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Despite the many advances in the use of weakly-compressible smoothed particle hydrodynamics (SPH) for the simulation of incompressible fluid flow, it is still challenging to obtain second-order convergence numerically. In this paper we perform a systematic numerical study of convergence and accuracy of kernel-based approximation, discretization operators, and weakly-compressible SPH (WCSPH) schemes. We explore the origins of the errors and issues preventing second-order convergence. Based on the study, we propose several new variations of the basic WCSPH scheme that are all second-order accurate. Additionally, we investigate the linear and angular momentum conservation property of the WCSPH schemes. Our results show that one may construct accurate WCSPH schemes that demonstrate second-order convergence through a judicious choice of kernel, smoothing length, and discretization operators in the discretization of the governing equations.

This work provides a theoretical framework for the pose estimation problem using total least squares for vector observations from landmark features. First, the optimization framework is formulated with observation vectors extracted from point cloud features. Then, error-covariance expressions are derived. The attitude and position solutions obtained via the derived optimization framework are proven to reach the bounds defined by the Cram\'er-Rao lower bound under the small-angle approximation of attitude errors. The measurement data for the simulation of this problem is provided through a series of vector observation scans, and a fully populated observation noise-covariance matrix is assumed as the weight in the cost function to cover the most general case of the sensor uncertainty. Here, previous derivations are expanded for the pose estimation problem to include more generic correlations in the errors than previous cases involving an isotropic noise assumption. The proposed solution is simulated in a Monte-Carlo framework to validate the error-covariance analysis.

This study proposes an efficient Newton-type method for the optimal control of switched systems under a given mode sequence. A mesh-refinement-based approach is utilized to discretize continuous-time optimal control problems (OCPs) using the direct multiple-shooting method to formulate a nonlinear program (NLP), which guarantees the local convergence of a Newton-type method. A dedicated structure-exploiting algorithm (Riccati recursion) is proposed that efficiently performs a Newton-type method for the NLP because its sparsity structure is different from a standard OCP. The proposed method computes each Newton step with linear time-complexity for the total number of discretization grids as the standard Riccati recursion algorithm. Additionally, it can always solve the Karush-Kuhn-Tucker (KKT) systems arising in the Newton-type method if the solution is sufficiently close to a local minimum. Conversely, general quadratic programming (QP) solvers cannot accomplish this because the Hessian matrix is inherently indefinite. Moreover, a modification on the reduced Hessian matrix is proposed using the nature of the Riccati recursion algorithm as the dynamic programming for a QP subproblem to enhance the convergence. A numerical comparison is conducted with off-the-shelf NLP solvers, which demonstrates that the proposed method is up to two orders of magnitude faster. Whole-body optimal control of quadrupedal gaits is also demonstrated and shows that the proposed method can achieve the whole-body model predictive control (MPC) of robotic systems with rigid contacts.

We present an advection-pressure flux-vector splitting method for the one and two- dimensional shallow water equations following the approach first proposed by Toro and V\'azquez for the compressible Euler equations. The resulting first-order schemes turn out to be exceedingly simple, with accuracy and robustness comparable to that of the sophisticated Godunov upwind method used in conjunction with complete non- linear Riemann solvers. The technique splits the full system into two subsystems, namely an advection system and a pressure system. The sought numerical flux results from fluxes for each of the subsystems. The basic methodology, extended on 2D unstructured meshes, constitutes the building block for the construction of numerical schemes of very high order of accuracy following the ADER approach. The presented numerical schemes are systematically assessed on a carefully selected suite of test problems with reference solutions, in one and two space dimensions.The applicabil- ity of the schemes is illustrated through simulations of tsunami wave propagation in the Pacific Ocean.

We show, that the complex step approximation $\mathrm{Im}(f(A+ihE))/h$ to the Fr\'echet derivative of matrix functions $f:\mathbb{R}^{m,n}\rightarrow\mathbb{R}^{m,n}$ is applicable to the matrix sign, square root and polar mapping using iterative schemes. While this property was already discovered for the matrix sign using Newtons method, we extend the research to the family of Pad\'e iterations, that allows us to introduce iterative schemes for finding function and derivative values while approximately preserving automorphism group structure.

