A Two-Grid Block-Centered Finite Difference Method for the Nonlinear Time-Fractional Parabolic Equation

In this article, a two-grid block-centered finite difference scheme is introduced and analyzed to solve the nonlinear time-fractional parabolic equation. This method is considered where the nonlinear problem is solved only on a coarse grid of size H and a linear problem is solved on a fine grid of size h. Stability results are proven rigorously. Error estimates are established on non-uniform rectangular grid which show that the discrete \(L^{\infty }(L^2)\) and \(L^2(H^1)\) errors are \(O(\triangle t^{2-\alpha }+h^2+H^3)\). Finally, some numerical experiments are presented to show the efficiency of the two-grid method and verify that the convergence rates are in agreement with the theoretical analysis.     

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.     

Tight bounds for parallel randomized load balancing

Given a distributed system of \(n\) balls and \(n\) bins, how evenly can we distribute the balls to the bins, minimizing communication? The fastest non-adaptive and symmetric algorithm achieving a constant maximum bin load requires \(\varTheta (\log \log n)\) rounds, and any such algorithm running for \(r\in {\mathcal {O}}(1)\) rounds incurs a bin load of \(\varOmega ((\log n/\log \log n)^{1/r})\). In this work, we explore the fundamental limits of the general problem. We present a simple adaptive symmetric algorithm that achieves a bin load of 2 in \(\log ^* n+{\mathcal {O}}(1)\) communication rounds using \({\mathcal {O}}(n)\) messages in total. Our main result, however, is a matching lower bound of \((1-o(1))\log ^* n\) on the time complexity of symmetric algorithms that guarantee small bin loads. The essential preconditions of the proof are (i) a limit of \({\mathcal {O}}(n)\) on the total number of messages sent by the algorithm and (ii) anonymity of bins, i.e., the port numberings of balls need not be globally consistent. In order to show that our technique yields indeed tight bounds, we provide for each assumption an algorithm violating it, in turn achieving a constant maximum bin load in constant time.     

Exact exponential algorithms for 3-machine flowshop scheduling problems

In this paper, we focus on the design of an exact exponential time algorithm with a proved worst-case running time for 3-machine flowshop scheduling problems considering worst-case scenarios. For the minimization of the makespan criterion, a Dynamic Programming algorithm running in \({\mathcal {O}}^*(3^n)\) is proposed, which improves the current best-known time complexity \(2^{{\mathcal {O}}(n)}\times \Vert I\Vert ^{{\mathcal {O}}(1)}\) in the literature. The idea is based on a dominance condition and the consideration of the Pareto Front in the criteria space. The algorithm can be easily generalized to other problems that have similar structures. The generalization on two problems, namely the \(F3\Vert f_\mathrm{max}\) and \(F3\Vert \sum f_i\) problems, is discussed.     

Two Mixed Finite Element Methods for Time-Fractional Diffusion Equations

Based on spatial conforming and nonconforming mixed finite element methods combined with classical L1 time stepping method, two fully-discrete approximate schemes with unconditional stability are first established for the time-fractional diffusion equation with Caputo derivative of order \(0<\alpha <1\). As to the conforming scheme, the spatial global superconvergence and temporal convergence order of \(O(h^2+\tau ^{2-\alpha })\) for both the original variable u in \(H^1\)-norm and the flux \(\vec {p}=\nabla u\) in \(L^2\)-norm are derived by virtue of properties of bilinear element and interpolation postprocessing operator, where h and \(\tau \) are the step sizes in space and time, respectively. At the same time, the optimal convergence rates in time and space for the nonconforming scheme are also investigated by some special characters of \(\textit{EQ}_1^{\textit{rot}}\) nonconforming element, which manifests that convergence orders of \(O(h+\tau ^{2-\alpha })\) and \(O(h^2+\tau ^{2-\alpha })\) for the original variable u in broken \(H^1\)-norm and \(L^2\)-norm, respectively, and approximation for the flux \(\vec {p}\) converging with order \(O(h+\tau ^{2-\alpha })\) in \(L^2\)-norm. Numerical examples are provided to demonstrate the theoretical analysis.     

