A HYBRID ALGORITHM FOR THE UNCERTAIN INVERSE p-MEDIAN LOCATION PROBLEM

In this paper, we investigate the inverse p-median location  problem with variable edge lengths and variable vertex weights on networks in which the vertex weights and modification costs are the independent uncertain variables. We propose a  model for the uncertain inverse p-median location problem  with tail value at risk objective. Then, we show that  it  is NP-hard. Therefore,  a hybrid particle swarm optimization  algorithm is presented  to obtain   the approximate optimal solution of the proposed model. The algorithm contains expected value simulation and tail value at risk simulation

Principles of Neural Network Architecture Design - Invertibility and Domain Knowledge

Neural networks architectures allow a tremendous variety of design choices. In this work, we study two principles underlying these architectures: First, the design and application of invertible neural networks (INNs). Second, the incorporation of domain knowledge into neural network architectures. After introducing the mathematical foundations of deep learning, we address the invertibility of standard feedforward neural networks from a mathematical perspective. These results serve as a motivation for our proposed invertible residual networks (i-ResNets). This architecture class is then studied in two scenarios: First, we propose ways to use i-ResNets as a normalizing flow and demonstrate the applicability for high-dimensional generative modeling. Second, we study the excessive invariance of common deep image classifiers and discuss consequences for adversarial robustness. We finish with a study of convolutional neural networks for tumor classification based on imaging mass spectrometry (IMS) data. For this application, we propose an adapted architecture guided by our knowledge of the domain of IMS data and show its superior performance on two challenging tumor classification datasets

Quantifying Outlierness of Funds from their Categories using Supervised Similarity

Mutual fund categorization has become a standard tool for the investment management industry and is extensively used by allocators for portfolio construction and manager selection, as well as by fund managers for peer analysis and competitive positioning. As a result, a (unintended) miscategorization or lack of precision can significantly impact allocation decisions and investment fund managers. Here, we aim to quantify the effect of miscategorization of funds utilizing a machine learning based approach. We formulate the problem of miscategorization of funds as a distance-based outlier detection problem, where the outliers are the data-points that are far from the rest of the data-points in the given feature space. We implement and employ a Random Forest (RF) based method of distance metric learning, and compute the so-called class-wise outlier measures for each data-point to identify outliers in the data. We test our implementation on various publicly available data sets, and then apply it to mutual fund data. We show that there is a strong relationship between the outlier measures of the funds and their future returns and discuss the implications of our findings.Comment: 8 pages, 5 tables, 8 figure

Reservoir Flooding Optimization by Control Polynomial Approximations

In this dissertation, we provide novel parametrization procedures for water-flooding production optimization problems, using polynomial approximation techniques. The methods project the original infinite dimensional controls space into a polynomial subspace. Our contribution includes new parameterization formulations using natural polynomials, orthogonal Chebyshev polynomials and Cubic spline interpolation. We show that the proposed methods are well suited for black-box approach with stochastic global-search method as they tend to produce smooth control trajectories, while reducing the solution space size. We demonstrate their efficiency on synthetic two-dimensional problems and on a realistic 3-dimensional problem. By contributing with a new adjoint method formulation for polynomial approximation, we implemented the methods also with gradient-based algorithms. In addition to fine-scale simulation, we also performed reduced order modeling, where we demonstrated a synergistic effect when combining polynomial approximation with model order reduction, that leads to faster optimization with higher gains in terms of Net Present Value. Finally, we performed gradient-based optimization under uncertainty. We proposed a new multi-objective function with three components, one that maximizes the expected value of all realizations, and two that maximize the averages of distribution tails from both sides. The new objective provides decision makers with the flexibility to choose the amount of risk they are willing to take, while deciding on production strategy or performing reserves estimation (P10;P50;P90)

A Parallel Geometric Multigrid Method for Adaptive Finite Elements

Applications in a variety of scientific disciplines use systems of Partial Differential Equations (PDEs) to model physical phenomena. Numerical solutions to these models are often found using the Finite Element Method (FEM), where the problem is discretized and the solution of a large linear system is required, containing millions or even billions of unknowns. Often times, the domain of these solves will contain localized features that require very high resolution of the underlying finite element mesh to accurately solve, while a mesh with uniform resolution would require far too much computational time and memory overhead to be feasible on a modern machine. Therefore, techniques like adaptive mesh refinement, where one increases the resolution of the mesh only where it is necessary, must be used. Even with adaptive mesh refinement, these systems can still be on the order of much more than a million unknowns (large mantle convection applications like the ones in [90] show simulations on over 600 billion unknowns), and attempting to solve on a single processing unit is infeasible due to limited computational time and memory required. For this reason, any application code aimed at solving large problems must be built using a parallel framework, allowing the concurrent use of multiple processing units to solve a single problem, and the code must exhibit efficient scaling to large amounts of processing units. Multigrid methods are currently the only known optimal solvers for linear systems arising from discretizations of elliptic boundary valued problems. These methods can be represented as an iterative scheme with contraction number less than one, independent of the resolution of the discretization [24, 54, 25, 103], with optimal complexity in the number of unknowns in the system [29]. Geometric multigrid (GMG) methods, where the hierarchy of spaces are defined by linear systems of finite element discretizations on meshes of decreasing resolution, have been shown to be robust for many different problem formulations, giving mesh independent convergence for highly adaptive meshes [26, 61, 83, 18], but these methods require specific implementations for each type of equation, boundary condition, mesh, etc., required by the specific application. The implementation in a massively parallel environment is not obvious, and research into this topic is far from exhaustive. We present an implementation of a massively parallel, adaptive geometric multigrid (GMG) method used in the open-source finite element library deal.II [5], and perform extensive tests showing scaling of the v-cycle application on systems with up to 137 billion unknowns run on up to 65,536 processors, and demonstrating low communication overhead of the algorithms proposed. We then show the flexibility of the GMG by applying the method to four different PDE systems: the Poisson equation, linear elasticity, advection-diffusion, and the Stokes equations. For the Stokes equations, we implement a fully matrix-free, adaptive, GMG-based solver in the mantle convection code ASPECT [13], and give a comparison to the current matrix-based method used. We show improvements in robustness, parallel scaling, and memory consumption for simulations with up to 27 billion unknowns and 114,688 processors. Finally, we test the performance of IDR(s) methods compared to the FGMRES method currently used in ASPECT, showing the effects of the flexible preconditioning used for the Stokes solves in ASPECT, and the demonstrating the possible reduction in memory consumption for IDR(s) and the potential for solving large scale problems. Parts of the work in this thesis has been submitted to peer reviewed journals in the form of two publications ([36] and [34]), and the implementations discussed have been integrated into two open-source codes, deal.II and ASPECT. From the contributions to deal.II, including a full length tutorial program, Step-63 [35], the author is listed as a contributing author to the newest deal.II release (see [5]). The implementation into ASPECT is based on work from the author and Timo Heister. The goal for the work here is to enable the community of geoscientists using ASPECT to solve larger problems than currently possible. Over the course of this thesis, the author was partially funded by the NSF Award OAC-1835452 and by the Computational Infrastructure in Geodynamics initiative (CIG), through the NSF under Award EAR-0949446 and EAR-1550901 and The University of California -- Davis

