In uncertainty quantification, variance-based global sensitivity analysis quantitatively determines the effect of each input random variable on the output by partitioning the total output variance into contributions from each input. However, computing conditional expectations can be prohibitively costly when working with expensive-to-evaluate models. Surrogate models can accelerate this, yet their accuracy depends on the quality and quantity of training data, which is expensive to generate (experimentally or computationally) for complex engineering systems. Thus, methods that work with limited data are desirable. We propose a diffeomorphic modulation under observable response preserving homotopy (D-MORPH) regression to train a polynomial dimensional decomposition surrogate of the output that minimizes the number of training data. The new method first computes a sparse Lasso solution and uses it to define the cost function. A subsequent D-MORPH regression minimizes the difference between the D-MORPH and Lasso solution. The resulting D-MORPH surrogate is more robust to input variations and more accurate with limited training data. We illustrate the accuracy and computational efficiency of the new surrogate for global sensitivity analysis using mathematical functions and an expensive-to-simulate model of char combustion. The new method is highly efficient, requiring only 15% of the training data compared to conventional regression.
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The nonlocality of the fractional operator causes numerical difficulties for long time computation of the time-fractional evolution equations. This paper develops a high-order fast time-stepping discontinuous Galerkin finite element method for the time-fractional diffusion equations, which saves storage and computational time. The optimal error estimate $O(N^{-p-1} + h^{m+1} + \varepsilon N^{r\alpha})$ of the current time-stepping discontinuous Galerkin method is rigorous proved, where $N$ denotes the number of time intervals, $p$ is the degree of polynomial approximation on each time subinterval, $h$ is the maximum space step, $r\ge1$, $m$ is the order of finite element space, and $\varepsilon>0$ can be arbitrarily small. Numerical simulations verify the theoretical analysis.
We prove a discrete analogue for the composition of the fractional integral and Caputo derivative. This result is relevant in numerical analysis of fractional PDEs when one discretizes the Caputo derivative with the so-called L1 scheme. The proof is based on asymptotic evaluation of the discrete sums with the use of the Euler-Maclaurin summation formula.
Stochastic optimization methods have been hugely successful in making large-scale optimization problems feasible when computing the full gradient is computationally prohibitive. Using the theory of modified equations for numerical integrators, we propose a class of stochastic differential equations that approximate the dynamics of general stochastic optimization methods more closely than the original gradient flow. Analyzing a modified stochastic differential equation can reveal qualitative insights about the associated optimization method. Here, we study mean-square stability of the modified equation in the case of stochastic coordinate descent.
Estimating parameters from data is a fundamental problem in physics, customarily done by minimizing a loss function between a model and observed statistics. In scattering-based analysis, researchers often employ their domain expertise to select a specific range of wavevectors for analysis, a choice that can vary depending on the specific case. We introduce another paradigm that defines a probabilistic generative model from the beginning of data processing and propagates the uncertainty for parameter estimation, termed ab initio uncertainty quantification (AIUQ). As an illustrative example, we demonstrate this approach with differential dynamic microscopy (DDM) that extracts dynamical information through Fourier analysis at a selected range of wavevectors. We first show that DDM is equivalent to fitting a temporal variogram in the reciprocal space using a latent factor model as the generative model. Then we derive the maximum marginal likelihood estimator, which optimally weighs information at all wavevectors, therefore eliminating the need to select the range of wavevectors. Furthermore, we substantially reduce the computational cost by utilizing the generalized Schur algorithm for Toeplitz covariances without approximation. Simulated studies validate that AIUQ significantly improves estimation accuracy and enables model selection with automated analysis. The utility of AIUQ is also demonstrated by three distinct sets of experiments: first in an isotropic Newtonian fluid, pushing limits of optically dense systems compared to multiple particle tracking; next in a system undergoing a sol-gel transition, automating the determination of gelling points and critical exponent; and lastly, in discerning anisotropic diffusive behavior of colloids in a liquid crystal. These outcomes collectively underscore AIUQ's versatility to capture system dynamics in an efficient and automated manner.
