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We carry out a stability and convergence analysis for the fully discrete scheme obtained by combining a finite or virtual element spatial discretization with the upwind-discontinuous Galerkin time-stepping applied to the time-dependent advection-diffusion equation. A space-time streamline-upwind Petrov-Galerkin term is used to stabilize the method. More precisely, we show that the method is inf-sup stable with constant independent of the diffusion coefficient, which ensures the robustness of the method in the convection- and diffusion-dominated regimes. Moreover, we prove optimal convergence rates in both regimes for the error in the energy norm. An important feature of the presented analysis is the control in the full $L^2(0,T;L^2(\Omega))$ norm without the need of introducing an artificial reaction term in the model. We finally present some numerical experiments in $(3 + 1)$-dimensions that validate our theoretical results.

We develop and analyze stochastic inexact Gauss-Newton methods for nonlinear least-squares problems and inexact Newton methods for nonlinear systems of equations. Random models are formed using suitable sampling strategies for the matrices involved in the deterministic models. The analysis of the expected number of iterations needed in the worst case to achieve a desired level of accuracy in the first-order optimality condition provides guidelines for applying sampling and enforcing, with fixed probability, a suitable accuracy in the random approximations. Results of the numerical validation of the algorithms are presented.

Efficient estimation of nonlinear functions of quantum states is crucial for various key tasks in quantum computing, such as entanglement spectroscopy, fidelity estimation, and feature analysis of quantum data. Conventional methods using state tomography and estimating numerous terms of the series expansion are computationally expensive, while alternative approaches based on a purified query oracle impose practical constraints. In this paper, we introduce the quantum state function (QSF) framework by extending the SWAP test via linear combination of unitaries and parameterized quantum circuits. Our framework enables the implementation of arbitrary degree-$n$ polynomial functions of quantum states with precision $\varepsilon$ using $\mathcal{O}(n/\varepsilon^2)$ copies. We further apply QSF for developing quantum algorithms of fundamental tasks, achieving a sample complexity of $\tilde{\mathcal{O}}(1/(\varepsilon^2\kappa))$ for both von Neumann entropy estimation and quantum state fidelity calculations, where $\kappa$ represents the minimal nonzero eigenvalue. Our work establishes a concise and unified paradigm for estimating and realizing nonlinear functions of quantum states, paving the way for the practical processing and analysis of quantum data.

Many partial differential equations (PDEs) such as Navier--Stokes equations in fluid mechanics, inelastic deformation in solids, and transient parabolic and hyperbolic equations do not have an exact, primal variational structure. Recently, a variational principle based on the dual (Lagrange multiplier) field was proposed. The essential idea in this approach is to treat the given PDE as constraints, and to invoke an arbitrarily chosen auxiliary potential with strong convexity properties to be optimized. This leads to requiring a convex dual functional to be minimized subject to Dirichlet boundary conditions on dual variables, with the guarantee that even PDEs that do not possess a variational structure in primal form can be solved via a variational principle. The vanishing of the first variation of the dual functional is, up to Dirichlet boundary conditions on dual fields, the weak form of the primal PDE problem with the dual-to-primal change of variables incorporated. We derive the dual weak form for the linear, one-dimensional, transient convection-diffusion equation. A Galerkin discretization is used to obtain the discrete equations, with the trial and test functions chosen as linear combination of either RePU activation functions (shallow neural network) or B-spline basis functions; the corresponding stiffness matrix is symmetric. For transient problems, a space-time Galerkin implementation is used with tensor-product B-splines as approximating functions. Numerical results are presented for the steady-state and transient convection-diffusion equation, and transient heat conduction. The proposed method delivers sound accuracy for ODEs and PDEs and rates of convergence are established in the $L^2$ norm and $H^1$ seminorm for the steady-state convection-diffusion problem.

