Laplacians may generate spurious eigenvalues on non-convex domains. To overcome this difficulty, we adopt a recently developed mixed method, which decomposes the biharmonic equation into three Poisson equations and still recovers the original solution. Using this idea, we design an efficient biharmonic eigenvalue algorithm, which contains only Poisson solvers. With this approach, eigenfunctions can be confined in the correct space and thereby spurious modes in non-convex domains are avoided. A priori error estimates for both eigenvalues and eigenfunctions on quasi-uniform meshes are obtained; in particular, a convergence rate of $\mathcal{O}({h}^{2\alpha})$ ($ 0<\alpha<\pi/\omega$, $\omega > \pi$ is the angle of the reentrant corner) is proved for the linear finite element. Surprisingly, numerical evidence demonstrates a $\mathcal{O}({h}^{2})$ convergent rate for the quasi-uniform mesh with the regular refinement strategy even on non-convex polygonal domains.
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In this paper, we analyze the convergence of the operator-compressed multiscale finite element method (OC MsFEM) for Schr\"{o}dinger equations with general multiscale potentials in the semiclassical regime. In the OC MsFEM the multiscale basis functions are constructed by solving a constrained energy minimization. Under a mild assumption on the mesh size $H$, we prove the exponential decay of the multiscale basis functions so that localized multiscale basis functions can be constructed, which achieve the same accuracy as the global ones if the oversampling size $m = O(\log(1/H))$. We prove the first-order convergence in the energy norm and second-order convergence in the $L^2$ norm for the OC MsFEM and super convergence rates can be obtained if the solution possesses sufficiently high regularity. By analysing the regularity of the solution, we also derive the dependence of the error estimates on the small parameters of the Schr\"{o}dinger equation. We find that the OC MsFEM outperforms the finite element method (FEM) due to the super convergence behavior for high-regularity solutions and weaker dependence on the small parameters for low-regularity solutions in the presence of the multiscale potential. Finally, we present numerical results to demonstrate the accuracy and robustness of the OC MsFEM.
We propose a monotone discretization for the integral fractional Laplace equation on bounded Lipschitz domains with the homogenous Dirichlet boundary condition. The method is inspired by a quadrature-based finite difference method of Huang and Oberman, but is defined on unstructured grids in arbitrary dimensions with a more flexible domain for approximating singular integral. The scale of the singular integral domain not only depends on the local grid size, but also on the distance to the boundary, since the H\"{o}lder coefficient of the solution deteriorates as it approaches the boundary. By using a discrete barrier function that also reflects the distance to the boundary, we show optimal pointwise convergence rates in terms of the H\"{o}lder regularity of the data on both quasi-uniform and graded grids. Several numerical examples are provided to illustrate the sharpness of the theoretical results.
Trefftz methods are high-order Galerkin schemes in which all discrete functions are elementwise solution of the PDE to be approximated. They are viable only when the PDE is linear and its coefficients are piecewise constant. We introduce a 'quasi-Trefftz' discontinuous Galerkin method for the discretisation of the acoustic wave equation with piecewise-smooth wavespeed: the discrete functions are elementwise approximate PDE solutions. We show that the new discretisation enjoys the same excellent approximation properties as the classical Trefftz one, and prove stability and high-order convergence of the DG scheme. We introduce polynomial basis functions for the new discrete spaces and describe a simple algorithm to compute them. The technique we propose is inspired by the generalised plane waves previously developed for time-harmonic problems with variable coefficients; it turns out that in the case of the time-domain wave equation under consideration the quasi-Trefftz approach allows for polynomial basis functions.
When a physical system is modeled by a nonlinear function, the unknown parameters can be estimated by fitting experimental observations by a least-squares approach. Newton's method and its variants are often used to solve problems of this type. In this paper, we are concerned with the computation of the minimal-norm solution of an underdetermined nonlinear least-squares problem. We present a Gauss-Newton type method, which relies on two relaxation parameters to ensure convergence, and which incorporates a procedure to dynamically estimate the two parameters, as well as the rank of the Jacobian matrix, along the iterations. Numerical results are presented.
We develop a class of mixed virtual volume methods for elliptic problems on polygonal/polyhedral grids. Unlike the mixed virtual element methods introduced in \cite{brezzi2014basic,da2016mixed}, our methods are reduced to symmetric, positive definite problems for the primary variable without using Lagrangian multipliers. We start from the usual way of changing the given equation into a mixed system using the Darcy's law, $\bu=-{\cal K} \nabla p$. By integrating the system of equations with some judiciously chosen test spaces on each element, we define new mixed virtual volume methods of all orders. We show that these new schemes are equivalent to the nonconforming virtual element methods for the primal variable $p$. Once the primary variable is computed solving the symmetric, positive definite system, all the degrees of freedom for the Darcy velocity are locally computed. Also, the $L^2$-projection onto the polynomial space is easy to compute. Hence our work opens an easy way to compute Darcy velocity on the polygonal/polyhedral grids. For the lowest order case, we give a formula to compute a Raviart-Thomas space like representation which satisfies the conservation law. An optimal error analysis is carried out and numerical results are presented which support the theory.
