arXiv subjects

Mihai Anitescu

Publications and source records attributed to Mihai Anitescu.

72 records · Page 4Linked to original sources

arXiv2020

A fast algorithm for maximum likelihood estimation of mixture proportions using sequential quadratic programming

Maximum likelihood estimation of mixture proportions has a long history, and continues to play an important role in modern statistics, including in development of nonparametric empirical Bayes methods. Maximum likelihood of mixture proportions has traditionally been solved using the expectation maximization (EM) algorithm, but recent work by Koenker & Mizera shows that modern convex optimization techniques -- in particular, interior point methods -- are substantially faster and more accurate than EM. Here, we develop a new solution based on sequential quadratic programming (SQP). It is substantially faster than the interior point method, and just as accurate. Our approach combines several ideas: first, it solves a reformulation of the original problem; second, it uses an SQP approach to make the best use of the expensive gradient and Hessian computations; third, the SQP iterations are implemented using an active set method to exploit the sparse nature of the quadratic subproblems; fourth, it uses accurate low-rank approximations for more efficient gradient and Hessian computations. We illustrate the benefits of our approach in experiments on synthetic data sets as well as a large genetic association data set. In large data sets (n = 1,000,000 observations, m = 1,000 mixture components), our implementation achieves at least 100-fold reduction in runtime compared with a state-of-the-art interior point solver. Our methods are implemented in Julia, and in an R package available on CRAN (see https://CRAN.R-project.org/package=mixsqp).

stat.CO↗

arXiv2020

Overlapping Schwarz Decomposition for Constrained Quadratic Programs

We present an overlapping Schwarz decomposition algorithm for constrained quadratic programs (QPs). Schwarz algorithms have been traditionally used to solve linear algebra systems arising from partial differential equations, but we have recently shown that they are also effective at solving structured optimization problems. In the proposed scheme, we consider QPs whose algebraic structure can be represented by graphs. The graph domain is partitioned into overlapping subdomains (yielding a set of coupled subproblems), solutions for the subproblems are computed in parallel, and convergence is enforced by updating primal-dual information in the overlapping regions. We show that convergence is guaranteed if the overlap is sufficiently large and that the convergence rate improves exponentially with the size of the overlap. Convergence results rely on a key property of graph-structured problems that is known as exponential decay of sensitivity. Here, we establish conditions under which this property holds for constrained QPs (as those found in network optimization and optimal control), thus extending existing work that addresses unconstrained QPs. The numerical behavior of the Schwarz scheme is demonstrated by using a DC optimal power flow problem defined over a network with 9,241 nodes.

math.OC↗

arXiv2020

Flexible nonstationary spatio-temporal modeling of high-frequency monitoring data

Many physical datasets are generated by collections of instruments that make measurements at regular time intervals. For such regular monitoring data, we extend the framework of half-spectral covariance functions to the case of nonstationarity in space and time and demonstrate that this method provides a natural and tractable way to incorporate complex behaviors into a covariance model. Further, we use this method with fully time-domain computations to obtain bona fide maximum likelihood estimators---as opposed to using Whittle-type likelihood approximations, for example---that can still be computed efficiently. We apply this method to very high-frequency Doppler LIDAR vertical wind velocity measurements, demonstrating that the model can expressively capture the extreme nonstationarity of dynamics above and below the atmospheric boundary layer and, more importantly, the interaction of the process dynamics across it.

stat.ME↗

arXiv2020

A Machine-Learning-Based Importance Sampling Method to Compute Rare Event Probabilities

We develop a novel computational method for evaluating the extreme excursion probabilities arising from random initialization of nonlinear dynamical systems. The method uses excursion probability theory to formulate a sequence of Bayesian inverse problems that, when solved, yields the biasing distribution. Solving multiple Bayesian inverse problems can be expensive; more so in higher dimensions. To alleviate the computational cost, we build machine-learning-based surrogates to solve the Bayesian inverse problems that give rise to the biasing distribution. This biasing distribution can then be used in an importance sampling procedure to estimate the extreme excursion probabilities.

physics.comp-ph↗

arXiv2020

Sequential Bayesian Parameter Estimation of Stochastic Dynamic Load Models

In this paper we focus on the parameter estimation of dynamic load models with stochastic terms, in particular, load models where protection settings are uncertain, such as in aggregated air conditioning units. We show how the uncertainty in the aggregated protection characteristics can be formulated as a stochastic differential equation with process noise. We cast the parameter inversion within a Bayesian parameter estimation framework, and we present methods to include process noise. We demonstrate the benefits of considering stochasticity in the parameter estimation and the risks of ignoring it.

math.OC↗

arXiv2020

Efficient computation of extreme excursion probabilities for dynamical systems

We develop a novel computational method for evaluating the extreme excursion probabilities arising for random initialization of nonlinear dynamical systems. The method uses a Markov chain Monte Carlo or a Laplace approximation approach to construct a biasing distribution that in turn is used in an importance sampling procedure to estimate the extreme excursion probabilities. The prior and likelihood of the biasing distribution are obtained by using Rice's formula from excursion probability theory. We use Gaussian mixture biasing distributions and approximate the non-Gaussian initial excitation by the method of moments to circumvent the linearity and Gaussianity assumptions needed by excursion probability theory. We demonstrate the effectiveness of this computational framework for nonlinear dynamical systems of up to 100 dimensions.

