College of Liberal Arts & Sciences

# International Workshop on Applied Mathematics & Statistics 2020

**International Workshop on Applied Mathematics and Statistics**

(Online Conference)

This workshop aims to provide a platform for the Ph.D. students to communicate their collective knowledge and interact. About 10 Ph.D. students will present their recent work on applied mathematics and statistics. Everyone is welcome.

**Practical information: **

- Zoom conference number: 922-2358-9615
- Passcode: 18121950

**Date and Time:**

UTC/GMT-06:00, Iowa Local Time

Date: 28^{th} December 2020 (Mon)

Time: 18:45-22:00 and 23:30-2:00

UTC/GMT+08:00, Shanghai Local Time

Date: 29^{th} December 2020 (TUE)

Time: 8:45-12:00 and 13:30-16:00

**Organizing committee: **

Weimin Han (UI)

Tong Li (UI)

Lihe Wang(UI)

Xueye Zhu(UI)

Samuel Drapeau (SJTU)

Yiqing Lin (SJTU)

**Sponsor:**

Department of Mathematics, University of Iowa

**Program**

**Session I**

18 : 45 – 18 : 50 **Opening address by Prof. Lihe Wang (UI)**

**Hosted by Weimin Han (UI)**

18 : 50—19 : 15 **Mingxiu Sui (UI)**

Title: Finite Element Method for Stochastic Variational Inequalities.

19 : 15 – 19 : 40 **Bryanna Petentler (UI)**

Title: Integral estimates for a class of degenerate equations

19 : 40 – 20 : 05 **Mati Ur Rahman (SJTU)**

Title: Investigation of fractional order tuberculosis (TB) model via Caputo derivative

20 : 05 – 20 : 30 **Chuan Lu (UI)**

Title: Bi-Fidelity Data-Assisted Neural Networks in Nonintrusive Reduced-Order Modeling

**Hosted by Xueyu Zhu (UI)**

20 : 30 – 20 : 55 **Manuel Rissel (SJTU) **

Title: Exponential stability for a coupled system of elasticity and thermoelasticity with second sound

20 : 55 – 21 : 20 **Xin Gao (SJTU) **

Title: On admissible positions of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle

21 : 20 – 21 : 45 **Yasir Nadeem Anjam (SJTU) **

Title: Geometric singularities of the solution of electromagnetic fields in two-dimensional non-smooth domains

21 : 45 – 22 : 10 **Zehang Li (SJTU) **

Title: Diagnostic accuracy of a fast computational approach to derive fractional flow reserve from coronary CT angiography

**Session II**

**Hosted by Samuel Drapeau and Yiqing Lin (SJTU)**

23 : 30 – 23 : 55 **Abdullah Qayed (SJTU)**

Title: Homogeneity test of several high-dimensional covariance matrices for stationary processes under non-normality

23 : 55 – 0 : 20 **Yunbo Zhang (SJTU)**

Title: Pricing and hedging performance on pegged FX markets based on a regime switching model

0: 20 – 0 : 45 **Xuan Tao (SJTU)**

Title: On detecting spoofing strategies in high frequency trading

**Hosted by Shan Luo (SJTU)**

0 : 45 – 1 : 10 **Jiyuan Tu (SJTU) **

Title: Variance reduced median-of-means estimator for Byzantine-robust distributed inference

1 : 10 – 1 : 35 **Zeyu Wu (SJTU) **

Title: High dimensional precision matrix estimation under asymptotic sparsity

1 : 35 – 2: 00 **Lei Qiao (SJTU) **

Title: Optimal sequential tests for detection of changes under finite measure space for finite sequences of networks

**Abstract**

**Finite element method for stochastic variational inequalities**

*Mingxiu Sui (UI)*

**Abstract.** I consider a stochastic variational inequality of the second kind with additive random noise in the force term, where the random noise is represented by a series in terms of standard or truncated standard normal variables and orthogonal basis functions. Solution existence and uniqueness are explored and statistical error bounds for finite element approximations are derived. A numerical example is included to illustrate the effectiveness of the method.

**Integral estimates for a class of degenerate equations**

*Bryanna Petentler (UI)*

**Abstract.** We prove Caldron-Zygmund type estimates for a class of degenerate equations that have subelliptic type of a nonnegative real number. These estimates are known only when the type was an integer before this work.

**Investigation of fractional order tuberculosis (TB) model via Caputo derivative**

*Mati Ur Rahman (SJTU)*

**Abstract.** In this article, we investigate the dynamics of Tuberculosis-(TB) with the help of fractional order model. To obtain the numerical results, we use the Laplace transform with Adomian decomposition method (LADM) and Homotopy perturbation method (HPM). Both the obtained numerical solutions are compared with each other. The concerned results are simulated corresponding to some real data for various frac- tional order, using Matlab. The simulations show the suitability of the considered Tuberculosis-(TB).

