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We consider the Cauchy problem for the Helmholtz equation with a domain in R^d, d>2 with N cylindrical outlets to infinity with bounded inclusions in R^{d-1}. Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Maz'ya proposed an alternating iterative method for solving Cauchy problems associated with elliptic,self-adjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Mpinganzima et al. for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in R^2 that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters, the Robin-Dirichlet alternating iterative procedure is convergent.

Two novel parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are here constructed, analyzed and tested numerically for implicit time discretizations of the three-dimensional Bidomain system of equations. This model represents the most advanced mathematical description of the cardiac bioelectrical activity and it consists of a degenerate system of two non-linear reaction-diffusion partial differential equations (PDEs), coupled with a stiff system of ordinary differential equations (ODEs). A finite element discretization in space and a segregated implicit discretization in time, based on decoupling the PDEs from the ODEs, yields at each time step the solution of a non-linear algebraic system. The Jacobian linear system at each Newton iteration is solved by a Krylov method, accelerated by BDDC or FETI-DP preconditioners, both augmented with the recently introduced {\em deluxe} scaling of the dual variables. A polylogarithmic convergence rate bound is proven for the resulting parallel Bidomain solvers. Extensive numerical experiments on linux clusters up to two thousands processors confirm the theoretical estimates, showing that the proposed parallel solvers are scalable and quasi-optimal.

The biharmonic equation with Dirichlet and Neumann boundary conditions discretized using the mixed finite element method and piecewise linear (with the possible exception of boundary triangles) finite elements on triangular elements has been well-studied for domains in R2. Here we study the analogous problem on polyhedral surfaces. In particular, we provide a convergence proof of discrete solutions to the corresponding smooth solution of the biharmonic equation. We obtain convergence rates that are identical to the ones known for the planar setting. Our proof focuses on three different problems: solving the biharmonic equation on the surface, solving the biharmonic equation in a discrete space in the metric of the surface, and solving the biharmonic equation in a discrete space in the metric of the polyhedral approximation of the surface. We employ inverse discrete Laplacians to bound the error between the solutions of the two discrete problems, and generalize a flat strategy to bound the remaining error between the discrete solutions and the exact solution on the curved surface.

We employ kernel-based approaches that use samples from a probability distribution to approximate a Kolmogorov operator on a manifold. The self-tuning variable-bandwidth kernel method [Berry & Harlim, Appl. Comput. Harmon. Anal., 40(1):68--96, 2016] computes a large, sparse matrix that approximates the differential operator. Here, we use the eigendecomposition of the discretization to (i) invert the operator, solving a differential equation, and (ii) represent gradient vector fields on the manifold. These methods only require samples from the underlying distribution and, therefore, can be applied in high dimensions or on geometrically complex manifolds when spatial discretizations are not available. We also employ an efficient $k$-$d$ tree algorithm to compute the sparse kernel matrix, which is a computational bottleneck.

Population dynamics is the study of temporal and spatial variation in the size of populations of organisms and is a major part of population ecology. One of the main difficulties in analyzing population dynamics is that we can only obtain observation data with coarse time intervals from fixed-point observations due to experimental costs or other constraints. Recently, modeling population dynamics by using continuous normalizing flows (CNFs) and dynamic optimal transport has been proposed to infer the expected trajectory of samples from a fixed-point observed population. While the sample behavior in CNF is deterministic, the actual sample in biological systems moves in an essentially random yet directional manner. Moreover, when a sample moves from point A to point B in dynamical systems, its trajectory is such that the corresponding action has the smallest possible value, known as the principle of least action. To satisfy these requirements of the sample trajectories, we formulate the Lagrangian Schr\"odinger bridge (LSB) problem and propose to solve it approximately using neural SDE with regularization. We also develop a model architecture that enables faster computation. Our experiments show that our solution to the LSB problem can approximate the dynamics at the population level and that using the prior knowledge introduced by the Lagrangian enables us to estimate the trajectories of individual samples with stochastic behavior.