math.NA↗

arXiv2020

Decentralized Schemes with Overlap for Solving Graph-Structured Optimization Problems

We present a new algorithmic paradigm for the decentralized solution of graph-structured optimization problems that arise in the estimation and control of network systems. A key and novel design concept of the proposed approach is that it uses overlapping subdomains to promote and accelerate convergence. We show that the algorithm converges if the size of the overlap is sufficiently large and that the convergence rate improves exponentially with the size of the overlap. The proposed approach provides a bridge between fully decentralized and centralized architectures and is flexible in that it enables the implementation of asynchronous schemes, handling of constraints, and balancing of computing, communication, and data privacy needs. The proposed scheme is tested in an estimation problem for a 9241-node power network and we show that it outperforms the alternating direction method of multipliers.

math.OC↗

arXiv2019

Solving Optimal Experimental Design with Sequential Quadratic Programming and Chebyshev Interpolation

We propose an optimization algorithm to compute the optimal sensor locations in experimental design in the formulation of Bayesian inverse problems, where the parameter-to-observable mapping is described through an integral equation and its discretization results in a continuously indexed matrix whose size depends on the mesh size n. By approximating the gradient and Hessian of the objective design criterion from Chebyshev interpolation, we solve a sequence of quadratic programs and achieve the complexity $\mathcal{O}(n\log^2(n))$. An error analysis guarantees the integrality gap shrinks to zero as $n\to\infty$, and we apply the algorithm on a two-dimensional advection-diffusion equation, to determine the LIDAR's optimal sensing directions for data collection.

stat.CO↗

arXiv2019

A Kinetic Monte Carlo Approach for Simulating Cascading Transmission Line Failure

In this work, cascading transmission line failures are studied through a dynamical model of the power system operating under fixed conditions. The power grid is modeled as a stochastic dynamical system where first-principles electromechanical dynamics are excited by small Gaussian disturbances in demand and generation around a specified operating point. In this context, a single line failure is interpreted in a large deviation context as a first escape event across a surface in phase space defined by line security constraints. The resulting system of stochastic differential equations admits a transverse decomposition of the drift, which leads to considerable simplification in evaluating the quasipotential (rate function) and, consequently, computation of exit rates. Tractable expressions for the rate of transmission line failure in a restricted network are derived from large deviation theory arguments and validated against numerical simulations. Extensions to realistic settings are considered, and individual line failure models are aggregated into a Markov model of cascading failure inspired by chemical kinetics. Cascades are generated by traversing a graph composed of weighted edges representing transitions to degraded network topologies. Numerical results indicate that the Markov model can produce cascades with qualitative power-law properties similar to those observed in empirical cascades.

eess.SY↗

arXiv2019

Exponential Decay in the Sensitivity Analysis of Nonlinear Dynamic Programming

In this paper, we study the sensitivity of discrete-time dynamic programs with nonlinear dynamics and objective to perturbations in the initial conditions and reference parameters. Under uniform controllability and boundedness assumptions for the problem data, we prove that the directional derivative of the optimal state and control at time $k$, $\boldsymbol{x}^*_{k}$ and $\boldsymbol{u}^*_{k}$, with respect to the reference signal at time $i$, $\boldsymbol{d}_{i}$, will have exponential decay in terms of $|k-i|$ with a decay rate $ρ$ independent of the temporal horizon length. The key technical step is to prove that a version of the convexification approach proposed by Verschueren et al. can be applied to the KKT conditions and results in a convex quadratic program with uniformly bounded data. In turn, Riccati techniques can be further employed to obtain the sensitivity result, borne from the observation that the directional derivatives are solutions of quadratic programs with structure similar to the KKT conditions themselves. We validate our findings with numerical experiments on a small nonlinear, nonconvex, dynamic program.

math.NA↗

arXiv2019

Distributionally Robust Optimization with Correlated Data from Vector Autoregressive Processes

We present a distributionally robust formulation of a stochastic optimization problem for non-i.i.d vector autoregressive data. We use the Wasserstein distance to define robustness in the space of distributions and we show, using duality theory, that the problem is equivalent to a finite convex-concave saddle point problem. The performance of the method is demonstrated on both synthetic and real data.

math.OC↗

arXiv2019

Scalable Gaussian Process Computations Using Hierarchical Matrices

We present a kernel-independent method that applies hierarchical matrices to the problem of maximum likelihood estimation for Gaussian processes. The proposed approximation provides natural and scalable stochastic estimators for its gradient and Hessian, as well as the expected Fisher information matrix, that are computable in quasilinear $O(n \log^2 n)$ complexity for a large range of models. To accomplish this, we (i) choose a specific hierarchical approximation for covariance matrices that enables the computation of their exact derivatives and (ii) use a stabilized form of the Hutchinson stochastic trace estimator. Since both the observed and expected information matrices can be computed in quasilinear complexity, covariance matrices for MLEs can also be estimated efficiently. After discussing the associated mathematics, we demonstrate the scalability of the method, discuss details of its implementation, and validate that the resulting MLEs and confidence intervals based on the inverse Fisher information matrix faithfully approach those obtained by the exact likelihood.

