An Advection-Robust Hybrid High-Order Method for the Oseen Problem

In this work, we study advection-robust Hybrid High-Order discretizations of the Oseen equations. For a given integer \(k\geqslant 0\), the discrete velocity unknowns are vector-valued polynomials of total degree \(\leqslant \, k\) on mesh elements and faces, while the pressure unknowns are discontinuous polynomials of total degree \(\leqslant \,k\) on the mesh. From the discrete unknowns, three relevant quantities are reconstructed inside each element: a velocity of total degree \(\leqslant \,(k+1)\), a discrete advective derivative, and a discrete divergence. These reconstructions are used to formulate the discretizations of the viscous, advective, and velocity–pressure coupling terms, respectively. Well-posedness is ensured through appropriate high-order stabilization terms. We prove energy error estimates that are advection-robust for the velocity, and show that each mesh element T of diameter \(h_T\) contributes to the discretization error with an \(\mathcal {O}(h_{T}^{k+1})\)-term in the diffusion-dominated regime, an \(\mathcal {O}(h_{T}^{k+\frac{1}{2}})\)-term in the advection-dominated regime, and scales with intermediate powers of \(h_T\) in between. Numerical results complete the exposition.     

Two-Grid Mixed Finite-Element Approximations to the Navier–Stokes Equations Based on a Newton-Type Step

A two-grid scheme to approximate the evolutionary Navier–Stokes equations is introduced and analyzed. A standard mixed finite element approximation is first obtained over a coarse mesh of size H at any positive time \(T>0\). Then, the approximation is postprocessed by means of solving a steady problem based on one step of a Newton iteration over a finer mesh of size \(h<H\). The method increases the rate of convergence of the standard Galerkin method in one unit in terms of H and equals the rate of convergence of the standard Galerkin method over the fine mesh h. However, the computational cost is essentially the cost of approaching the Navier–Stokes equations with the plain Galerkin method over the coarse mesh of size H since the cost of solving one single steady problem is negligible compared with the cost of computing the Galerkin approximation over the full time interval (0, T]. For the analysis we take into account the loss of regularity at initial time of the solution of the Navier–Stokes equations in the absence of nonlocal compatibility conditions. Some numerical experiments are shown.     

Planar collision-type dependence on incident angle and on friction coefficient

This work addresses the ‘hard collision’ approach to the solution of planar, simple non-holonomic systems undergoing a one-point collision-with-friction problem, showing that (i) there are no coherent types of collision whereby forward sliding follows sticking, unless the initial relative tangential velocity of the colliding points vanishes; and (ii) the type of collision can be determined directly, given the collision angle of incidence \(\alpha\) and Coulomb’s coefficient of friction \(\mu\) between the colliding points. The classic hitting rod problem is used to illustrate the \(\alpha \)–\(\mu\) collision-type dependence. Finally, the relation between collision with friction and tangential impact problems in multibody systems is briefly discussed.     

$$\mathcal $$-FAINV: hierarchically factored approximate inverse preconditioners

Given a sparse matrix, its LU-factors, inverse and inverse factors typically suffer from substantial fill-in, leading to non-optimal complexities in their computation as well as their storage. In the past, several computationally efficient methods have been developed to compute approximations to these otherwise rather dense matrices. Many of these approaches are based on approximations through sparse matrices, leading to well-known ILU, sparse approximate inverse or factored sparse approximate inverse techniques and their variants. A different approximation approach is based on blockwise low rank approximations and is realized, for example, through hierarchical (\(\mathcal H\)-) matrices. While \(\mathcal H\)-inverses and \(\mathcal H\)-LU factors have been discussed in the literature, this paper will consider the construction of an approximation of the factored inverse through \(\mathcal H\)-matrices (\(\mathcal H\)-FAINV). We will describe a blockwise approach that permits to replace (exact) matrix arithmetic through approximate efficient \(\mathcal H\)-arithmetic. We conclude with numerical results in which we use approximate factored inverses as preconditioners in the iterative solution of the discretized convection–diffusion problem.     