Numerical Algorithm With High Spatial Accuracy for the Fractional Diffusion-Wave Equation With Neumann Boundary Conditions

A fourth-order compact algorithm is discussed for solving the time fractional diffusion-wave equation with Neumann boundary conditions. The \(L1\) discretization is applied for the time-fractional derivative and the compact difference approach for the spatial discretization. The unconditional stability and the global convergence of the compact difference scheme are proved rigorously, where a new inner product is introduced for the theoretical analysis. The convergence order is \(\mathcal{O }(\tau ^{3-\alpha }+h^4)\) in the maximum norm, where \(\tau \) is the temporal grid size and \(h\) is the spatial grid size, respectively. In addition, a Crank–Nicolson scheme is presented and the corresponding error estimates are also established. Meanwhile, a compact ADI difference scheme for solving two-dimensional case is derived and the global convergence order of \(\mathcal{O }(\tau ^{3-\alpha }+h_1^4+h_2^4)\) is given. Then extension to the case with Robin boundary conditions is also discussed. Finally, several numerical experiments are included to support the theoretical results, and some comparisons with the Crank–Nicolson scheme are presented to show the effectiveness of the compact scheme.     

Indistinguishability of pure orthogonal product states by LOCC

We construct two sets of incomplete and extendible quantum pure orthogonal product states (POPS) in general bipartite high-dimensional quantum systems, which are all indistinguishable by local operations and classical communication. The first set of POPS is composed of two parts which are \(\mathcal {C}^m\otimes \mathcal {C}^{n_1}\) with \(5\le m\le n_1\) and \(\mathcal {C}^m\otimes \mathcal {C}^{n_2}\) with \(5\le m \le n_2\), where \(n_1\) is odd and \(n_2\) is even. The second one is in \(\mathcal {C}^m\otimes \mathcal {C}^n\) \((m, n\ge 4)\). Some subsets of these two sets can be extended into complete sets that local indistinguishability can be decided by noncommutativity which quantifies the quantumness of a quantum ensemble. Our study shows quantum nonlocality without entanglement.     

Data assimilation and sampling in Banach spaces

This paper studies the problem of approximating a function f in a Banach space \(\mathcal{X}\) from measurements \(l_j(f)\), \(j=1,\ldots ,m\), where the \(l_j\) are linear functionals from \(\mathcal{X}^*\). Quantitative results for such recovery problems require additional information about the sought after function f. These additional assumptions take the form of assuming that f is in a certain model class \(K\subset \mathcal{X}\). Since there are generally infinitely many functions in K which share these same measurements, the best approximation is the center of the smallest ball B, called the Chebyshev ball, which contains the set \(\bar{K}\) of all f in K with these measurements. Therefore, the problem is reduced to analytically or numerically approximating this Chebyshev ball. Most results study this problem for classical Banach spaces \(\mathcal{X}\) such as the \(L_p\) spaces, \(1\le p\le \infty \), and for K the unit ball of a smoothness space in \(\mathcal{X}\). Our interest in this paper is in the model classes \(K=\mathcal{K}(\varepsilon ,V)\), with \(\varepsilon >0\) and V a finite dimensional subspace of \(\mathcal{X}\), which consists of all \(f\in \mathcal{X}\) such that \(\mathrm{dist}(f,V)_\mathcal{X}\le \varepsilon \). These model classes, called approximation sets, arise naturally in application domains such as parametric partial differential equations, uncertainty quantification, and signal processing. A general theory for the recovery of approximation sets in a Banach space is given. This theory includes tight a priori bounds on optimal performance and algorithms for finding near optimal approximations. It builds on the initial analysis given in Maday et al. (Int J Numer Method Eng 102:933–965, 2015) for the case when \(\mathcal{X}\) is a Hilbert space, and further studied in Binev et al. (SIAM UQ, 2015). It is shown how the recovery problem for approximation sets is connected with well-studied concepts in Banach space theory such as liftings and the angle between spaces. Examples are given that show how this theory can be used to recover several recent results on sampling and data assimilation.     

