The transparent boundary condition for the free Schr\"{o}dinger equation on a rectangular computational domain requires implementation of an operator of the form $\sqrt{\partial_t-i\triangle_{\Gamma}}$ where $\triangle_{\Gamma}$ is the Laplace-Beltrami operator. It is known that this operator is nonlocal in time as well as space which poses a significant challenge in developing an efficient numerical method of solution. The computational complexity of the existing methods scale with the number of time-steps which can be attributed to the nonlocal nature of the boundary operator. In this work, we report an effectively local approximation for the boundary operator such that the resulting complexity remains independent of number of time-steps. At the heart of this algorithm is a Pad\'e approximant based rational approximation of certain fractional operators that handles corners of the domain adequately. For the spatial discretization, we use a Legendre-Galerkin spectral method with a new boundary adapted basis which ensures that the resulting linear system is banded. A compatible boundary-lifting procedure is also presented which accommodates the segments as well as the corners on the boundary. The proposed novel scheme can be implemented within the framework of any one-step time marching schemes. In particular, we demonstrate these ideas for two one-step methods, namely, the backward-differentiation formula of order 1 (BDF1) and the trapezoidal rule (TR). For the sake of comparison, we also present a convolution quadrature based scheme conforming to the one-step methods which is computationally expensive but serves as a golden standard. Finally, several numerical tests are presented to demonstrate the effectiveness of our novel method as well as to verify the order of convergence empirically.
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We provide an algorithm for deciding simple grammar bisimilarity whose complexity is polynomial in the valuation of the grammar (maximum seminorm among production rules). Since the valuation is at most exponential in the size of the grammar, this gives rise to a single-exponential running time. Previously only a doubly-exponential algorithm was known. As an application, we provide a conversion from context-free session types to simple grammars whose valuation is linear in the size of the type. In this way, we provide the first polynomial-time algorithm for deciding context-free session type equivalence.
In this paper, we effectively solve the inverse source problem of the fractional Poisson equation using MC-fPINNs. We construct two neural networks $ u_{NN}(x;\theta )$ and $f_{NN}(x;\psi)$ to approximate the solution $u^{*}(x)$ and the forcing term $f^{*}(x)$ of the fractional Poisson equation. To optimize these two neural networks, we use the Monte Carlo sampling method mentioned in MC-fPINNs and define a new loss function combining measurement data and the underlying physical model. Meanwhile, we present a comprehensive error analysis for this method, along with a prior rule to select the appropriate parameters of neural networks. Several numerical examples are given to demonstrate the great precision and robustness of this method in solving high-dimensional problems up to 10D, with various fractional order $\alpha$ and different noise levels of the measurement data ranging from 1$\%$ to 10$\%$.
In this work, we use the monolithic convex limiting (MCL) methodology to enforce relevant inequality constraints in implicit finite element discretizations of the compressible Euler equations. In this context, preservation of invariant domains follows from positivity preservation for intermediate states of the density and internal energy. To avoid spurious oscillations, we additionally impose local maximum principles on intermediate states of the density, velocity components, and specific total energy. For the backward Euler time stepping, we show the invariant domain preserving (IDP) property of the fully discrete MCL scheme by constructing a fixed-point iteration that is IDP and converges under a strong time step restriction. Our iterative solver for the nonlinear discrete problem employs a more efficient fixed-point iteration. The matrix of the associated linear system is a robust low-order Jacobian approximation that exploits the homogeneity property of the flux function. The limited antidiffusive terms are treated explicitly. We use positivity preservation as a stopping criterion for nonlinear iterations. The first iteration yields the solution of a linearized semi-implicit problem. This solution possesses the discrete conservation property but is generally not IDP. Further iterations are performed if any non-IDP states are detected. The existence of an IDP limit is guaranteed by our analysis. To facilitate convergence to steady-state solutions, we perform adaptive explicit underrelaxation at the end of each time step. The calculation of appropriate relaxation factors is based on an approximate minimization of nodal entropy residuals. The performance of proposed algorithms and alternative solution strategies is illustrated by the convergence history for standard two-dimensional test problems.
