and magnetic fields. We focus on two types of block preconditioners. The first type is based on the well-posedness results of the discrete problem. The second uses an exact block factorization of the linear system, for which the structure-preserving discretization yields sparse Schur complements. We prove robustness and optimality of these block preconditioners, and provide supporting numerical tests., Comment: 18 pages, 2 figures

A stable finite element scheme that avoids pressure oscillations for a three-field Biot's model in poroelasticity is considered. The involved variables are the displacements, fluid flux (Darcy velocity), and the pore pressure, and they are discretized by using the lowest possible approximation order: Crouzeix-Raviart finite elements for the displacements, lowest order Raviart-Thomas-Nedelec elements for the Darcy velocity, and piecewise constant approximation for the pressure. Mass lumping technique is introduced for the Raviart-Thomas-Nedelec elements in order to eliminate the Darcy velocity and, therefore, reduce the computational cost. We show convergence of the discrete scheme which is implicit in time and use these types of elements in space with and without mass lumping. Finally, numerical experiments illustrate the convergence of the method and show its effectiveness to avoid spurious pressure oscillations when mass lumping for the Raviart-Thomas-Nedelec elements is used.

This note proposes embedding a time dependent PDE into a convection-diffusion type PDE (in one space dimension higher) with singularity, for which two discretization schemes, the classical streamline-diffusion and the EAFE (edge average finite element) one, are investigated in terms of stability and error analysis. The EAFE scheme, in particular, is extended to be arbitrary order which is of interest on its own. Numerical results, in combined space-time domain demonstrate the feasibility of the proposed approach., Comment: 20 pages 3 figures

Spectral graph bisections are a popular heuristic aimed at approximating the solution of the NP-complete graph bisection problem. This technique, however, does not always provide a robust tool for graph partitioning. Using a special class of graphs, we prove that the standard spectral graph bisection can produce bisections that are far from optimal. In particular, we show that the maximum error in the spectral approximation of the optimal bisection (partition sizes exactly equal) cut for such graphs is bounded below by a constant multiple of the order of the graph squared., Comment: To appear in Linear and Multilinear Algebra

In this paper we design and analyze a uniform preconditioner for a class of high order Discontinuous Galerkin schemes. The preconditioner is based on a space splitting involving the high order conforming subspace and results from the interpretation of the problem as a nearly-singular problem. We show that the proposed preconditioner exhibits spectral bounds that are uniform with respect to the discretization parameters, i.e., the mesh size, the polynomial degree and the penalization coefficient. The theoretical estimates obtained are supported by several numerical simulations.

In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic multigrid method in the geometric setting for elliptic eigenvalue problems and show its uniform convergence under certain assumptions. Numerical tests are presented for computing the Fiedler vector of several practical graphs, and numerical results show the efficiency and optimality of our proposed cascadic multigrid algorithm., Comment: 16 pages

This paper contains construction and analysis a finite element approximation for convection dominated diffusion problems with full coefficient matrix on general simplicial partitions in $R^d$, $d=2,3$. This construction is quite close to the scheme of Xu and Zikatanov (Math. Comp. 1999) where a diagonal coefficient matrix has been considered. The scheme is of the class of exponentially fitted methods that does not use upwind or checking the flow direction. It is stable for sufficiently small discretization step-size assuming that the boundary value problem for the convection-diffusion equation is uniquely solvable. Further, it is shown that, under certain conditions on the mesh the scheme is monotone. Convergence of first order is derived under minimal smoothness of the solution., Comment: 12 pages, no figures; (Obchysljuval'na ta prykladna matematyka, Kiev)

This work is on the numerical approximation of incoming solutions to Maxwell's equations with dissipative boundary conditions whose energy decays exponentially with time. Such solutions are called asymptotically disappearing (ADS) and they play an importarnt role in inverse back-scatering problems. The existence of ADS is a difficult mathematical problem. For the exterior of a sphere, such solutions have been constructed analytically by Colombini, Petkov and Rauch [7] by specifying appropriate initial conditions. However, for general domains of practical interest (such as Lipschitz polyhedra), the existence of such solutions is not evident. This paper considers a finite-element approximation of Maxwell's equations in the exterior of a polyhedron, whose boundary approximates the sphere. Standard Nedelec-Raviart-Thomas elements are used with a Crank-Nicholson scheme to approximate the electric and magnetic fields. Discrete initial conditions interpolating the ones chosen in [7] are modified so that they are (weakly) divergence-free. We prove that with such initial conditions, the approximation to the electric field is weakly divergence-free for all time. Finally, we show numerically that the finite-element approximations of the ADS also decay exponentially with time when the mesh size and the time step become small., Comment: 15 pages, 3 figures

We derive a three-term recurrence relation for computing the polynomial of best approximation in the uniform norm to $x^{-1}$ on a finite interval with positive endpoints. As application, we consider two-level methods for scalar elliptic partial differential equation (PDE), where the relaxation on the fine grid uses the aforementioned polynomial of best approximation. Based on a new smoothing property of this polynomial smoother that we prove, combined with a proper choice of the coarse space, we obtain as a corollary, that the convergence rate of the resulting two-level method is uniform with respect to the mesh parameters, coarsening ratio and PDE coefficient variation., Comment: 23 pages 5 tables and 3 figures