A sparse-grid isogeometric solver

Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS as a basis for the approximation of the solution of PDEs. In this work, we investigate to which extent IGA solvers can benefit from the so-called sparse-grids construction in its combination technique form, which was first introduced in the early 90s in the context of the approximation of high-dimensional PDEs. The tests that we report show that, in accordance to the literature, a sparse-grid construction can indeed be useful if the solution of the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the case of non-smooth solutions when some a-priori knowledge on the location of the singularities of the solution can be exploited to devise suitable non-equispaced meshes. Finally, we remark that sparse grids can be seen as a simple way to parallelize pre-existing serial IGA solvers in a straightforward fashion, which can be beneficial in many practical situations.Comment: updated version after revie

A reduced conjugate gradient basis method for fractional diffusion

This work is on a fast and accurate reduced basis method for solving discretized fractional elliptic partial differential equations (PDEs) of the form $\mathcal{A}^su=f$ by rational approximation. A direct computation of the action of such an approximation would require solving multiple (20$\sim$30) large-scale sparse linear systems. Our method constructs the reduced basis using the first few directions obtained from the preconditioned conjugate gradient method applied to one of the linear systems. As shown in the theory and experiments, only a small number of directions (5$\sim$10) are needed to approximately solve all large-scale systems on the reduced basis subspace. This reduces the computational cost dramatically because: (1) We only use one of the large-scale problems to construct the basis; and (2) all large-scale problems restricted to the subspace have much smaller sizes. We test our algorithms for fractional PDEs on a 3d Euclidean domain, a 2d surface, and random combinatorial graphs. We also use a novel approach to construct the rational approximation for the fractional power function by the orthogonal greedy algorithm (OGA)

Efficient Reordered Nonlinear Gauss-Seidel Solvers With Higher Order For Black-Oil Models

The fully implicit method is the most commonly used approach to solve black-oil problems in reservoir simulation. The method requires repeated linearization of large nonlinear systems and produces ill-condi\-tioned linear systems. We present a strategy to reduce computational time that relies on two key ideas: (\textit{i}) a sequential formulation that decouples flow and transport into separate subproblems, and (\textit{ii}) a highly efficient Gauss--Seidel solver for the transport problems. This solver uses intercell fluxes to reorder the grid cells according to their upstream neighbors, and groups cells that are mutually dependent because of counter-current flow into local clusters. The cells and local clusters can then be solved in sequence, starting from the inflow and moving gradually downstream, since each new cell or local cluster will only depend on upstream neighbors that have already been computed. Altogether, this gives optimal localization and control of the nonlinear solution process. This method has been successfully applied to real-field problems using the standard first-order finite volume discretization. Here, we extend the idea to first-order dG methods on fully unstructured grids. We also demonstrate proof of concept for the reordering idea by applying it to the full simulation model of the Norne oil field, using a prototype variant of the open-source OPM Flow simulator.Comment: Comput Geosci (2019

An implicit local time-stepping method based on cell reordering for multiphase flow in porous media

We discuss how to introduce local time-step refinements in a sequential implicit method for multiphase flow in porous media. Our approach relies heavily on causality-based optimal ordering, which implies that cells can be ordered according to total fluxes after the pressure field has been computed, leaving the transport problem as a sequence of ordinary differential equations, which can be solved cell-by-cell or block-by-block. The method is suitable for arbitrary local time steps and grids, is mass-conservative, and reduces to the standard implicit upwind finite-volume method in the case of equal time steps in adjacent cells. The method is validated by a series of numerical simulations. We discuss various strategies for selecting local time steps and demonstrate the efficiency of the method and several of these strategies by through a series of numerical examples.publishedVersio

An adjustable-ratio flow dividing hydraulic valve

This thesis proposes a new type of hydraulic valve: an adjustable-ratio flow divider. This valve attempts to split one input flow into two output flows in a predetermined ratio, independent of load pressure or total flow. The valve uses a two dimensional structure to form a two-stage valve with only one moving part; the pilot stage uses the spool s rotary position, and the main stage uses its linear position. This arrangement allows for a cheaper, simpler valve with smaller volumes (translating into faster response). The ratio of outlet flows can be set on the fly by the angular position of the spool, driven by a stepper motor or other low-power input. In order to evaluate the initial feasibility of the concept, steady state and dynamic models were developed and the effects of the physical parameters were studied. Two non-linear non-derivative multiobjective optimization strategies were used to determine the optimum parameters for a prototype. Finally, the prototype s performance was experimentally examined and appears to work as expected

Learning Generic Solutions for Multiphase Transport in Porous Media via the Flux Functions Operator

Traditional numerical schemes for simulating fluid flow and transport in porous media can be computationally expensive. Advances in machine learning for scientific computing have the potential to help speed up the simulation time in many scientific and engineering fields. DeepONet has recently emerged as a powerful tool for accelerating the solution of partial differential equations (PDEs) by learning operators (mapping between function spaces) of PDEs. In this work, we learn the mapping between the space of flux functions of the Buckley-Leverett PDE and the space of solutions (saturations). We use Physics-Informed DeepONets (PI-DeepONets) to achieve this mapping without any paired input-output observations, except for a set of given initial or boundary conditions; ergo, eliminating the expensive data generation process. By leveraging the underlying physical laws via soft penalty constraints during model training, in a manner similar to Physics-Informed Neural Networks (PINNs), and a unique deep neural network architecture, the proposed PI-DeepONet model can predict the solution accurately given any type of flux function (concave, convex, or non-convex) while achieving up to four orders of magnitude improvements in speed over traditional numerical solvers. Moreover, the trained PI-DeepONet model demonstrates excellent generalization qualities, rendering it a promising tool for accelerating the solution of transport problems in porous media.Comment: 23 pages, 11 figure

Exponential integrators: tensor structured problems and applications

The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems

Numerical Recipes in Python

Numerical Recipes in Python is to serve as Laboratory Manual of Simplified Numerical Analysis (Python Version): A companion book of the principal book: Simplified Numerical Analysis (Fourth Edition) by Dr. Amjad Ali