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The Bochner integral is a generalization of the Lebesgue integral, for functions taking their values in a Banach space. Therefore, both its mathematical definition and its formalization in the Coq proof assistant are more challenging as we cannot rely on the properties of real numbers. Our contributions include an original formalization of simple functions, Bochner integrability defined by a dependent type, and the construction of the proof of the integrability of measurable functions under mild hypotheses (weak separability). Then, we define the Bochner integral and prove several theorems, including dominated convergence and the equivalence with an existing formalization of Lebesgue integral for nonnegative functions.

A contiguous area cartogram is a geographic map in which the area of each region is rescaled to be proportional to numerical data (e.g., population size) while keeping neighboring regions connected. Few studies have investigated whether readers can make accurate quantitative assessments using contiguous area cartograms. Therefore, we conducted an experiment to determine the accuracy, speed, and confidence with which readers infer numerical data values for the mapped regions. We investigated whether including an area-to-value legend (in the form of a square symbol next to the value represented by the square's area) makes it easier for map readers to estimate magnitudes. We also evaluated the effectiveness of two additional features: grid lines and an interactive area-to-value legend that allows participants to select the value represented by the square. Without any legends and only informed about the total numerical value represented by the whole cartogram, the distribution of estimates for individual regions was centered near the true value with substantial spread. Selectable legends with grid lines significantly reduced the spread but led to a tendency to underestimate the values. When comparing differences between regions or between cartograms, legends and grid lines made estimation slower but not more accurate. However, legends and grid lines made it more likely that participants completed the tasks. We recommend considering the cartogram's use case and purpose before deciding whether to include grid lines or an interactive legend.

In this paper we develop a new simple and effective isogeometric analysis for modeling thermal buckling of stiffened laminated composite plates with cutouts using level sets. We employ a first order shear deformation theory to approximate the displacement field of the stiffeners and the plate. Numerical modeling with a treatment of trimmed objects, such as internal cutouts in terms of NURBS-based isogeometric analysis presents several challenges, primarily due to need for using the tensor product of the NURBS basis functions. Due to this feature, the refinement operations can only be performed globally on the domain and not locally around the cutout. The new approach can overcome the drawbacks in modeling complex geometries with multiple-patches as the level sets are used to describe the internal cutouts; while the numerical integration is used only inside the physical domain. Results of parametric studies are presented which show the influence of ply orientation, size and orientation of the cutout and the position and profile of the curvilinear stiffeners. The numerical examples show high reliability and efficiency of the present method compared with other published solutions and ABAQUS.

In this article we investigate a system of geometric evolution equations describing a curvature driven motion of a family of 3D curves in the normal and binormal directions. Evolving curves may be subject of mutual interactions having both local or nonlocal character where the entire curve may influence evolution of other curves. Such an evolution and interaction can be found in applications. We explore the direct Lagrangian approach for treating the geometric flow of such interacting curves. Using the abstract theory of nonlinear analytic semi-flows, we are able to prove local existence, uniqueness and continuation of classical H\"older smooth solutions to the governing system of nonlinear parabolic equations. Using the finite volume method, we construct an efficient numerical scheme solving the governing system of nonlinear parabolic equations. Additionally, a nontrivial tangential velocity is considered allowing for redistribution of discretization nodes. We also present several computational studies of the flow combining the normal and binormal velocity and considering nonlocal interactions.

Optimal experimental design (OED) plays an important role in the problem of identifying uncertainty with limited experimental data. In many applications, we seek to minimize the uncertainty of a predicted quantity of interest (QoI) based on the solution of the inverse problem, rather than the inversion model parameter itself. In these scenarios, we develop an efficient method for goal-oriented optimal experimental design (GOOED) for large-scale Bayesian linear inverse problem that finds sensor locations to maximize the expected information gain (EIG) for a predicted QoI. By deriving a new formula to compute the EIG, exploiting low-rank structures of two appropriate operators, we are able to employ an online-offline decomposition scheme and a swapping greedy algorithm to maximize the EIG at a cost measured in model solutions that is independent of the problem dimensions. We provide detailed error analysis of the approximated EIG, and demonstrate the efficiency, accuracy, and both data- and parameter-dimension independence of the proposed algorithm for a contaminant transport inverse problem with infinite-dimensional parameter field.