High-Order Compact Difference Methods for Caputo-Type Variable Coefficient Fractional Sub-diffusion Equations in Conservative Form

A set of high-order compact finite difference methods is proposed for solving a class of Caputo-type fractional sub-diffusion equations in conservative form. The diffusion coefficient of the equation may be spatially variable, and the proposed methods have the global convergence order \(\mathcal{O}(\tau ^{r}+h^{4})\), where \(r\ge 2\) is a positive integer and \(\tau \) and h are the temporal and spatial steps. Such new high-order compact difference methods greatly improve the known methods in the literature. The local truncation error and the solvability of the methods are discussed in detail. By applying a discrete energy technique to the matrix form of the methods, a rigorous theoretical analysis of the stability and convergence of the methods is carried out for the case of \(2\le r\le 6\), and the optimal error estimates in the weighted \(H^{1}\), \(L^{2}\) and \(L^{\infty }\) norms are obtained for the general case of variable coefficient. Applications are given to two model problems, and some numerical results are presented to illustrate the various convergence orders of the methods.     

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.     

A Weak Galerkin Method with an Over-Relaxed Stabilization for Low Regularity Elliptic Problems

A new weak Galerkin (WG) finite element method is developed and analyzed for solving second order elliptic problems with low regularity solutions in the Sobolev space \(W^{2,p}(\Omega )\) with \(p\in (1,2)\). A WG stabilizer was introduced by Wang and Ye (Math Comput 83:2101–2126, 2014) for a simpler variational formulation, and it has been commonly used since then in the WG literature. In this work, for the purpose of dealing with low regularity solutions, we propose to generalize the stabilizer of Wang and Ye by introducing a positive relaxation index to the mesh size h. The relaxed stabilization gives rise to a considerable flexibility in treating weak continuity along the interior element edges. When the norm index \(p\in (1,2]\), we strictly derive that the WG error in energy norm has an optimal convergence order \(O(h^{l+1-\frac{1}{p}-\frac{p}{4}})\) by taking the relaxed factor \(\beta =1+\frac{2}{p}-\frac{p}{2}\), and it also has an optimal convergence order \(O(h^{l+2-\frac{2}{p}})\) in \(L^2\) norm when the solution \(u\in W^{l+1,p}\) with \(p\in [1,1+\frac{2}{p}-\frac{p}{2}]\) and \(l\ge 1\). It is recovered for \(p=2\) that with the choice of \(\beta =1\), error estimates in the energy and \(L^2\) norms are optimal for the source term in the sobolev space \(L^2\). Weak variational forms of the WG method give rise to desirable flexibility in enforcing boundary conditions and can be easily implemented without requiring a sufficiently large penalty factor as in the usual discontinuous Galerkin methods. In addition, numerical results illustrate that the proposed WG method with an over-relaxed factor \(\beta (\ge 1)\) converges at optimal algebraic rates for several low regularity elliptic problems.     

Producibility in hierarchical self-assembly

Three results are shown on producibility in the hierarchical model of tile self-assembly. It is shown that a simple greedy polynomial-time strategy decides whether an assembly α is producible. The algorithm can be optimized to use \(O(|\alpha | \log ^2 |\alpha |)\) time. Cannon et al. (STACS 2013: proceedings of the thirtieth international symposium on theoretical aspects of computer science. pp 172–184, 2013) showed that the problem of deciding if an assembly α is the unique producible terminal assembly of a tile system \({\mathcal {T}}\) can be solved in \(O(|\alpha |^2 |{\mathcal {T}}| + |\alpha | |{\mathcal {T}}|^2)\) time for the special case of noncooperative “temperature 1” systems. It is shown that this can be improved to \(O(|\alpha | |{\mathcal {T}}| \log |{\mathcal {T}}|)\) time. Finally, it is shown that if two assemblies are producible, and if they can be overlapped consistently—i.e., if the positions that they share have the same tile type in each assembly—then their union is also producible.     

A Fractional Order Collocation Method for Second Kind Volterra Integral Equations with Weakly Singular Kernels

In this paper, we develop a fractional order spectral collocation method for solving second kind Volterra integral equations with weakly singular kernels. It is well known that the original solution of second kind Volterra integral equations with weakly singular kernels usually can be split into two parts, the first is the singular part and the second is the smooth part with the assumption that the integer m being its smooth order. On the basis of this characteristic of the solution, we first choose the fractional order Lagrange interpolation function of Chebyshev type as the basis of the approximate space in the collocation method, and then construct a simple quadrature rule to obtain a fully discrete linear system. Consequently, with the help of the Lagrange interpolation approximate theory we establish that the fully discrete approximate equation has a unique solution for sufficiently large n, where \(n+1\) denotes the dimension of the approximate space. Moreover, we prove that the approximate solution arrives at an optimal convergence order \(\mathcal{O}(n^{-m}\log n)\) in the infinite norm and \(\mathcal{O}(n^{-m})\) in the weighted square norm. In addition, we prove that for sufficiently large n, the infinity-norm condition number of the coefficient matrix corresponding to the linear system is \(\mathcal{O}(\log ^2 n)\) and its spectral condition number is \(\mathcal{O}(1)\). Numerical examples are presented to demonstrate the effectiveness of the proposed method.     