This work explores multi-modal inference in a high-dimensional simplified model, analytically quantifying the performance gain of multi-modal inference over that of analyzing modalities in isolation. We present the Bayes-optimal performance and weak recovery thresholds in a model where the objective is to recover the latent structures from two noisy data matrices with correlated spikes. The paper derives the approximate message passing (AMP) algorithm for this model and characterizes its performance in the high-dimensional limit via the associated state evolution. The analysis holds for a broad range of priors and noise channels, which can differ across modalities. The linearization of AMP is compared numerically to the widely used partial least squares (PLS) and canonical correlation analysis (CCA) methods, which are both observed to suffer from a sub-optimal recovery threshold.
Determining the complexity of computing Gr\"{o}bner bases is an important problem both in theory and in practice, and for that the solving degree plays a key role. In this paper, we study the solving degrees of affine semi-regular sequences and their homogenized sequences. Some of our results are considered to give mathematically rigorous proofs of the correctness of methods for computing Gr\"{o}bner bases of the ideal generated by an affine semi-regular sequence. This paper is a sequel of the authors' previous work and gives additional results on the solving degrees and important behaviors of Gr\"obner basis computation. We also define the generalized degree of regularity for a sequence of homogeneous polynomials. For the homogenization of an affine semi-regular sequence, we relate its generalized degree of regularity with its maximal Gr\"{o}bner basis degree (i.e., the solving degree of the homogenized sequence). The definition of a generalized (cryptographic) semi-regular sequence is also given, and it derives a new cryptographic assumption to estimate the security of cryptosystems and signature schemes. From our experimental observation, we raise a conjecture and some questions related to this generalized semi-regularity. These new definitions and our results provide a theoretical formulation of (somehow heuristic) discussions done so far in the cryptographic community.
Highly resolved finite element simulations of a laser beam welding process are considered. The thermomechanical behavior of this process is modeled with a set of thermoelasticity equations resulting in a nonlinear, nonsymmetric saddle point system. Newton's method is used to solve the nonlinearity and suitable domain decomposition preconditioners are applied to accelerate the convergence of the iterative method used to solve all linearized systems. Since a onelevel Schwarz preconditioner is in general not scalable, a second level has to be added. Therefore, the construction and numerical analysis of a monolithic, twolevel overlapping Schwarz approach with the GDSW (Generalized Dryja-Smith-Widlund) coarse space and an economic variant thereof are presented here.
Efficiently enumerating all the extreme points of a polytope identified by a system of linear inequalities is a well-known challenge issue.We consider a special case and present an algorithm that enumerates all the extreme points of a bisubmodular polyhedron in $\mathcal{O}(n^4|V|)$ time and $\mathcal{O}(n^2)$ space complexity, where $ n$ is the dimension of underlying space and $V$ is the set of outputs. We use the reverse search and signed poset linked to extreme points to avoid the redundant search. Our algorithm is a generalization of enumerating all the extreme points of a base polyhedron which comprises some combinatorial enumeration problems.
A finite element method is introduced to track interface evolution governed by the level set equation. The method solves for the level set indicator function in a narrow band around the interface. An extension procedure, which is essential for a narrow band level set method, is introduced based on a finite element $L^2$- or $H^1$-projection combined with the ghost-penalty method. This procedure is formulated as a linear variational problem in a narrow band around the surface, making it computationally efficient and suitable for rigorous error analysis. The extension method is combined with a discontinuous Galerkin space discretization and a BDF time-stepping scheme. The paper analyzes the stability and accuracy of the extension procedure and evaluates the performance of the resulting narrow band finite element method for the level set equation through numerical experiments.
We present difference schemes for stochastic transport equations with low-regularity velocity fields. We establish $L^2$ stability and convergence of the difference approximations under conditions that are less strict than those required for deterministic transport equations. The $L^2$ estimate, crucial for the analysis, is obtained through a discrete duality argument and a comprehensive examination of a class of backward parabolic difference schemes.
We propose a new full discretization of the Biot's equations in poroelasticity. The construction is driven by the inf-sup theory, which we recently developed. It builds upon the four-field formulation of the equations obtained by introducing the total pressure and the total fluid content. We discretize in space with Lagrange finite elements and in time with backward Euler. We establish inf-sup stability and quasi-optimality of the proposed discretization, with robust constants with respect to all material parameters. We further construct an interpolant showing how the error decays for smooth solutions.
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