Unconditionally Optimal Error Analysis of Crank–Nicolson Galerkin FEMs for a Strongly Nonlinear Parabolic System

In this paper, we present unconditionally optimal error estimates of linearized Crank–Nicolson Galerkin finite element methods for a strongly nonlinear parabolic system in \(\mathbb {R}^d\ (d=2,3)\). However, all previous works required certain time-step conditions that were dependent on the spatial mesh size. In order to overcome several entitative difficulties caused by the strong nonlinearity of the system, the proof takes two steps. First, by using a temporal-spatial error splitting argument and a new technique, optimal \(L^2\) error estimates of the numerical schemes can be obtained under the condition \(\tau \ge h\), where \(\tau \) denotes the time-step size and h is the spatial mesh size. Second, we obtain the boundedness of numerical solutions by mathematical induction and inverse inequality when \(\tau \le h\). Then, optimal \(L^2\) and \(H^1\) error estimates are proved in a different way for such case. Numerical results are given to illustrate our theoretical analyses.     

A Hybrid High-Order Method for the Steady Incompressible Navier–Stokes Problem

In this work we introduce and analyze a novel Hybrid High-Order method for the steady incompressible Navier–Stokes equations. The proposed method is inf-sup stable on general polyhedral meshes, supports arbitrary approximation orders, and is (relatively) inexpensive thanks to the possibility of statically condensing a subset of the unknowns at each nonlinear iteration. We show under general assumptions the existence of a discrete solution, which is also unique provided a data smallness condition is verified. Using a compactness argument, we prove convergence of the sequence of discrete solutions to minimal regularity exact solutions for general data. For more regular solutions, we prove optimal convergence rates for the energy-norm of the velocity and the \(L^2\)-norm of the pressure under a standard data smallness assumption. More precisely, when polynomials of degree \(k\ge 0\) at mesh elements and faces are used, both quantities are proved to converge as \(h^{k+1}\) (with h denoting the meshsize).     

A parameterized complexity view on non-preemptively scheduling interval-constrained jobs: few machines,small looseness,and small slack

We study the problem of non-preemptively scheduling n jobs, each job j with a release time \(t_j\), a deadline \(d_j\), and a processing time \(p_j\), on m parallel identical machines. Cieliebak et al. (2004) considered the two constraints \(|d_j-t_j|\le \lambda {}p_j\) and \(|d_j-t_j|\le p_j +\sigma \) and showed the problem to be NP-hard for any \(\lambda >1\) and for any \(\sigma \ge 2\). We complement their results by parameterized complexity studies: we show that, for any \(\lambda >1\), the problem remains weakly NP-hard even for \(m=2\) and strongly W[1]-hard parameterized by m. We present a pseudo-polynomial-time algorithm for constant m and \(\lambda \) and a fixed-parameter tractability result for the parameter m combined with \(\sigma \).     

A Constructive Solution to the Ranking Problem in Partial Order Optimality Theory

Partial order optimality theory (PoOT) (Anttila and Cho in Lingua 104:31–56, 1998) is a conservative generalization of classical optimality theory (COT) (Prince and Smolensky in Optimality theory: constraint interaction in generative grammar, Blackwell Publishers, Malden, 1993/2004) that makes possible the modeling of free variation and quantitative regularities without any numerical parameters. Solving the ranking problem for PoOT has so far remained an outstanding problem: allowing for free variation, given a finite set of input/output pairs, i.e., a dataset, \(\Delta \) that a speaker S knows to be part of some language L, how can S learn the set of all grammars G under some constraint set C compatible with \(\Delta \)?. Here, allowing for free variation, given the set of all PoOT grammars GPoOT over a constraint set C , for an arbitrary \(\Delta \), I provide set-theoretic means for constructing the actual set G compatible with \(\Delta \). Specifically, I determine the set of all STRICT ORDERS of C that are compatible with \(\Delta \). As every strict total order is a strict order, our solution is applicable in both PoOT and COT, showing that the ranking problem in COT is a special instance of a more general one in PoOT.     