We study matrix sensing, which is the problem of reconstructing a low-rank matrix from a few linear measurements. It can be formulated as an overparameterized regression problem, which can be solved by factorized gradient descent when starting from a small random initialization. Linear neural networks, and in particular matrix sensing by factorized gradient descent, serve as prototypical models of non-convex problems in modern machine learning, where complex phenomena can be disentangled and studied in detail. Much research has been devoted to studying special cases of asymmetric matrix sensing, such as asymmetric matrix factorization and symmetric positive semi-definite matrix sensing. Our key contribution is introducing a continuous differential equation that we call the $\textit{perturbed gradient flow}$. We prove that the perturbed gradient flow converges quickly to the true target matrix whenever the perturbation is sufficiently bounded. The dynamics of gradient descent for matrix sensing can be reduced to this formulation, yielding a novel proof of asymmetric matrix sensing with factorized gradient descent. Compared to directly analyzing the dynamics of gradient descent, the continuous formulation allows bounding key quantities by considering their derivatives, often simplifying the proofs. We believe the general proof technique may prove useful in other settings as well.
We aim to establish Bowen's equations for upper capacity invariance pressure and Pesin-Pitskel invariance pressure of discrete-time control systems. We first introduce a new invariance pressure called induced invariance pressure on partitions that specializes the upper capacity invariance pressure on partitions, and then show that the two types of invariance pressures are related by a Bowen's equation. Besides, to establish Bowen's equation for Pesin-Pitskel invariance pressure on partitions we also introduce a new notion called BS invariance dimension on subsets. Moreover, a variational principle for BS invariance dimension on subsets is established.
Multivariate imputation by chained equations (MICE) is one of the most popular approaches to address missing values in a data set. This approach requires specifying a univariate imputation model for every variable under imputation. The specification of which predictors should be included in these univariate imputation models can be a daunting task. Principal component analysis (PCA) can simplify this process by replacing all of the potential imputation model predictors with a few components summarizing their variance. In this article, we extend the use of PCA with MICE to include a supervised aspect whereby information from the variables under imputation is incorporated into the principal component estimation. We conducted an extensive simulation study to assess the statistical properties of MICE with different versions of supervised dimensionality reduction and we compared them with the use of classical unsupervised PCA as a simpler dimensionality reduction technique.
In this paper, efficient alternating direction implicit (ADI) schemes are proposed to solve three-dimensional heat equations with irregular boundaries and interfaces. Starting from the well-known Douglas-Gunn ADI scheme, a modified ADI scheme is constructed to mitigate the issue of accuracy loss in solving problems with time-dependent boundary conditions. The unconditional stability of the new ADI scheme is also rigorously proven with the Fourier analysis. Then, by combining the ADI schemes with a 1D kernel-free boundary integral (KFBI) method, KFBI-ADI schemes are developed to solve the heat equation with irregular boundaries. In 1D sub-problems of the KFBI-ADI schemes, the KFBI discretization takes advantage of the Cartesian grid and preserves the structure of the coefficient matrix so that the fast Thomas algorithm can be applied to solve the linear system efficiently. Second-order accuracy and unconditional stability of the KFBI-ADI schemes are verified through several numerical tests for both the heat equation and a reaction-diffusion equation. For the Stefan problem, which is a free boundary problem of the heat equation, a level set method is incorporated into the ADI method to capture the time-dependent interface. Numerical examples for simulating 3D dendritic solidification phenomenons are also presented.
Determining the number of factors in high-dimensional factor modeling is essential but challenging, especially when the data are heavy-tailed. In this paper, we introduce a new estimator based on the spectral properties of Spearman sample correlation matrix under the high-dimensional setting, where both dimension and sample size tend to infinity proportionally. Our estimator is robust against heavy tails in either the common factors or idiosyncratic errors. The consistency of our estimator is established under mild conditions. Numerical experiments demonstrate the superiority of our estimator compared to existing methods.
A standard approach to solve ordinary differential equations, when they describe dynamical systems, is to adopt a Runge-Kutta or related scheme. Such schemes, however, are not applicable to the large class of equations which do not constitute dynamical systems. In several physical systems, we encounter integro-differential equations with memory terms where the time derivative of a state variable at a given time depends on all past states of the system. Secondly, there are equations whose solutions do not have well-defined Taylor series expansion. The Maxey-Riley-Gatignol equation, which describes the dynamics of an inertial particle in nonuniform and unsteady flow, displays both challenges. We use it as a test bed to address the questions we raise, but our method may be applied to all equations of this class. We show that the Maxey-Riley-Gatignol equation can be embedded into an extended Markovian system which is constructed by introducing a new dynamical co-evolving state variable that encodes memory of past states. We develop a Runge-Kutta algorithm for the resultant Markovian system. The form of the kernels involved in deriving the Runge-Kutta scheme necessitates the use of an expansion in powers of $t^{1/2}$. Our approach naturally inherits the benefits of standard time-integrators, namely a constant memory storage cost, a linear growth of operational effort with simulation time, and the ability to restart a simulation with the final state as the new initial condition.
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