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University of Kansas
1.
Lindsey, Theodore S.
On the Kalman Filter and Its Variations.
Degree: MA, Mathematics, 2014, University of Kansas
URL: http://hdl.handle.net/1808/14535 
► The objective of this paper is to explore the standard Kalman filter and two non-linear variations. Additionally, we will discuss the derivation of the Kalman…
(more)
▼ The objective of this paper is to explore the standard Kalman filter and two non-linear variations. Additionally, we will discuss the derivation of the Kalman filter using Newton's method. Next we will consider the implementation of both the Extended Kalman filter and the Unscented Kalman filter, paying special attention to the cases where the Unscented Kalman filter performs better than the Extended Kalman filter. Finally, we will make a comparison between these two Kalman filter variations and consider a few other modifications to the standard Kalman filter.
Advisors/Committee Members: Pasik-Duncan, Bozenna (advisor), Duncan, Tyrone (cmtemember), Tu, Xuemin (cmtemember).
Subjects/Keywords: Mathematics; Kalman filter
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APA (6th Edition):
Lindsey, T. S. (2014). On the Kalman Filter and Its Variations. (Masters Thesis). University of Kansas. Retrieved from http://hdl.handle.net/1808/14535
Chicago Manual of Style (16th Edition):
Lindsey, Theodore S. “On the Kalman Filter and Its Variations.” 2014. Masters Thesis, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/14535.
MLA Handbook (7th Edition):
Lindsey, Theodore S. “On the Kalman Filter and Its Variations.” 2014. Web. 25 Jan 2021.
Vancouver:
Lindsey TS. On the Kalman Filter and Its Variations. [Internet] [Masters thesis]. University of Kansas; 2014. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/14535.
Council of Science Editors:
Lindsey TS. On the Kalman Filter and Its Variations. [Masters Thesis]. University of Kansas; 2014. Available from: http://hdl.handle.net/1808/14535
University of Kansas
2.
Wang, Yao.
Efficient Tunnel Detection with Waveform Inversion of Back-scattered Surface Waves.
Degree: MA, Mathematics, 2019, University of Kansas
URL: http://hdl.handle.net/1808/30104
► An efficient subsurface imaging method employing back-scattered surface waves is developed to detect near-surface underground elastic-wave velocity anomalies, such as tunnels, sinkholes, fractures, faults, and…
(more)
▼ An efficient subsurface imaging method employing back-scattered surface waves is developed to detect near-surface underground elastic-wave velocity anomalies, such as tunnels, sinkholes, fractures, faults, and abandoned manmade infrastructures. The back-scattered surface waves are generated by seismic waves impinging on the velocity anomalies and diffracting back toward the source. These wave events contain plentiful information of the subsurface velocity anomalies including spatial location, shape, size, and velocity of the interior medium. Studies have demonstrated that the back-scattered surface waves can be easily distinguished in the frequency-wavenumber (F-k) domain and have less interference by other wave modes. Based on these features, a near-surface velocity anomaly detection method by using waveform inversion of the back-scattered surface waves (BSWI) is proposed. The main objective of this thesis is to review the theoretical background and study the feasibility of the proposed BSWI method. The proposed BSWI method is tested with numerical and real-world examples. First, the numerical example uses the conventional full-waveform inversion (FWI) method as a benchmark to demonstrate the efficiency of BSWI method in detecting shallow velocity anomalies. Then, the BSWI method is tested with field data. In this study, 2D seismic data were acquired over a manmade concrete tunnel located on the main campus of the
University of
Kansas (KU). Different workflows including FWI method and BSWI method are applied to the acquired data and tested for imaging the known tunnel. The field example demonstrates that BSWI can accurately image the tunnel. Compared with FWI, BSWI is less demanding in data processing. Finally, this thesis concludes that the proposed BSWI method is capable of efficiently detecting a near-surface tunnel with the minimum amount of data processing which lends it as a method suitable for application in the field.
Advisors/Committee Members: Tu, Xuemin (advisor), Xu, Hongguo (cmtemember), Tsoflias, Georgios P (cmtemember).