Entanglement witness criteria of strong-<Emphasis Type="Italic">k</Emphasis>-separability for multipartite quantum states

Let \(H_{1}, H_{2},\ldots ,H_{n}\) be separable complex Hilbert spaces with \(\dim H_{i}\ge 2\) and \(n\ge 2\). Assume that \(\rho \) is a state in \(H=H_1\otimes H_2\otimes \cdots \otimes H_n\). \(\rho \) is called strong-k-separable \((2\le k\le n)\) if \(\rho \) is separable for any k-partite division of H. In this paper, an entanglement witnesses criterion of strong-k-separability is obtained, which says that \(\rho \) is not strong-k-separable if and only if there exist a k-division space \(H_{m_{1}}\otimes \cdots \otimes H_{m_{k}}\) of H, a finite-rank linear elementary operator positive on product states \(\Lambda :\mathcal {B}(H_{m_{2}}\otimes \cdots \otimes H_{m_{k}})\rightarrow \mathcal {B}(H_{m_{1}})\) and a state \(\rho _{0}\in \mathcal {S}(H_{m_{1}}\otimes H_{m_{1}})\), such that \(\mathrm {Tr}(W\rho )<0\), where \(W=(\mathrm{Id}\otimes \Lambda ^{\dagger })\rho _{0}\) is an entanglement witness. In addition, several different methods of constructing entanglement witnesses for multipartite states are also given.     

Throughput maximization for speed scaling with agreeable deadlines

We study the following energy-efficient scheduling problem. We are given a set of n jobs which have to be scheduled by a single processor whose speed can be varied dynamically. Each job \(J_j\) is characterized by a processing requirement (work) \(p_j\), a release date \(r_j\), and a deadline \(d_j\). We are also given a budget of energy E which must not be exceeded and our objective is to maximize the throughput (i.e., the number of jobs which are completed on time). We show that the problem can be solved optimally via dynamic programming in \(O(n^4 \log n \log P)\) time when all jobs have the same release date, where P is the sum of the processing requirements of the jobs. For the more general case with agreeable deadlines where the jobs can be ordered so that, for every \(i < j\), it holds that \(r_i \le r_j\) and \(d_i \le d_j\), we propose an optimal dynamic programming algorithm which runs in \(O(n^6 \log n \log P)\) time. In addition, we consider the weighted case where every job \(J_j\) is also associated with a weight \(w_j\) and we are interested in maximizing the weighted throughput (i.e., the total weight of the jobs which are completed on time). For this case, we show that the problem becomes \(\mathcal{NP}\)-hard in the ordinary sense even when all jobs have the same release date and we propose a pseudo-polynomial time algorithm for agreeable instances.     

