Efficient computation of highly oscillatory integrals by using QTT tensor approximation

We propose a new method for the efficient approximation of a class of highly oscillatory weighted integrals where the oscillatory function depends on the frequency parameter $\omega \geq 0$, typically varying in a large interval. Our approach is based, for fixed but arbitrary oscillator, on the pre-computation and low-parametric approximation of certain $\omega$-dependent prototype functions whose evaluation leads in a straightforward way to recover the target integral. The difficulty that arises is that these prototype functions consist of oscillatory integrals and are itself oscillatory which makes them both difficult to evaluate and to approximate. Here we use the quantized-tensor train (QTT) approximation method for functional $m$-vectors of logarithmic complexity in $m$ in combination with a cross-approximation scheme for TT tensors. This allows the accurate approximation and efficient storage of these functions in the wide range of grid and frequency parameters. Numerical examples illustrate the efficiency of the QTT-based numerical integration scheme on various examples in one and several spatial dimensions.Comment: 20 page

Non-Intrusive, High-Dimensional Uncertainty Quantification for the Robust Simulation of Fluid Flows

Uncertainty Quantification is the field of mathematics that focuses on the propagation and influence of uncertainties on models. Mostly complex numerical models are considered with uncertain parameters or uncertain model properties. Several methods exist to model the uncertain parameters of numerical models. Stochastic Collocation is a method that samples the random variables of the input parameters using a deterministic procedure and then interpolates or integrates the unknown quantity of interest using the samples. Because moments of the distribution of the unknown quantity are essentially integrals of the quantity, the main focus will be on calculating integrals. Calculating an integral using samples can be done efficiently using a quadrature or cubature rule. Both sample the space of integration in a deterministic way and several algorithms to determine the samples exist, each with its own advantages and disadvantages. In the one-dimensional case a method is proposed that has all relevant advantages (positive weights, nested points and dependency on the input distribution). The principle of the introduced quadrature rule can also be applied to a multi-dimensional setting. However, if negative weights are allowed in the multi-dimensional case a cubature rule can be generated that has a very small number of points compared to the methods described in literature. The new method uses the fact that the tensor product of several quadrature rules has many points with the same weight that can be considered as on

Uncertainty quantification of coal seam gas production prediction using Polynomial Chaos

A surrogate model approximates a computationally expensive solver. Polynomial Chaos is a method to construct surrogate models by summing combinations of carefully chosen polynomials. The polynomials are chosen to respect the probability distributions of the uncertain input variables (parameters); this allows for both uncertainty quantification and global sensitivity analysis. In this paper we apply these techniques to a commercial solver for the estimation of peak gas rate and cumulative gas extraction from a coal seam gas well. The polynomial expansion is shown to honour the underlying geophysics with low error when compared to a much more complex and computationally slower commercial solver. We make use of advanced numerical integration techniques to achieve this accuracy using relatively small amounts of training data

Efficient Methods for Multidimensional Global Polynomial Approximation with Applications to Random PDEs

In this work, we consider several ways to overcome the challenges associated with polynomial approximation and integration of smooth functions depending on a large number of inputs. We are motivated by the problem of forward uncertainty quantification (UQ), whereby inputs to mathematical models are considered as random variables. With limited resources, finding more efficient and accurate ways to approximate the multidimensional solution to the UQ problem is of crucial importance, due to the “curse of dimensionality” and the cost of solving the underlying deterministic problem. The first way we overcome the complexity issue is by exploiting the structure of the approximation schemes used to solve the random partial differential equations (PDE), thereby significantly reducing the overall cost of the approximation. We do this first using multilevel approximations in the physical variables, and second by exploiting the hierarchy of nested sparse grids in the random parameter space. With these algorithmic advances, we provably decrease the complexity of collocation methods for solving random PDE problems. The second major theme in this work is the choice of efficient points for multidimensional interpolation and interpolatory quadrature. A major consideration in interpolation in multiple dimensions is the balance between stability, i.e., the Lebesgue constant of the interpolant, and the granularity of the approximation, e.g., the ability to choose an arbitrary number of interpolation points or to adaptively refine the grid. For these reasons, the Leja points are a popular choice for approximation on both bounded and unbounded domains. Mirroring the best-known results for interpolation on compact domains, we show that Leja points, defined for weighted interpolation on R, have a Lebesgue constant which grows subexponentially in the number of interpolation nodes. Regarding multidimensional quadratures, we show how certain new rules, generated from conformal mappings of classical interpolatory rules, can be used to increase the efficiency in approximating multidimensional integrals. Specifically, we show that the convergence rate for the novel mapped sparse grid interpolatory quadratures is improved by a factor that is exponential in the dimension of the underlying integral

Some quadrature methods for general and singular integrals in one and two dimensions

In this thesis numerical integration in one and two dimensions is considered. In chapter two transformation methods are considered primarily for singular integrals and methods of computing the transformations themselves are derived. The well-known transformation based on the IMT rule and error function are extended to non-standard functions. The implementation of these rules and their performances are demonstrated. These transformations are then extended to two-dimensions and are used to develop accurate rules for integrating singular integrals. In addition to this, a polynomial transformation with the aim of the reduction in the number of function evaluations is also considered and the resultant product rule is applied to two-dimensional non-singular integrals. Finally, the use of monomials in the construction of integration rules for non-singular two-dimensional integrals is considered and some rules developed. In all these situations the rules developed are tested and compared with existing methods. The results show that the new rules compare favourably with existing ones

An Inference-Driven Branch and Bound Optimization Strategy for Planning Ambulance Services

Strategic placement of ambulances is important to the efficient functioning of emergency services. As part of an ongoing collaboration with Wayne County 911, we developed a strategy to optimize the placement of ambulances throughout Wayne County based on de-identified call and response data from 2016 and 2017. The primary goals of the optimization were minimizing annual operating cost and mean response time, as well as providing a constructive solution that could naturally evolve from the existing plan. This thesis details the derivation and implementation of one of the optimization strategies used in this project. It is based on parametric statistical inference and the “Branch and Bound” pattern that is often used in non-linear global optimization

Adaptive sparse grid algorithms with applications to electromagnetic scattering under uncertainty

We discuss adaptive sparse grid algorithms for stochastic differential equations with a particular focus on applications to electromagnetic scattering by structures with holes of uncertain size, location, and quantity. Stochastic collocation (SC) methods are used in combination with an adaptive sparse grid approach based on nested Gauss-Patterson grids. As an error estimator we demonstrate how the nested structure allows an effective error estimation through Richardson extrapolation. This is shown to allow excellent error estimation and it also provides an efficient means by which to estimate the solution at the next level of the refinement. We introduce an adaptive approach for the computation of problems with discrete random variables and demonstrate its efficiency for scattering problems with a random number of holes. The results are compared with results based on Monte Carlo methods and with Stroud based integration, confirming the accuracy and efficiency of the proposed techniques