**Bi-Fidelity Data-Assisted Neural Networks in Nonintrusive Reduced-Order Modeling **

*Chuan Lu (UI)*

**Abstract.** We present a new nonintrusive reduced basis method when a cheap low-fidelity model and an expensive high-fidelity model are available. The method employs proper orthogonal decomposition method (POD) to generate the high-fidelity reduced basis and a shallow multilayer perceptron to learn the high-fidelity reduced coefficients. In contrast to previously proposed methods, besides the model parameters, we also augmented the features extracted from the data generated by an efficient bi-fidelity surrogate as the input feature of the proposed neural network. By incorporating relevant bi-fidelity features, we demonstrate that such an approach can improve the predictive capability and robustness of the neural network via several benchmark examples.

**Exponential stability for a coupled system of elasticity and thermoelasticity with second sound**

*Manuel Rissel (SJTU)*

**Abstract.** This talk is concerned with exponential stability for solutions to a linear transmission problem in one spatial dimension, modeling the evolution of an elastic-thermoelastic-elastic bar. The thermoelastic middle part of the bar is subject to either Cattaneo’s or Fourier’s law for heat conduction while the elastic parts are described by a wave equation. Therefore, a natural question, which is answered positively, is whether the dissipation localized in the middle part of the bar is sufficient for rendering the whole system exponentially stable. This issue is investigated from the linear operator semigroup point of view, with an emphasis on the case with second sound, namely when Cattaneo’s law for heat conduction is employed. In particular, by means of uniform resolvent bounds for the underlying generator, it is shown that, as time goes to infinity, every solution converges with an exponential rate to a stationary state of the system. This talk is based on a joint work with Ya-Guang Wang.

**On admissible positions of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle **

*Xin Gao (SJTU)*

**Abstract.** This paper concerns with the existence of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle, which are governed by the Euler equations with the slip boundary condition on the wall of the nozzle and a receiver pressure at the exit. Mathematically, it can be formulated as a free boundary problem with the shock front being the free boundary to be determined. In dealing with the free boundary problem, one of the key points is determining the position of the shock front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution gives an initial approximating position of the shock front. Compared with 2-D case, new difficulties arise due to the additional 0-order terms and singularities along the symmetric axis. New observation and careful analysis will be done to overcome these difficulties. Once the initial approximation is obtained, a nonlinear iteration scheme can be carried out, which converges to a transonic shock solution to the problem.

**Geometric singularities of the solution of electromagnetic fields in two-dimensional non-smooth domains **

*Yasir Nadeem Anjam (SJTU)*

**Abstract.** In this talk, we will present a constructive mathematical framework for treating the H2-regularity analysis of the solutions of the generalized boundary value problem for the classical time-harmonic Maxwell's equations and regularized Maxwell's equations in two-dimensional domains with corners. We will use the discrete Fourier transform to investigate the singular behavior of the solution structure near the corners. It is shown that the solution is split into the form of singular and regular parts near the corner points for the values of non-acute angles, and does not belong locally to space H2. The explicit formulas for the singularity functions and their corresponding coefficients are given. We will finish by giving some regularity results.

**Diagnostic accuracy of a fast computational approach to derive fractional flow reserve from coronary CT angiography **

*Zehang Li (SJTU)*

**Abstract.** Aims: The goal of this study was to evaluate the diagnostic accuracy of a new computational method for identifying hemodynamically significant stenoses in patients with coronary artery disease (CAD) from coronary CT angiography (CTA), using fractional flow reserve (FFR) as the reference standard. Methods and Results: Lumen borders of the coronary arterial tree were first delineated using a validated software package. Coronary flow was estimated based on the size of reference coronary trees as if there was no stenosis. Finally, the quantitative flow ratio (QFR), a novel computational algorithm to calculate FFR from imaging data without using a pressure wire, was applied to derive CTA-based QFR (CT-QFR). Comparison of CT-QFR and FFR was retrospectively performed in 156 vessels from 134 consecutive patients who underwent both coronary CTA and FFR assessment. Average FFR was 0.82±0.10 and 41.7% vessels were hemodynamically significant. AUC of CT-QFR for identifying vessels with FFR≤0.80 was 0.92, substantially higher than diameter stenosis (increase=0.16,p<0.001) and minimal lumen area (increase=0.20,p<0.001). Accuracy, sensitivity, specificity, positive predictive value and negative predictive value were 87.2%[95%CI:81.9%–92.5%], 87.7%[95%CI:77.2%-94.5%], 86.8%[95%CI:78.1%-93.0%], 82.6%[95%CI:71.6%-90.7%] and 90.8%[95%CI:82.7%-95.9%], respectively. CT-QFR in extensively calcified vessels had numerically lower correlation with FFR (r=0.76 versus r=0.81,p=0.444) and higher variability in repeat analysis (0.06 versus 0.04,p=0.013) compared with that in vessels with no or less calcification. Conclusion: Integration of CT-QFR with coronary CTA imaging improved its diagnostic performance in identifying hemodynamically significant coronary stenosis. The diagnostic performance and reproducibility of CT-QFR were influenced by extensively calcified lesions.

