We present a space--time ultra-weak discontinuous Galerkin discretization of the linear Schr\"odinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal~$h$-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials, or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method.
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We consider two-phase fluid deformable surfaces as model systems for biomembranes. Such surfaces are modeled by incompressible surface Navier-Stokes-Cahn-Hilliard-like equations with bending forces. We derive this model using the Lagrange-D'Alembert principle considering various dissipation mechanisms. The highly nonlinear model is solved numerically to explore the tight interplay between surface evolution, surface phase composition, surface curvature and surface hydrodynamics. It is demonstrated that hydrodynamics can enhance bulging and furrow formation, which both can further develop to pinch-offs. The numerical approach builds on a Taylor-Hood element for the surface Navier-Stokes part, a semi-implicit approach for the Cahn-Hilliard part, higher order surface parametrizations, appropriate approximations of the geometric quantities, and mesh redistribution. We demonstrate convergence properties that are known to be optimal for simplified sub-problems.
High-dimensional transport equations frequently occur in science and engineering. Computing their numerical solution, however, is challenging due to its high dimensionality. In this work we develop an algorithm to efficiently solve the transport equation in moderately complex geometrical domains using a Galerkin method stabilized by streamline diffusion. The ansatz spaces are a tensor product of a sparse grid in space and discontinuous piecewise polynomials in time. Here, the sparse grid is constructed upon nested multilevel finite element spaces to provide geometric flexibility. This results in an implicit time-stepping scheme which we prove to be stable and convergent. If the solution has additional mixed regularity, the convergence of a $2d$-dimensional problem equals that of a $d$-dimensional one up to logarithmic factors. For the implementation, we rely on the representation of sparse grids as a sum of anisotropic full grid spaces. This enables us to store the functions and to carry out the computations on a sequence regular full grids exploiting the tensor product structure of the ansatz spaces. In this way existing finite element libraries and GPU acceleration can be used. The combination technique is used as a preconditioner for an iterative scheme to solve the transport equation on the sequence of time strips. Numerical tests show that the method works well for problems in up to six dimensions. Finally, the method is also used as a building block to solve nonlinear Vlasov-Poisson equations.
In the \emph{graph matching} problem we observe two graphs $G,H$ and the goal is to find an assignment (or matching) between their vertices such that some measure of edge agreement is maximized. We assume in this work that the observed pair $G,H$ has been drawn from the correlated Wigner model -- a popular model for correlated weighted graphs -- where the entries of the adjacency matrices of $G$ and $H$ are independent Gaussians and each edge of $G$ is correlated with one edge of $H$ (determined by the unknown matching) with the edge correlation described by a parameter $\sigma\in [0,1)$. In this paper, we analyse the performance of the \emph{projected power method} (PPM) as a \emph{seeded} graph matching algorithm where we are given an initial partially correct matching (called the seed) as side information. We prove that if the seed is close enough to the ground-truth matching, then with high probability, PPM iteratively improves the seed and recovers the ground-truth matching (either partially or exactly) in $\mathcal{O}(\log n)$ iterations. Our results prove that PPM works even in regimes of constant $\sigma$, thus extending the analysis in \citep{MaoRud} for the sparse Erd\H{o}s-R\'enyi model to the (dense) Wigner model. As a byproduct of our analysis, we see that the PPM framework generalizes some of the state-of-art algorithms for seeded graph matching. We support and complement our theoretical findings with numerical experiments on synthetic data.
We are interested in numerically solving a transitional model derived from the Bloch model. The Bloch equation describes the time evolution of the density matrix of a quantum system forced by an electromagnetic wave. In a high frequency and low amplitude regime, it asymptotically reduces to a non-stiff rate equation. As a middle ground, the transitional model governs the diagonal part of the density matrix. It fits in a general setting of linear problems with a high-frequency quasi-periodic forcing and an exponentially decaying forcing. The numerical resolution of such problems is challenging. Adapting high-order averaging techniques to this setting, we separate the slow (rate) dynamics from the fast (oscillatory and decay) dynamics to derive a new micro-macro problem. We derive estimates for the size of the micro part of the decomposition, and of its time derivatives, showing that this new problem is non-stiff. As such, we may solve this micro-macro problem with uniform accuracy using standard numerical schemes. To validate this approach, we present numerical results first on a toy problem and then on the transitional Bloch model.