In this paper we consider the spatial semi-discretization of conservative PDEs. Such finite dimensional approximations of infinite dimensional dynamical systems can be described as flows in suitable matrix spaces, which in turn leads to the need to solve polynomial matrix equations, a classical and important topic both in theoretical and in applied mathematics. Solving numerically these equations is challenging due to the presence of several conservation laws which our finite models incorporate and which must be retained while integrating the equations of motion. In the last thirty years, the theory of geometric integration has provided a variety of techniques to tackle this problem. These numerical methods require to solve both direct and inverse problems in matrix spaces. We present two algorithms to solve a cubic matrix equation arising in the geometric integration of isospectral flows. This type of ODEs includes finite models of ideal hydrodynamics, plasma dynamics, and spin particles, which we use as test problems for our algorithms.

In this paper, we establish error estimates for the numerical approximation of the parabolic optimal control problem with measure data in a two-dimensional nonconvex polygonal domain. Due to the presence of measure data in the state equation and the nonconvex nature of the domain, the finite element error analysis is not straightforward. Regularity results for the control problem based on the first-order optimality system are discussed. The state variable and co-state variable are approximated by continuous piecewise linear finite element, and the control variable is approximated by piecewise constant functions. A priori error estimates for the state and control variable are derived for spatially discrete control problem and fully discrete control problem in $L^2(L^2)$-norm. A numerical experiment is performed to illustrate our theoretical findings.

In this paper, we introduce the Hessian-Schatten total-variation (HTV) -- a novel seminorm that quantifies the total "rugosity" of multivariate functions. Our motivation for defining HTV is to assess the complexity of supervised learning schemes. We start by specifying the adequate matrix-valued Banach spaces that are equipped with suitable classes of mixed-norms. We then show that HTV is invariant to rotations, scalings, and translations. Additionally, its minimum value is achieved for linear mappings, supporting the common intuition that linear regression is the least complex learning model. Next, we present closed-form expressions for computing the HTV of two general classes of functions. The first one is the class of Sobolev functions with a certain degree of regularity, for which we show that HTV coincides with the Hessian-Schatten seminorm that is sometimes used as a regularizer for image reconstruction. The second one is the class of continuous and piecewise linear (CPWL) functions. In this case, we show that the HTV reflects the total change in slopes between linear regions that have a common facet. Hence, it can be viewed as a convex relaxation (l1-type) of the number of linear regions (l0-type) of CPWL mappings. Finally, we illustrate the use of our proposed seminorm with some concrete examples.

In this paper, we develop a Monte Carlo algorithm named the Frozen Gaussian Sampling (FGS) to solve the semiclassical Schr\"odinger equation based on the frozen Gaussian approximation. Due to the highly oscillatory structure of the wave function, traditional mesh-based algorithms suffer from "the curse of dimensionality", which gives rise to more severe computational burden when the semiclassical parameter \(\ep\) is small. The Frozen Gaussian sampling outperforms the existing algorithms in that it is mesh-free in computing the physical observables and is suitable for high dimensional problems. In this work, we provide detailed procedures to implement the FGS for both Gaussian and WKB initial data cases, where the sampling strategies on the phase space balance the need of variance reduction and sampling convenience. Moreover, we rigorously prove that, to reach a certain accuracy, the number of samples needed for the FGS is independent of the scaling parameter \(\ep\). Furthermore, the complexity of the FGS algorithm is of a sublinear scaling with respect to the microscopic degrees of freedom and, in particular, is insensitive to the dimension number. The performance of the FGS is validated through several typical numerical experiments, including simulating scattering by the barrier potential, formation of the caustics and computing the high-dimensional physical observables without mesh.

This article is concerned with the nonconforming finite element method for distributed elliptic optimal control problems with pointwise constraints on the control and gradient of the state variable. We reduce the minimization problem into a pure state constraint minimization problem. In this case, the solution of the minimization problem can be characterized as fourth-order elliptic variational inequalities of the first kind. To discretize the control problem we have used the bubble enriched Morley finite element method. To ensure the existence of the solution to discrete problems three bubble functions corresponding to the mean of the edge are added to the discrete space. We derive the error in the state variable in $H^2$-type energy norm. Numerical results are presented to illustrate our analytical findings.