Advanced finite-element discretizations and preconditioners for models of poroelasticity have attracted significant attention in recent years. The equations of poroelasticity offer significant challenges in both areas, due to the potentially strong coupling between unknowns in the system, saddle-point structure, and the need to account for wide ranges of parameter values, including limiting behavior such as incompressible elasticity. This paper was motivated by an attempt to develop monolithic multigrid preconditioners for the discretization developed in [48]; we show here why this is a difficult task and, as a result, we modify the discretization in [48] through the use of a reduced quadrature approximation, yielding a more "solver-friendly" discretization. Local Fourier analysis is used to optimize parameters in the resulting monolithic multigrid method, allowing a fair comparison between the performance and costs of methods based on Vanka and Braess-Sarazin relaxation. Numerical results are presented to validate the LFA predictions and demonstrate efficiency of the algorithms. Finally, a comparison to existing block-factorization preconditioners is also given.

For the general class of residual distribution (RD) schemes, including many finite element (such as continuous/discontinuous Galerkin) and flux reconstruction methods, an approach to construct entropy conservative/ dissipative semidiscretizations by adding suitable correction terms has been proposed by Abgrall (J.~Comp.~Phys. 372: pp. 640--666, 2018). In this work, the correction terms are characterized as solutions of certain optimization problems and are adapted to the SBP-SAT framework, focusing on discontinuous Galerkin methods. Novel generalizations to entropy inequalities, multiple constraints, and kinetic energy preservation for the Euler equations are developed and tested in numerical experiments. For all of these optimization problems, explicit solutions are provided. Additionally, the correction approach is applied for the first time to obtain a fully discrete entropy conservative/dissipative RD scheme. Here, the application of the deferred correction (DeC) method for the time integration is essential. This paper can be seen as describing a systematic method to construct structure preserving discretization, at least for the considered example.

Motivated by the goal of achieving long-term drift-free camera pose estimation in complex scenarios, we propose a global positioning framework fusing visual, inertial and Global Navigation Satellite System (GNSS) measurements in multiple layers. Different from previous loosely- and tightly- coupled methods, the proposed multi-layer fusion allows us to delicately correct the drift of visual odometry and keep reliable positioning while GNSS degrades. In particular, local motion estimation is conducted in the inner-layer, solving the problem of scale drift and inaccurate bias estimation in visual odometry by fusing the velocity of GNSS, pre-integration of Inertial Measurement Unit (IMU) and camera measurement in a tightly-coupled way. The global localization is achieved in the outer-layer, where the local motion is further fused with GNSS position and course in a long-term period in a loosely-coupled way. Furthermore, a dedicated initialization method is proposed to guarantee fast and accurate estimation for all state variables and parameters. We give exhaustive tests of the proposed framework on indoor and outdoor public datasets. The mean localization error is reduced up to 63%, with a promotion of 69% in initialization accuracy compared with state-of-the-art works. We have applied the algorithm to Augmented Reality (AR) navigation, crowd sourcing high-precision map update and other large-scale applications.

In this paper we discuss a reduced basis method for linear evolution PDEs, which is based on the application of the Laplace transform. The main advantage of this approach consists in the fact that, differently from time stepping methods, like Runge-Kutta integrators, the Laplace transform allows to compute the solution directly at a given instant, which can be done by approximating the contour integral associated to the inverse Laplace transform by a suitable quadrature formula. In terms of the reduced basis methodology, this determines a significant improvement in the reduction phase - like the one based on the classical proper orthogonal decomposition (POD) - since the number of vectors to which the decomposition applies is drastically reduced as it does not contain all intermediate solutions generated along an integration grid by a time stepping method. We show the effectiveness of the method by some illustrative parabolic PDEs arising from finance and also provide some evidence that the method we propose, when applied to a simple advection equation, does not suffer the problem of slow decay of singular values which instead affects methods based on time integration of the Cauchy problem arising from space discretization.

In a wide range of practical problems, such as forming operations and impact tests, assuming that one of the contacting bodies is rigid is an excellent approximation to the physical phenomenon. In this work, the well-established dual mortar method is adopted to enforce interface constraints in the finite deformation frictionless contact of rigid and deformable bodies. The efficiency of the nonlinear contact algorithm proposed here is based on two main contributions. Firstly, a variational formulation of the method using the so-called Petrov-Galerkin scheme is investigated, as it unlocks a significant simplification by removing the need to explicitly evaluate the dual basis functions. The corresponding first-order dual mortar interpolation is presented in detail. Particular focus is, then, placed on the extension for second-order interpolation by employing a piecewise linear interpolation scheme, which critically retains the geometrical information of the finite element mesh. Secondly, a new definition for the nodal orthonormal moving frame attached to each contact node is suggested. It reduces the geometrical coupling between the nodes and consequently decreases the stiffness matrix bandwidth. The proposed contributions decrease the computational complexity of dual mortar methods for rigid/deformable interaction, especially in the three-dimensional setting, while preserving accuracy and robustness.