In this paper, we study the probabilistic stability analysis of a subclass of stochastic hybrid systems, called the Planar Probabilistic Piecewise Constant Derivative Systems (Planar PPCD), where the continuous dynamics is deterministic, constant rate and planar, the discrete switching between the modes is probabilistic and happens at boundary of the invariant regions, and the continuous states are not reset during switching. These aptly model piecewise linear behaviors of planar robots. Our main result is an exact algorithm for deciding absolute and almost sure stability of Planar PPCD under some mild assumptions on mutual reachability between the states and the presence of non-zero probability self-loops. Our main idea is to reduce the stability problems on planar PPCD into corresponding problems on Discrete Time Markov Chains with edge weights. Our experimental results on planar robots with faulty angle actuator demonstrate the practical feasibility of this approach.
Recently, one of us proposed a quantum algorithm called belief propagation with quantum messages (BPQM) for decoding classical data encoded using a binary linear code with tree Tanner graph that is transmitted over a pure-state CQ channel [Renes, NJP 19 072001 (2017)]. This algorithm presents a genuine quantum counterpart to decoding based on classical belief propagation, which has found wide success in classical coding theory when used in conjunction with LDPC or Turbo codes. More recently Rengaswamy et al. [npj Quantum Information 7 97 (2021)] numerically observed that, for a small example code, BPQM implements the optimal decoder for determining the entire input codeword. Here we significantly expand the understanding, formalism, and applicability of the BPQM algorithm with the following contributions. First, we prove analytically that BPQM realizes optimal decoding for any binary linear code with tree Tanner graph. We also provide the first formal description of the BPQM algorithm in full detail and without any ambiguity. In so doing, we identify a key flaw overlooked in the original algorithm and subsequent works which implies quantum circuit realizations will be exponentially large in the code size. Although BPQM passes quantum messages, other information required by the algorithm is processed globally. We remedy this problem by formulating a truly message-passing algorithm which approximates BPQM and has circuit complexity $\mathcal{O}(\text{poly } n, \text{polylog } \frac{1}{\epsilon})$, where $n$ is the code length and $\epsilon$ is the approximation error. Finally, we also propose a novel method for extending BPQM to factor graphs containing cycles by making use of approximate cloning. We show some promising numerical results that indicate that BPQM on factor graphs with cycles can significantly outperform the best possible classical decoder.
We study the phase reconstruction of signals $f$ belonging to complex Gaussian shift-invariant spaces $V^\infty(\varphi)$ from spectrogram measurements $|\mathcal{G}f(X)|$ where $\mathcal{G}$ is the Gabor transform and $X \subseteq \mathbb{R}^2$. An explicit reconstruction formula will demonstrate that such signals can be recovered from measurements located on parallel lines in the time-frequency plane by means of a Riesz basis expansion. Moreover, connectedness assumptions on $|f|$ result in stability estimates in the situation where one aims to reconstruct $f$ on compact intervals. Driven by a recent observation that signals in Gaussian shift-invariant spaces are determined by lattice measurements [Grohs, P., Liehr, L., Injectivity of Gabor phase retrieval from lattice measurements, arXiv:2008.07238] we prove a sampling result on the stable approximation from finitely many spectrogram samples. The resulting algorithm provides a non-iterative, provably stable and convergent approximation technique. In addition, it constitutes a method of approximating signals in function spaces beyond $V^\infty(\varphi)$, such as Paley-Wiener spaces.
We consider boundary element methods where the Calder\'on projector is used for the system matrix and boundary conditions are weakly imposed using a particular variational boundary operator designed using techniques from augmented Lagrangian methods. Regardless of the boundary conditions, both the primal trace variable and the flux are approximated. We focus on the imposition of Dirichlet conditions on the Helmholtz equation, and extend the analysis of the Laplace problem from \emph{Boundary element methods with weakly imposed boundary conditions} to this case. The theory is illustrated by a series of numerical examples.
Under some regularity assumptions, we report an a priori error analysis of a dG scheme for the Poisson and Stokes flow problem in their dual mixed formulation. Both formulations satisfy a Babu\v{s}ka-Brezzi type condition within the space H(div) x L2. It is well known that the lowest order Crouzeix-Raviart element paired with piecewise constants satisfies such a condition on (broken) H1 x L2 spaces. In the present article, we use this pair. The continuity of the normal component is weakly imposed by penalizing jumps of the broken H(div) component. For the resulting methods, we prove well-posedness and convergence with constants independent of data and mesh size. We report error estimates in the methods natural norms and optimal local error estimates for the divergence error. In fact, our finite element solution shares for each triangle one DOF with the CR interpolant and the divergence is locally the best-approximation for any regularity. Numerical experiments support the findings and suggest that the other errors converge optimally even for the lowest regularity solutions and a crack-problem, as long as the crack is resolved by the mesh.
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