In this paper we get error bounds for fully discrete approximations of infinite horizon problems via the dynamic programming approach. It is well known that considering a time discretization with a positive step size $h$ an error bound of size $h$ can be proved for the difference between the value function (viscosity solution of the Hamilton-Jacobi-Bellman equation corresponding to the infinite horizon) and the value function of the discrete time problem. However, including also a spatial discretization based on elements of size $k$ an error bound of size $O(k/h)$ can be found in the literature for the error between the value functions of the continuous problem and the fully discrete problem. In this paper we revise the error bound of the fully discrete method and prove, under similar assumptions to those of the time discrete case, that the error of the fully discrete case is in fact $O(h+k)$ which gives first order in time and space for the method. This error bound matches the numerical experiments of many papers in the literature in which the behaviour $1/h$ from the bound $O(k/h)$ have not been observed.

This paper makes the first attempt to apply newly developed upwind GFDM for the meshless solution of two-phase porous flow equations. In the presented method, node cloud is used to flexibly discretize the computational domain, instead of complicated mesh generation. Combining with moving least square approximation and local Taylor expansion, spatial derivatives of oil-phase pressure at a node are approximated by generalized difference operators in the local influence domain of the node. By introducing the first-order upwind scheme of phase relative permeability, and combining the discrete boundary conditions, fully-implicit GFDM-based nonlinear discrete equations of the immiscible two-phase porous flow are obtained and solved by the nonlinear solver based on the Newton iteration method with the automatic differentiation, to avoid the additional computational cost and possible computational instability caused by sequentially coupled scheme. Two numerical examples are implemented to test the computational performances of the presented method. Detailed error analysis finds the two sources of the calculation error, roughly studies the convergence order thus find that the low-order error of GFDM makes the convergence order of GFDM lower than that of FDM when node spacing is small, and points out the significant effect of the symmetry or uniformity of the node collocation in the node influence domain on the accuracy of generalized difference operators, and the radius of the node influence domain should be small to achieve high calculation accuracy, which is a significant difference between the studied hyperbolic two-phase porous flow problem and the elliptic problems when GFDM is applied.

We study the numerical approximation by space-time finite element methods of a multi-physics system coupling hyperbolic elastodynamics with parabolic transport and modelling poro- and thermoelasticity. The equations are rewritten as a first-order system in time. Discretizations by continuous Galerkin methods in space and time with inf-sup stable pairs of finite elements for the spatial approximation of the unknowns are investigated. Optimal order error estimates of energy-type are proven. Superconvergence at the time nodes is addressed briefly. The error analysis can be extended to discontinuous and enriched Galerkin space discretizations. The error estimates are confirmed by numerical experiments.

The minimum energy path (MEP) describes the mechanism of reaction, and the energy barrier along the path can be used to calculate the reaction rate in thermal systems. The nudged elastic band (NEB) method is one of the most commonly used schemes to compute MEPs numerically. It approximates an MEP by a discrete set of configuration images, where the discretization size determines both computational cost and accuracy of the simulations. In this paper, we consider a discrete MEP to be a stationary state of the NEB method and prove an optimal convergence rate of the discrete MEP with respect to the number of images. Numerical simulations for the transitions of some several proto-typical model systems are performed to support the theory.

In this paper, a third order gas kinetic scheme is developed on the three dimensional hybrid unstructured meshes for the numerical simulation of compressible inviscid and viscous flows. A third-order WENO reconstruction is developed on the hybrid unstructured meshes, including tetrahedron, pyramid, prism and hexahedron. A simple strategy is adopted for the selection of big stencil and sub-stencils. Incorporate with the two-stage fourth-order temporal discretization and lower-upper symmetric Gauss-Seidel methods, both explicit and implicit high-order gas-kinetic schemes are developed. A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy and robustness for both inviscid and viscous flows.