An Adaptive Staggered Discontinuous Galerkin Method for the Steady State Convection–Diffusion Equation

Staggered grid techniques have been applied successfully to many problems. A distinctive advantage is that physical laws arising from the corresponding partial differential equations are automatically preserved. Recently, a staggered discontinuous Galerkin (SDG) method was developed for the convection–diffusion equation. In this paper, we are interested in solving the steady state convection–diffusion equation with a small diffusion coefficient \(\epsilon \). It is known that the exact solution may have large gradient in some regions and thus a very fine mesh is needed. For convection dominated problems, that is, when \(\epsilon \) is small, exact solutions may contain sharp layers. In these cases, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, a new SDG method is proposed and the proof of its stability is provided. In order to construct an adaptive mesh refinement strategy for this new SDG method, we derive an a-posteriori error estimator and prove its efficiency and reliability under a boundedness assumption on \(h/\epsilon \), where h is the mesh size. Moreover, we will present some numerical results with singularities and sharp layers to show the good performance of the proposed error estimator as well as the adaptive mesh refinement strategy.     

Numerical Methods and Comparison for the Dirac Equation in the Nonrelativistic Limit Regime

We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter \(0<\varepsilon \ll 1\) which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength \(O(\varepsilon ^2)\) and O(1) in time and space, respectively. We begin with several frequently used finite difference time domain (FDTD) methods and obtain rigorously their error estimates in the nonrelativistic limit regime by paying particular attention to how error bounds depend explicitly on mesh size h and time step \(\tau \) as well as the small parameter \(\varepsilon \). Based on the error bounds, in order to obtain ‘correct’ numerical solutions in the nonrelativistic limit regime, i.e. \(0<\varepsilon \ll 1\), the FDTD methods share the same \(\varepsilon \)-scalability on time step and mesh size as: \(\tau =O(\varepsilon ^3)\) and \(h=O(\sqrt{\varepsilon })\). Then we propose and analyze two numerical methods for the discretization of the Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the symmetric exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their \(\varepsilon \)-scalability is improved to \(\tau =O(\varepsilon ^2)\) and \(h=O(1)\) when \(0<\varepsilon \ll 1\). Extensive numerical results are reported to support our error estimates.     

A Neumann series of Bessel functions representation for solutions of Sturm–Liouville equations

A Neumann series of Bessel functions (NSBF) representation for solutions of Sturm–Liouville equations and for their derivatives is obtained. The representation possesses an attractive feature for applications: for all real values of the spectral parameter \(\omega \) the estimate of the difference between the exact solution and the approximate one (the truncated NSBF) depends on N (the truncation parameter) and the coefficients of the equation and does not depend on \(\omega \). A similar result is valid when \(\omega \in {\mathbb {C}}\) belongs to a strip \(\left| \hbox {Im }\omega \right| <C\). This feature makes the NSBF representation especially useful for applications requiring computation of solutions for large intervals of \(\omega \). Error and decay rate estimates are obtained. An algorithm for solving initial value, boundary value or spectral problems for the Sturm–Liouville equation is developed and illustrated on a test problem.     

The local–global conjecture for scheduling with non-linear cost

We consider the classical scheduling problem on a single machine, on which we need to schedule sequentially n given jobs. Every job j has a processing time \(p_j\) and a priority weight \(w_j\), and for a given schedule a completion time \(C_j\). In this paper, we consider the problem of minimizing the objective value \(\sum _j w_j C_j^\beta \) for some fixed constant \(\beta >0\). This non-linearity is motivated for example by the learning effect of a machine improving its efficiency over time, or by the speed scaling model. For \(\beta =1\), the well-known Smith’s rule that orders job in the non-increasing order of \(w_j/p_j\) gives the optimum schedule. However, for \(\beta \ne 1\), the complexity status of this problem is open. Among other things, a key issue here is that the ordering between a pair of jobs is not well defined, and might depend on where the jobs lie in the schedule and also on the jobs between them. We investigate this question systematically and substantially generalize the previously known results in this direction. These results lead to interesting new dominance properties among schedules which lead to huge speed up in exact algorithms for the problem. An experimental study evaluates the impact of these properties on the exact algorithm A*.     