Subjects/Keywords: Mathematics; Geophysical engineering; back-scatter; surface waves; waveform inversion
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APA (6th Edition):
Wang, Y. (2019). Efficient Tunnel Detection with Waveform Inversion of Back-scattered Surface Waves. (Masters Thesis). University of Kansas. Retrieved from http://hdl.handle.net/1808/30104
Chicago Manual of Style (16th Edition):
Wang, Yao. “Efficient Tunnel Detection with Waveform Inversion of Back-scattered Surface Waves.” 2019. Masters Thesis, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/30104.
MLA Handbook (7th Edition):
Wang, Yao. “Efficient Tunnel Detection with Waveform Inversion of Back-scattered Surface Waves.” 2019. Web. 25 Jan 2021.
Vancouver:
Wang Y. Efficient Tunnel Detection with Waveform Inversion of Back-scattered Surface Waves. [Internet] [Masters thesis]. University of Kansas; 2019. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/30104.
Council of Science Editors:
Wang Y. Efficient Tunnel Detection with Waveform Inversion of Back-scattered Surface Waves. [Masters Thesis]. University of Kansas; 2019. Available from: http://hdl.handle.net/1808/30104
University of Kansas
3.
Song, Xiaoming.
Malliavin calculus for backward stochastic differential equations and stochastic differential equations driven by fractional Brownian motion and numerical schemes.
Degree: PhD, Mathematics, 2011, University of Kansas
URL: http://hdl.handle.net/1808/7836
► In this dissertation, I investigate two types of stochastic differential equations driven by fractional Brownian motion and backward stochastic differential equations. Malliavin calculus is a…
(more)
▼ In this dissertation, I investigate two types of stochastic differential equations driven by
fractional Brownian motion and backward stochastic differential equations. Malliavin
calculus is a powerful tool in developing the main results in this dissertation.
This dissertation is organized as follows.
In Chapter 1, I introduce some notations and preliminaries on Malliavin Calculus
for both Brownian motion and fractional Brownian motion.
In Chapter 2, I study backward stochastic differential equations with general terminal
value and general random generator. In particular, the terminal value has not
necessary to be given by a forward diffusion equation. The randomness of the generator
does not need to be from a forward equation neither. Motivated from applications to
numerical simulations, first the Lp-H¨older continuity of the solution is obtained. Then,
several numerical approximation schemes for backward stochastic differential equations
are proposed and the rate of convergence of the schemes is established based on
the obtained Lp-H¨older continuity results.
Chapter 3 is concerned with a singular stochastic differential equation driven by
an additive one-dimensional fractional Brownian motion with Hurst parameter H > 1
2 .
Under some assumptions on the drift, we show that there is a unique solution, which
has moments of all orders. We also apply the techniques of Malliavin calculus to prove
that the solution has an absolutely continuous law at any time t > 0.
In Chapter 4, I am interested in some approximation solutions of a type of stochastic
differential equations driven by multi-dimensional fractional Brownian motion BH
with Hurst parameter H > 1
2 . In order to obtain an optimal rate of convergence, some
techniques are developed in the deterministic case. Some work in progress is contained
in this chapter.
The results obtained in Chapter 2 are accepted by the Annals of Applied Probability,
and the material contained in Chapter 3 has been published in Statistics and Probability
Letters 78 (2008) 2075-2085.
Advisors/Committee Members: Hu, Yaozhong (advisor), Nualart, David (advisor), Feng, Jin (cmtemember), Koch, Paul D. (cmtemember), Tu, Xuemin (cmtemember).
Subjects/Keywords: Mathematics
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APA (6th Edition):
Song, X. (2011). Malliavin calculus for backward stochastic differential equations and stochastic differential equations driven by fractional Brownian motion and numerical schemes. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/7836
Chicago Manual of Style (16th Edition):
Song, Xiaoming. “Malliavin calculus for backward stochastic differential equations and stochastic differential equations driven by fractional Brownian motion and numerical schemes.” 2011. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/7836.
MLA Handbook (7th Edition):
Song, Xiaoming. “Malliavin calculus for backward stochastic differential equations and stochastic differential equations driven by fractional Brownian motion and numerical schemes.” 2011. Web. 25 Jan 2021.
Vancouver:
Song X. Malliavin calculus for backward stochastic differential equations and stochastic differential equations driven by fractional Brownian motion and numerical schemes. [Internet] [Doctoral dissertation]. University of Kansas; 2011. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/7836.
