We present a hybrid method for time-dependent particle transport problems that combines Monte Carlo (MC) estimation with deterministic solutions based on discrete ordinates. For spatial discretizations, the MC algorithm computes a piecewise constant solution and the discrete ordinates uses bilinear discontinuous finite elements. From the hybridization of the problem, the resulting problem solved by Monte Carlo is scattering free, resulting in a simple, efficient solution procedure. Between time steps, we use a projection approach to ``relabel'' collided particles as uncollided particles. From a series of standard 2-D Cartesian test problems we observe that our hybrid method has improved accuracy and reduction in computational complexity of approximately an order of magnitude relative to standard discrete ordinates solutions.
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This manuscript summarizes the outcome of the focus groups at "The f(A)bulous workshop on matrix functions and exponential integrators", held at the Max Planck Institute for Dynamics of Complex Technical Systems in Magdeburg, Germany, on 25-27 September 2023. There were three focus groups in total, each with a different theme: knowledge transfer, high-performance and energy-aware computing, and benchmarking. We collect insights, open issues, and perspectives from each focus group, as well as from general discussions throughout the workshop. Our primary aim is to highlight ripe research directions and continue to build on the momentum from a lively meeting.
We design in this work a discrete de Rham complex on manifolds. This complex, written in the framework of exterior calculus, is applicable on meshes on the manifold with generic elements, and has the same cohomology as the continuous de Rham complex. Notions of local (full and trimmed) polynomial spaces are developed, with compatibility requirements between polynomials on mesh entities of various dimensions. Explicit examples of polynomials spaces are presented. The discrete de Rham complex is then used to set up a scheme for the Maxwell equations on a 2D manifold without boundary, and we show that a natural discrete version of the constraint linking the electric field and the electric charge density is satisfied. Numerical examples are provided on the sphere and the torus, based on a bespoke analytical solution and mesh design on each manifold.
We investigate the global existence of a solution for the stochastic fractional nonlinear Schr\"odinger equation with radially symmetric initial data in a suitable energy space $H^{\alpha}$. Using a variational principle, we demonstrate that the stochastic fractional nonlinear Schr\"odinger equation in the Stratonovich sense forms an infinite-dimensional stochastic Hamiltonian system, with its phase flow preserving symplecticity. We develop a structure-preserving algorithm for the stochastic fractional nonlinear Schr\"odinger equation from the perspective of symplectic geometry. It is established that the stochastic midpoint scheme satisfies the corresponding symplectic law in the discrete sense. Furthermore, since the midpoint scheme is implicit, we also develop a more effective mass-preserving splitting scheme. Consequently, the convergence order of the splitting scheme is shown to be $1$. Two numerical examples are conducted to validate the efficiency of the theory.
We derive bounds on the moduli of the eigenvalues of special type of matrix rational functions using the following techniques/methods: (1) the Bauer-Fike theorem on an associated block matrix of the given matrix rational function, (2) by associating a real rational function, along with Rouch$\text{\'e}$ theorem for the matrix rational function and (3) by a numerical radius inequality for a block matrix for the matrix rational function. These bounds are compared when the coefficients are unitary matrices. Numerical examples are given to illustrate the results obtained.
We present new Neumann-Neumann algorithms based on a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semi-discretization, the Lagrange multiplier approach provides a coupled forward-backward optimality system, which can be solved using a time domain decomposition. Due to the forward-backward structure of the optimality system, nine variants can be found for the Neumann-Neumann algorithms. We analyze their convergence behavior and determine the optimal relaxation parameter for each algorithm. Our analysis reveals that the most natural algorithms are actually only good smoothers, and there are better choices which lead to efficient solvers. We illustrate our analysis with numerical experiments.
This paper develops a class of robust weak Galerkin methods for the stationary incompressible convective Brinkman-Forchheimer equations. The methods adopt piecewise polynomials of degrees $m\ (m\geq1)$ and $m-1$ respectively for the approximations of velocity and pressure variables inside the elements and piecewise polynomials of degrees $k \ ( k=m-1,m)$ and $m$ respectively for their numerical traces on the interfaces of elements, and are shown to yield globally divergence-free velocity approximation. Existence and uniqueness results for the discrete schemes, as well as optimal a priori error estimates, are established. A convergent linearized iterative algorithm is also presented. Numerical experiments are provided to verify the performance of the proposed methods
This manuscript examines the problem of nonlinear stochastic fractional neutral integro-differential equations with weakly singular kernels. Our focus is on obtaining precise estimates to cover all possible cases of Abel-type singular kernels. Initially, we establish the existence, uniqueness, and continuous dependence on the initial value of the true solution, assuming a local Lipschitz condition and linear growth condition. Additionally, we develop the Euler-Maruyama method for the numerical solution of the equation and prove its strong convergence under the same conditions as the well-posedness. Moreover, we determine the accurate convergence rate of this method under global Lipschitz conditions and linear growth conditions. And also we have proven generalized Gronwall inequality with a multi-weakly singularity.
It is known that standard stochastic Galerkin methods encounter challenges when solving partial differential equations with high-dimensional random inputs, which are typically caused by the large number of stochastic basis functions required. It becomes crucial to properly choose effective basis functions, such that the dimension of the stochastic approximation space can be reduced. In this work, we focus on the stochastic Galerkin approximation associated with generalized polynomial chaos (gPC), and explore the gPC expansion based on the analysis of variance (ANOVA) decomposition. A concise form of the gPC expansion is presented for each component function of the ANOVA expansion, and an adaptive ANOVA procedure is proposed to construct the overall stochastic Galerkin system. Numerical results demonstrate the efficiency of our proposed adaptive ANOVA stochastic Galerkin method for both diffusion and Helmholtz problems.
In arXiv:2305.03945 [math.NA], a first-order optimization algorithm has been introduced to solve time-implicit schemes of reaction-diffusion equations. In this research, we conduct theoretical studies on this first-order algorithm equipped with a quadratic regularization term. We provide sufficient conditions under which the proposed algorithm and its time-continuous limit converge exponentially fast to a desired time-implicit numerical solution. We show both theoretically and numerically that the convergence rate is independent of the grid size, which makes our method suitable for large-scale problems. The efficiency of our algorithm has been verified via a series of numerical examples conducted on various types of reaction-diffusion equations. The choice of optimal hyperparameters as well as comparisons with some classical root-finding algorithms are also discussed in the numerical section.
We introduce a new model which can be considered as a extended version of the Hawkes process in a discrete sense. This model enables the integration of various residual distributions while preserving the fundamental properties of the original Hawkes process. The rich nature of this model enables a filtered historical simulation which incorporate the properties of original time series more accurately. The process naturally extends to multi-variate models with easy implementations of estimation and simulation. We investigate the effect of flexible residual distribution on estimation of high frequency financial data compared with the Hawkes process.
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