stat.CO↗

arXiv2018

Simulating the stochastic dynamics and cascade failure of power networks

For large-scale power networks, the failure of particular transmission lines can offload power to other lines and cause self-protection trips to activate, instigating a cascade of line failures. In extreme cases, this can bring down the entire network. Learning where the vulnerabilities are and the expected timescales for which failures are likely is an active area of research. In this article we present a novel stochastic dynamics model for a large-scale power network along with a framework for efficient computer simulation of the model including long timescale events such as cascade failure. We build on an existing Hamiltonian formulation and introduce stochastic forcing and damping components to simulate small perturbations to the network. Our model and simulation framework allow assessment of the particular weaknesses in a power network that make it susceptible to cascade failure, along with the timescales and mechanism for expected failures.

physics.soc-ph↗

arXiv2016

Stochastic simulation of predictive space-time scenarios of wind speed using observations and physical models

We propose a statistical space-time model for predicting atmospheric wind speed based on deterministic numerical weather predictions and historical measurements. We consider a Gaussian multivariate space-time framework that combines multiple sources of past physical model outputs and measurements along with model predictions in order to produce a probabilistic wind speed forecast within the prediction window. We illustrate this strategy on a ground wind speed forecast for several months in 2012 for a region near the Great Lakes in the United States. The results show that the prediction is improved in the mean-squared sense relative to the numerical forecasts as well as in probabilistic scores. Moreover, the samples are shown to produce realistic wind scenarios based on the sample spectrum.

stat.AP↗

arXiv2016

A Bayesian Approach for Parameter Estimation with Uncertainty for Dynamic Power Systems

We address the problem of estimating the uncertainty in the solution of power grid inverse problems within the framework of Bayesian inference. We investigate two approaches, an adjoint-based method and a stochastic spectral method. These methods are used to estimate the maximum a posteriori point of the parameters and their variance, which quantifies their uncertainty. Within this framework we estimate several parameters of the dynamic power system, such as generator inertias, which are not quantifiable in steady-state models. We illustrate the performance of these approaches on a 9-bus power grid example and analyze the dependence on measurement frequency, estimation horizon, perturbation size, and measurement noise. We assess the computational efficiency, and discuss the expected performance when these methods are applied to large systems.

math.OC↗

arXiv2015

A Stochastic Electricity Market Clearing Formulation with Consistent Pricing Properties

We argue that deterministic market clearing formulations introduce arbitrary distortions between day-ahead and expected real-time prices that bias economic incentives and block diversification. We extend and analyze the stochastic clearing formulation proposed by Pritchard et al. (2010) in which the social surplus function induces penalties between day-ahead and real-time quantities. We prove that the formulation yields price distortions that are bounded by the bid prices, and we show that adding a similar penalty term to transmission flows and phase angles ensures boundedness throughout the network. We prove that when the price distortions are zero, day-ahead quantities converge to the quantile of real-time counterparts. The undesired effects of price distortions suggest that stochastic settings provide significant benefits over deterministic ones that go beyond social surplus improvements. We propose additional metrics to evaluate these benefits.

econ.GN↗

arXiv2013

Stochastic approximation of score functions for Gaussian processes

We discuss the statistical properties of a recently introduced unbiased stochastic approximation to the score equations for maximum likelihood calculation for Gaussian processes. Under certain conditions, including bounded condition number of the covariance matrix, the approach achieves $O(n)$ storage and nearly $O(n)$ computational effort per optimization step, where $n$ is the number of data sites. Here, we prove that if the condition number of the covariance matrix is bounded, then the approximate score equations are nearly optimal in a well-defined sense. Therefore, not only is the approximation efficient to compute, but it also has comparable statistical properties to the exact maximum likelihood estimates. We discuss a modification of the stochastic approximation in which design elements of the stochastic terms mimic patterns from a $2^n$ factorial design. We prove these designs are always at least as good as the unstructured design, and we demonstrate through simulation that they can produce a substantial improvement over random designs. Our findings are validated by numerical experiments on simulated data sets of up to 1 million observations. We apply the approach to fit a space-time model to over 80,000 observations of total column ozone contained in the latitude band $40^{\circ}$-$50^{\circ}$N during April 2012.

stat.AP↗

arXiv2003

Unconditional Stability for Numerical Scheme Combining Implicit Timestepping for Local Effects and Explicit Timestepping for Nonlocal Effects

A combination of implicit and explicit timestepping is analyzed for a system of ODEs motivated by ones arising from spatial discretizations of evolutionary partial differential equations. Loosely speaking, the method we consider is implicit in local and stabilizing terms in the underlying PDE and explicit in nonlocal and unstabilizing terms. Unconditional stability and convergence of the numerical scheme are proven by the energy method and by algebraic techniques. This stability result is surprising because usually when different methods are combined, the stability properties of the least stable method plays a determining role in the combination.

math.NA↗