Jones proposed the study of two subfactors of a $II_1$ factor as a quantization of two closed subspaces in a Hilbert space. The Pimsner-Popa probabilistic constant, Sano-Watatani angle, interior and exterior angle, and Connes-St{\o}rmer relative entropy (along with a slight variant of it) are a few key invariants for pair of subfactors that analyze their relative position. In practice, however, the explicit computation of these invariants is often difficult. In this article, we provide an in-depth analysis of a special class of two subfactors, namely a pair of spin model subfactors of the hyperfinite type $II_1$ factor $R$. We first characterize when two distinct $n\times n$ complex Hadamard matrices give rise to distinct spin model subfactors. Then, a detailed investigation has been carried out for pairs of (Hadamard equivalent) complex Hadamard matrices of order $2\times 2$ as well as Hadamard inequivalent complex Hadamard matrices of order $4\times 4$. To the best of our knowledge, this article is the first instance in the literature where the exact value of the Pimsner-Popa probabilistic constant and the noncommutative relative entropy for pairs of (non-trivial) subfactors have been obtained. Furthermore, we prove the factoriality of the intersection of the corresponding pair of subfactors using the `commuting square technique'. En route, we construct an infinite family of potentially new subfactors of $R$. All these subfactors are irreducible with Jones index $4n,n\geq 2$. As a consequence, the rigidity of the interior angle between the spin model subfactors is established. Last but not least, we explicitly compute the Sano-Watatani angle between the spin model subfactors.
In this work, an exponential Discontinuous Galerkin (DG) method is proposed to solve numerically Vlasov type equations. The DG method is used for space discretization which is combined exponential Lawson Runge-Kutta method for time discretization to get high order accuracy in time and space. In addition to get high order accuracy in time, the use of Lawson methods enables to overcome the stringent condition on the time step induced by the linear part of the system. Moreover, it can be proved that a discrete Poisson equation is preserved. Numerical results on Vlasov-Poisson and Vlasov Maxwell equations are presented to illustrate the good behavior of the exponential DG method.
In this paper, we present an error estimate of a second-order linearized finite element (FE) method for the 2D Navier-Stokes equations with variable density. In order to get error estimates, we first introduce an equivalent form of the original system. Later, we propose a general BDF2-FE method for solving this equivalent form, where the Taylor-Hood FE space is used for discretizing the Navier-Stokes equations and conforming FE space is used for discretizing density equation. We show that our scheme ensures discrete energy dissipation. Under the assumption of sufficient smoothness of strong solutions, an error estimate is presented for our numerical scheme for variable density incompressible flow in two dimensions. Finally, some numerical examples are provided to confirm our theoretical results.
In this work, we study the convergence and performance of nonlinear solvers for the Bidomain equations after decoupling the ordinary and partial differential equations of the cardiac system. Firstly, we provide a rigorous proof of the global convergence of Quasi-Newton methods, such as BFGS, and nonlinear Conjugate-Gradient methods, such as Fletcher--Reeves, for the Bidomain system, by analyzing an auxiliary variational problem under physically reasonable hypotheses. Secondly, we compare several nonlinear Bidomain solvers in terms of execution time, robustness with respect to the data and parallel scalability. Our findings indicate that Quasi-Newton methods are the best choice for nonlinear Bidomain systems, since they exhibit faster convergence rates compared to standard Newton-Krylov methods, while maintaining robustness and scalability. Furthermore, first-order methods also demonstrate competitiveness and serve as a viable alternative, particularly for matrix-free implementations that are well-suited for GPU computing.
A class of averaging block nonlinear Kaczmarz methods is developed for the solution of the nonlinear system of equations. The convergence theory of the proposed method is established under suitable assumptions and the upper bounds of the convergence rate for the proposed method with both constant stepsize and adaptive stepsize are derived. Numerical experiments are presented to verify the efficiency of the proposed method, which outperforms the existing nonlinear Kaczmarz methods in terms of the number of iteration steps and computational costs.
We consider the numerical solution of the real time equilibrium Dyson equation, which is used in calculations of the dynamical properties of quantum many-body systems. We show that this equation can be written as a system of coupled, nonlinear, convolutional Volterra integro-differential equations, for which the kernel depends self-consistently on the solution. As is typical in the numerical solution of Volterra-type equations, the computational bottleneck is the quadratic-scaling cost of history integration. However, the structure of the nonlinear Volterra integral operator precludes the use of standard fast algorithms. We propose a quasilinear-scaling FFT-based algorithm which respects the structure of the nonlinear integral operator. The resulting method can reach large propagation times, and is thus well-suited to explore quantum many-body phenomena at low energy scales. We demonstrate the solver with two standard model systems: the Bethe graph, and the Sachdev-Ye-Kitaev model.
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