In this paper, for solving a class of linear parabolic equations in rectangular domains, we have proposed an efficient Parareal exponential integrator finite element method. The proposed method first uses the finite element approximation with continuous multilinear rectangular basis function for spatial discretization, and then takes the Runge-Kutta approach accompanied with Parareal framework for time integration of the resulting semi-discrete system to produce parallel-in-time numerical solution. Under certain regularity assumptions, fully-discrete error estimates in $L^2$-norm are derived for the proposed schemes with random interpolation nodes. Moreover, a fast solver can be provided based on tensor product spectral decomposition and fast Fourier transform (FFT), since the mass and coefficient matrices of the proposed method can be simultaneously diagonalized with an orthogonal matrix. A series of numerical experiments in various dimensions are also presented to validate the theoretical results and demonstrate the excellent performance of the proposed method.

The SE and DE formulas are known as efficient quadrature formulas for integrals with endpoint singularities. Especially for integrals with algebraic singularity, explicit error bounds in a computable form have been given, which are useful for computation with guaranteed accuracy. Such explicit error bounds have also given for integrals with logarithmic singularity. However, the error bounds have two points to be discussed. The first point is on overestimation of divergence speed of logarithmic singularity. The second point is on the case where there exist both logarithmic and algebraic singularity. To remedy these points, this study provides new error bounds for integrals with logarithmic and algebraic singularity. Although existing and new error bounds described above handle integrals over the finite interval, the SE and DE formulas may be applied to integrals over the semi-infinite interval. On the basis of the new results, this study provides new error bounds for integrals over the semi-infinite interval with logarithmic and algebraic singularity at the origin.

We present an efficient algorithm for the application of sequences of planar rotations to a matrix. Applying such sequences efficiently is important in many numerical linear algebra algorithms for eigenvalues. Our algorithm is novel in three main ways. First, we introduce a new kernel that is optimized for register reuse in a novel way. Second, we introduce a blocking and packing scheme that improves the cache efficiency of the algorithm. Finally, we thoroughly analyze the memory operations of the algorithm which leads to important theoretical insights and makes it easier to select good parameters. Numerical experiments show that our algorithm outperforms the state-of-the-art and achieves a flop rate close to the theoretical peak on modern hardware.

In this paper, in order to improve the spatial accuracy, the exponential integrator Fourier Galerkin method (EIFG) is proposed for solving semilinear parabolic equations in rectangular domains. In this proposed method, the spatial discretization is first carried out by the Fourier-based Galerkin approximation, and then the time integration of the resulting semi-discrete system is approximated by the explicit exponential Runge-Kutta approach, which leads to the fully-discrete numerical solution. With certain regularity assumptions on the model problem, error estimate measured in $H^2$-norm is explicitly derived for EIFG method with two RK stages. Several two and three dimensional examples are shown to demonstrate the excellent performance of EIFG method, which are coincident to the theoretical results.

We present a simple universal algorithm for high-dimensional integration which has the optimal error rate (independent of the dimension) in all weighted Korobov classes both in the randomized and the deterministic setting. Our theoretical findings are complemented by numerical tests.

In this paper, we develop monolithic limiting techniques for enforcing nonlinear stability constraints in enriched Galerkin (EG) discretizations of nonlinear scalar hyperbolic equations. To achieve local mass conservation and gain control over the cell averages, the space of continuous (multi-)linear finite element approximations is enriched with piecewise-constant functions. The resulting spatial semi-discretization has the structure of a variational multiscale method. For linear advection equations, it is inherently stable but generally not bound preserving. To satisfy discrete maximum principles and ensure entropy stability in the nonlinear case, we use limiters adapted to the structure of our locally conservative EG method. The cell averages are constrained using a flux limiter, while the nodal values of the continuous component are constrained using a clip-and-scale limiting strategy for antidiffusive element contributions. The design and analysis of our new algorithms build on recent advances in the fields of convex limiting and algebraic entropy fixes for finite element methods. In addition to proving the claimed properties of the proposed approach, we conduct numerical studies for two-dimensional nonlinear hyperbolic problems. The numerical results demonstrate the ability of our limiters to prevent violations of the imposed constraints, while preserving the optimal order of accuracy in experiments with smooth solutions.