Hybridized Discontinuous Galerkin Method for Elliptic Interface Problems: Error Estimates Under Low Regularity Assumptions of Solutions

New hybridized discontinuous Galerkin (HDG) methods for the interface problem for elliptic equations are proposed. Unknown functions of our schemes are \(u_h\) in elements and \(\hat{u}_h\) on inter-element edges. That is, we formulate our schemes without introducing the flux variable. We assume that subdomains \(\Omega _1\) and \(\Omega _2\) are polyhedral domains and that the interface \(\Gamma =\partial \Omega _1\cap \partial \Omega _2\) is polyhedral surface or polygon. Moreover, \(\Gamma \) is assumed to be expressed as the union of edges of some elements. We deal with the case where the interface is transversely connected with the boundary of the whole domain \(\overline{\Omega }=\overline{\Omega _1\cap \Omega _2}\). Consequently, the solution u of the interface problem may not have a sufficient regularity, say \(u\in H^2(\Omega )\) or \(u|_{\Omega _1}\in H^2(\Omega _1)\), \(u|_{\Omega _2}\in H^2(\Omega _2)\). We succeed in deriving optimal order error estimates in an HDG norm and the \(L^2\) norm under low regularity assumptions of solutions, say \(u|_{\Omega _1}\in H^{1+s}(\Omega _1)\) and \(u|_{\Omega _2}\in H^{1+s}(\Omega _2)\) for some \(s\in (1/2,1]\), where \(H^{1+s}\) denotes the fractional order Sobolev space. Numerical examples to validate our results are also presented.     

Validation of right coronary artery lumen area from cardiac computed tomography against intravascular ultrasound

Quantification of coronary artery disease (CAD) from cardiac computed tomography angiography (CTA) is important both structurally (lumen area stenosis) and functionally (combined with computational fluid dynamics to determine fractional flow reserve) for assessment of ischemic stenosis and to guide treatment. Hence, it is important to have CTA image processing technique for segmentation and reconstruction of coronary arteries. In this study, we developed segmentation and reconstruction techniques, based on fast marching and Runge–Kutta methods for centerline extraction, and surface mesh generation. The accuracy of the reconstructed models was validated with direct intravascular ultrasound (IVUS) measurements in 1950 cross sections within 4 arteries. High correlation was found between CTA and IVUS measurements for lumen areas (\(r=0.993\), \(p<0.001\)). Receiver-operating characteristic (ROC) curves showed excellent accuracies for detection of different cutoff values of cross-lumen area (5 \(\text {mm}^2\), 6 \(\text {mm}^2\), 7 \(\text {mm}^2\) and 8 \(\text {mm}^2\), all ROC values >0.99). We conclude that our technique has sufficient accuracy for quantifying coronary lumen area. The accuracy and efficiency demonstrated that our approach can facilitate quantitative evaluation of coronary stenosis and potentially help in real-time assessment of CAD.     

Pure state ‘really’ informationally complete with rank-1 POVM

What is the minimal number of elements in a rank-1 positive operator-valued measure (POVM) which can uniquely determine any pure state in d-dimensional Hilbert space \(\mathcal {H}_d\)? The known result is that the number is no less than \(3d-2\). We show that this lower bound is not tight except for \(d=2\) or 4. Then we give an upper bound \(4d-3\). For \(d=2\), many rank-1 POVMs with four elements can determine any pure states in \(\mathcal {H}_2\). For \(d=3\), we show eight is the minimal number by construction. For \(d=4\), the minimal number is in the set of \(\{10,11,12,13\}\). We show that if this number is greater than 10, an unsettled open problem can be solved that three orthonormal bases cannot distinguish all pure states in \(\mathcal {H}_4\). For any dimension d, we construct \(d+2k-2\) adaptive rank-1 positive operators for the reconstruction of any unknown pure state in \(\mathcal {H}_d\), where \(1\le k \le d\).     