**Homogeneity test of several high-dimensional covariance matrices for stationary processes under non-normality **

*Abdullah Qayed (SJTU)*

**Abstract.** We propose a test for testing the equality of several high-dimensional covariance matrices for stationary processes with a general distribution. The asymptotic distribution of the proposed test is proved to be $ \chi^2 $ distribution. Both the numerical simulation and empirical study illustrate that the proposed test has perfect performance, in particular, its power can approach to one uniformly on a set of covariance matrices with three known distributions.

**Pricing and hedging performance on pegged FX markets based on a regime switching model**

*Yunbo Zhang (SJTU)*

**Abstract.** This paper investigates the hedging performance of pegged foreign exchange market in a regime switching (RS) model introduced in \cite{drapeau2019}. We compare two prices, an exact solution and first order approximation and provide the bounds for the error. We provide exact RS delta, approximated RS delta as well as mean variance hedging strategies for this specific model and compare their performance. To improve the efficiency of the pricing and calibration procedure, the Fourier approach of this regime-switching model is developed in our work. It turns out that: 1 -- the calibration of the volatility surface with this regime switching model outperforms on real data the classical SABR model; 2 -- the Fourier approach is significantly faster than the direct approach; 3 -- in terms of hedging, the approximated RS delta hedge is a viable alternative to the exact RS delta hedge while significantly faster.

**On detecting spoofing strategies in high frequency trading**

*Xuan Tao (SJTU)*

**Abstract.** Spoofing is an illegal act of artificially modifying the supply to drive temporarily prices in a given direction for profit. In practice, detection of such an act is challenging due to the complexity of modern electronic platforms and the high frequency at which orders are channeled. We present a micro-structural study of spoofing in a simple static setting. A multilevel imbalance which influences the resulting price movement is introduced upon which we describe the optimization strategy of a potential spoofer. We provide conditions under which a market is more likely to admit spoofing behavior as a function of the characteristics of the market. We describe the optimal spoofing strategy after optimization which allows us to quantify the resulting impact on the imbalance after spoofing. Based on these results we calibrate the model to real Level 2 datasets from TMX, and provide some monitoring procedures based on the Wasserstein distance to detect spoofing strategies in real time.

**Variance reduced median-of-means estimator for Byzantine-robust distributed inference **

*Jiyuan Tu (SJTU)*

**Abstract.** This paper develops an efficient distributed inference algorithm, which is robust against a moderate fraction of Byzantine nodes, namely arbitrary and possibly adversarial machines in a distributed learning system. In robust statistics, the median-of-means (MOM) has been a popular approach to hedge against Byzantine failures due to its ease of implementation and computational efficiency. However, the MOM estimator has the shortcoming in terms of statistical efficiency. The first main contribution of the paper is to propose a variance reduced median-of-means (VRMOM) estimator, which improves the statistical efficiency over the vanilla MOM estimator and is computationally as efficient as the MOM. Based on the proposed VRMOM estimator, we develop a general distributed inference algorithm that is robust against Byzantine failures. Theoretically, our distributed algorithm achieves a fast convergence rate with only a constant number of rounds of communications. We also provide the asymptotic normality result for the purpose of statistical inference. To the best of our knowledge, this is the first normality result in the setting of Byzantine-robust distributed learning. The simulation results are also presented to illustrate the effectiveness of our method.

**High dimensional precision matrix estimation under asymptotic sparsity **

*Zeyu Wu (SJTU)*

**Abstract.** In this paper, we focus on estimating the high dimensional precision matrix under the asymptotic sparse condition. We revisit the SCIO method proposed by \cite{liu2015fast} and derive general error bounds under regular conditions. The common irrepresentable condition is relaxed and the results are applicable to the asymptotic sparse matrices. As applications, we also study the precision matrix estimation for the heavy-tailed data, the non-paranormal data, and the matrix data.

**Optimal sequential tests for detection of changes under finite measure space for finite sequences of networks **

*Lei Qiao (SJTU)*

**Abstract.** This paper considers the change-point problem for finite sequences of networks. To avoid the difficulty of computing the normalization coefficient in the models such as Exponential Random Graphical Model (ERGM) and Markov networks, we construct a finite measure space with measure ratio statistics. A new performance measure of detection delay is proposed to detect the changes in distribution of the network data. And under the performance measure we defined, an optimal sequential test is presented. The good performance of the optimal sequential test is illustrated numerically on ERGM and Erdős-Rényi network sequences.