MECA Worksop on Dust on Mars 2

Topics addressed include: sedimentary debris; mineralogy; Martian dust cycles; Mariner 9 mission; Viking observations; Mars Observer; atmospheric circulation; aeolian features; aerosols; and landslides

Reduced order modeling of convection-dominated flows, dimensionality reduction and stabilization

We present methodologies for reduced order modeling of convection dominated flows. Accordingly, three main problems are addressed. Firstly, an optimal manifold is realized to enhance reducibility of convection dominated flows. We design a low-rank auto-encoder to specifically reduce the dimensionality of solution arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Also, considering that the latent variables are often hard to interpret, many of these methods are dismissed in the reduced order modeling of dynamical systems governed by partial differential equations (PDEs). This deficiency is of importance to the extent that linear methods, such as principle component analysis (PCA) and Koopman operators, are still prevalent. Accordingly, we propose an interpretable nonlinear dimensionality reduction algorithm. An unsupervised learning problem is constructed that learns a diffeomorphic spatio-temporal grid which registers the output sequence of the PDEs on a non-uniform time-varying grid. The Kolmogorov n-width of the mapped data on the learned grid is minimized. Secondly, the reduced order models are constructed on the realized manifolds. We project the high fidelity models on the learned manifold, leading to a time-varying system of equations. Moreover, as a data-driven model free architecture, recurrent neural networks on the learned manifold are trained, showing versatility of the proposed framework. Finally, a stabilization method is developed to maintain stability and accuracy of the projection based ROMs on the learned manifold a posteriori. We extend the eigenvalue reassignment method of stabilization of linear time-invariant ROMs, to the more general case of linear time-varying systems. Through a post-processing step, the ROMs are controlled using a constrained nonlinear lease-square minimization problem. The controller and the input signals are defined at the algebraic level, using left and right singular vectors of the reduced system matrices. The proposed stabilization method is general and applicable to a large variety of linear time-varying ROMs

Optimal Foreign Reserves, The Dollar Trap and Demand for Global Safe Assets: A DSGE analysis for China

The recent surge of foreign reserves in emerging markets has sparked fierce debate about what level of reserves is the optimal amount for a country. Conventional models have achieved important advances in understanding the behaviour of central banks’ reserve policy, but fail to find convincing solutions to the puzzle of why emerging economies, and China in particular, would continue to accumulate massive reserves. With reference to China’s massive hoarding of foreign reserves, this thesis develops a representative agent model with elements of dynamic stochastic general equilibrium (DSGE) modelling. The model constructed in this thesis explicitly considers the risky steady state as the equilibrium point when agents take into account future uncertainty but when the shock realizations are zero. In this risky steady state we derive the optimal reserves for emerging markets, with particular reference to the Chinese case. The precautionary savings motivation for holding reserves is then analysed within this framework. This thesis derives the optimality of Chinese reserve accumulation, and provides a plausible explanation for reserve build-up in China and its underlying driving forces. In order to better understand the foreign reserves accumulation, this thesis further attempts to analyse current external wealth allocation in a portfolio perspective within a DSGE framework. A two-country model is employed, and a Value at Risk (VaR) constraint is introduced to reproduce the risk averse behaviour of investors. After accounting for risk diversification, our findings imply that an investor would shift their portfolio holding to bond related assets. Finally, China has accumulated a huge amount of foreign reserves. The majority of these assets are denominated in the US dollar. Furthermore, in terms of asset type, the US T-bill is the dominant investment instrument in China’s international portfolio choice. This raises questions as to why the central bank of China chooses to make such an investment decision, and what the global repercussions might be. Therefore, China’s role in the growing demand for global safe assets deserves exploration. Given the world-wide shortage of global safe assets, to what extent China will continue the current international investment decision, and the driving forces behind such policy inertia, are major concerns. In order to gain a better understanding, this thesis applies a global solving method, as well as a standard local solving method