On a Class of Implicit–Explicit Runge–Kutta Schemes for Stiff Kinetic Equations Preserving the Navier–Stokes Limit

Implicit–explicit (IMEX) Runge–Kutta (RK) schemes are popular high order time discretization methods for solving stiff kinetic equations. As opposed to the compressible Euler limit (leading order asymptotics of the Boltzmann equation as the Knudsen number \(\varepsilon \) goes to zero), their asymptotic behavior at the Navier–Stokes (NS) level (next order asymptotics) was rarely studied. In this paper, we analyze a class of existing IMEX RK schemes and show that, under suitable initial conditions, they can capture the NS limit without resolving the small parameter \(\varepsilon \), i.e., \(\varepsilon =o(\Delta t)\), \(\Delta t^m=o(\varepsilon )\), where m is the order of the explicit RK part in the IMEX scheme. Extensive numerical tests for BGK and ES-BGK models are performed to verify our theoretical results.     

Local Discontinuous Galerkin Method for Incompressible Miscible Displacement Problem in Porous Media

In this paper, we develop local discontinuous Galerkin method for the two-dimensional coupled system of incompressible miscible displacement problem. Optimal error estimates in \(L^{\infty }(0, T; L^{2})\) for concentration c, \(L^{2}(0, T; L^{2})\) for \(\nabla c\) and \(L^{\infty }(0, T; L^{2})\) for velocity \(\mathbf{u}\) are derived. The main techniques in the analysis include the treatment of the inter-element jump terms which arise from the discontinuous nature of the numerical method, the nonlinearity, and the coupling of the models. The main difficulty is how to treat the inter-element discontinuities of two independent solution variables (one from the flow equation and the other from the transport equation) at cell interfaces. Numerical experiments are shown to demonstrate the theoretical results.     

Online scheduling of moldable parallel tasks

In this paper, we study an online scheduling problem with moldable parallel tasks on m processors. Each moldable task can be processed simultaneously on any number of processors of a parallel computer, and the processing time of a moldable task depends on the number of processors allotted to it. Tasks arrive one by one. Upon arrival of each task, the scheduler has to determine both the number of processors and the starting time for the task. Moreover, these decisions cannot be changed in the future. The objective is to attain a schedule such that the longest completion time over all tasks, i.e., the makespan, is minimized. First, we provide a general framework to show that any \(\rho \)-bounded algorithm for scheduling of rigid parallel tasks (the number of processors for a task is fixed a prior) can be extended to yield an algorithm for scheduling of moldable tasks with a competitive ratio of \(4\rho \) if the ratio \(\rho \) is known beforehand. As a consequence, we achieve the first constant competitive ratio, 26.65, for the moldable parallel tasks scheduling problem. Next, we provide an improved algorithm with a competitive ratio of at most 16.74.     

Discontinuous Galerkin Methods for a Stationary Navier–Stokes Problem with a Nonlinear Slip Boundary Condition of Friction Type

In this work, several discontinuous Galerkin (DG) methods are introduced and analyzed to solve a variational inequality from the stationary Navier–Stokes equations with a nonlinear slip boundary condition of friction type. Existence, uniqueness and stability of numerical solutions are shown for the DG methods. Error estimates are derived for the velocity in a broken \(H^1\)-norm and for the pressure in an \(L^2\)-norm, with the optimal convergence order when linear elements for the velocity and piecewise constants for the pressure are used. Numerical results are reported to demonstrate the theoretically predicted convergence orders, as well as the capability in capturing the discontinuity, the ability in handling the shear layers, the capacity in dealing with the advection-dominated problem, and the application to the general polygonal mesh of the DG methods.