Cost-efficient design of a quantum multiplier–accumulator unit

This paper proposes a cost-efficient quantum multiplier–accumulator unit. The paper also presents a fast multiplication algorithm and designs a novel quantum multiplier device based on the proposed algorithm with the optimum time complexity as multiplier is the major device of a multiplier–accumulator unit. We show that the proposed multiplication technique has time complexity \(O((3 {\hbox {log}}_{2}n)+1)\), whereas the best known existing technique has \(O(n{\hbox {log}}_{2} n)\), where n is the number of qubits. In addition, our design proposes three new quantum circuits: a circuit representing a quantum full-adder, a circuit known as quantum ANDing circuit, which performs the ANDing operation and a circuit presenting quantum accumulator. Moreover, the proposed quantum multiplier–accumulator unit is the first ever quantum multiplier–accumulator circuit in the literature till now, which has reduced garbage outputs and ancillary inputs to a great extent. The comparative study shows that the proposed quantum multiplier performs better than the existing multipliers in terms of depth, quantum gates, delays, area and power with the increasing number of qubits. Moreover, we design the proposed quantum multiplier–accumulator unit, which performs better than the existing ones in terms of hardware and delay complexities, e.g., the proposed (\(n\times n\))—qubit quantum multiplier–accumulator unit requires \(O(n^{2})\) hardware and \(O({\hbox {log}}_{2}n)\) delay complexities, whereas the best known existing quantum multiplier–accumulator unit requires \(O(n^{3})\) hardware and \(O((n-1)^{2} +1+n)\) delay complexities. In addition, the proposed design achieves an improvement of 13.04, 60.08 and 27.2% for \(4\times 4\), 7.87, 51.8 and 27.1% for \(8\times 8\), 4.24, 52.14 and 27% for \(16\times 16\), 2.19, 52.15 and 27.26% for \(32 \times 32\) and 0.78, 52.18 and 27.28% for \(128 \times 128\)-qubit multiplications over the best known existing approach in terms of number of quantum gates, ancillary inputs and garbage outputs, respectively. Moreover, on average, the proposed design gains an improvement of 5.62% in terms of area and power consumptions over the best known existing approach.     

Improved constructions for quantum maximum distance separable codes

In this work, we further improve the distance of the quantum maximum distance separable (MDS) codes of length \(n=\frac{q^2+1}{10}\). This yields new families of quantum MDS codes. We also construct a family of new quantum MDS codes with parameters \([[\frac{q^2-1}{3}, \frac{q^2-1}{3}-2d+2, d]]_{q}\), where \(q=2^m\), \(2\le d\le \frac{q-1}{3}\) if \(3\mid (q+2)\), and \(2\le d\le \frac{2q-1}{3}\) if \(3\mid (q+1)\). Compared with the known quantum MDS codes, these quantum MDS codes have much larger minimum distance.     

Faster search by lackadaisical quantum walk

In the typical model, a discrete-time coined quantum walk searching the 2D grid for a marked vertex achieves a success probability of \(O(1/\log N)\) in \(O(\sqrt{N \log N})\) steps, which with amplitude amplification yields an overall runtime of \(O(\sqrt{N} \log N)\). We show that making the quantum walk lackadaisical or lazy by adding a self-loop of weight 4 / N to each vertex speeds up the search, causing the success probability to reach a constant near 1 in \(O(\sqrt{N \log N})\) steps, thus yielding an \(O(\sqrt{\log N})\) improvement over the typical, loopless algorithm. This improved runtime matches the best known quantum algorithms for this search problem. Our results are based on numerical simulations since the algorithm is not an instance of the abstract search algorithm.     

Fast Iterative Method with a Second-Order Implicit Difference Scheme for Time-Space Fractional Convection–Diffusion Equation

In this paper we intend to establish fast numerical approaches to solve a class of initial-boundary problem of time-space fractional convection–diffusion equations. We present a new unconditionally stable implicit difference method, which is derived from the weighted and shifted Grünwald formula, and converges with the second-order accuracy in both time and space variables. Then, we show that the discretizations lead to Toeplitz-like systems of linear equations that can be efficiently solved by Krylov subspace solvers with suitable circulant preconditioners. Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from \({\mathcal {O}}(N^2)\) to \({\mathcal {O}}(N)\) and the computational complexity from \({\mathcal {O}}(N^3)\) to \({\mathcal {O}}(N\log N)\) in each iterative step, where N is the number of grid nodes. Extensive numerical examples are reported to support our theoretical findings and show the utility of these methods over traditional direct solvers of the implicit difference method, in terms of computational cost and memory requirements.     

A greedy approximation algorithm for minimum-gap scheduling

We consider scheduling of unit-length jobs with release times and deadlines, where the objective is to minimize the number of gaps in the schedule. Polynomial-time algorithms for this problem are known, yet they are rather inefficient, with the best algorithm running in time \(O(n^4)\) and requiring \(O(n^3)\) memory. We present a greedy algorithm that approximates the optimum solution within a factor of 2 and show that our analysis is tight. Our algorithm runs in time \(O(n^2 \log n)\) and needs only O(n) memory. In fact, the running time is \(O(n (g^*+1)\log n)\), where \(g^*\) is the minimum number of gaps.