Council of Science Editors:
Song X. Malliavin calculus for backward stochastic differential equations and stochastic differential equations driven by fractional Brownian motion and numerical schemes. [Doctoral Dissertation]. University of Kansas; 2011. Available from: http://hdl.handle.net/1808/7836
University of Kansas
4.
Hu, Guannan.
Fractional Diffusion in Gaussian Noisy Environment.
Degree: PhD, Mathematics, 2015, University of Kansas
URL: http://hdl.handle.net/1808/21696
► Three types of stochastic partial differential equations are studied in this dissertation. We prove the existence and uniqueness of the solutions and obtain some properties…
(more)
▼ Three types of stochastic partial differential equations are studied in this dissertation. We prove the existence and uniqueness of the solutions and obtain some properties of the solutions. Chapter 3 studies the linear stochastic partial differential equation of fractional orders both in time and space variables. We prove the existence and uniqueness of the solution and calculate the moment bounds of the solution when the noise has Reisz kernel as space covariance. Along the way, we obtain some new properties of the fundamental solutions. Chapter 4 studies the time-fractional diffusion with fractional Gaussian noise. We obtain conditions so that the square integrable solution exists uniquely. Chapter 5 studies the time-fractional diffusion where the Gaussian noise is general in time with space covariance given by fractional, Riesz and Bessel kernel.
Advisors/Committee Members: Hu, Yaozhong (advisor), Nualart, David (cmtemember), Soo, Terry (cmtemember), Tu, Xuemin (cmtemember), Huan, Luke (cmtemember).
Subjects/Keywords: Mathematics; chaos expansion; Fox's H-function;
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APA ·
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MLA ·
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Export
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APA (6th Edition):
Hu, G. (2015). Fractional Diffusion in Gaussian Noisy Environment. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/21696
Chicago Manual of Style (16th Edition):
Hu, Guannan. “Fractional Diffusion in Gaussian Noisy Environment.” 2015. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/21696.
MLA Handbook (7th Edition):
Hu, Guannan. “Fractional Diffusion in Gaussian Noisy Environment.” 2015. Web. 25 Jan 2021.
Vancouver:
Hu G. Fractional Diffusion in Gaussian Noisy Environment. [Internet] [Doctoral dissertation]. University of Kansas; 2015. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/21696.
Council of Science Editors:
Hu G. Fractional Diffusion in Gaussian Noisy Environment. [Doctoral Dissertation]. University of Kansas; 2015. Available from: http://hdl.handle.net/1808/21696
University of Kansas
5.
Han, Zheng.
Reflected diffusions and applications to finance and operations management.
Degree: PhD, Mathematics, 2015, University of Kansas
URL: http://hdl.handle.net/1808/21698
► This dissertation provides explicit solutions to four special stochastic optimal control problems for reflected diffusions and Markov modulated reflected diffusions. The main mathematical tool that…
(more)
▼ This dissertation provides explicit solutions to four special stochastic optimal control problems for reflected diffusions and Markov modulated reflected diffusions. The main mathematical tool that we use is the ergodic theory for stochastic differential equations (SDE’s), in particular, the ergodic theory for reflected diffusions. First, we present some basic definitions of reflected diffusions, the Itô’s formulas. After this we study the ergodic theorems, namely, the law of large numbers, for reflected diffusions. The reflected diffusions and Markov modulated reflected diffusions are used to simulate bounded price dynamics regulated by policy makers. Motivated by this, we analyze several stochastic optimal control problems for reflected diffusions (or for Markov modulated reflected diffusions). These stochastic optimal control problems are solved in the sense that we can reduce them to some explicit optimization problems. Numerical computations are also given for the reduced optimization problems. Our work resulted in one accepted paper, one completed paper and two on-going projects.
Advisors/Committee Members: Hu, Yaozhong (advisor), Nualart, David (cmtemember), Liu, Weishi (cmtemember), Tu, Xuemin (cmtemember), Zhang, Jianbo (cmtemember).
Subjects/Keywords: Mathematics; Finance; Operations research; Ergodic theory; Optimal barrier; Reflected diffusions
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APA (6th Edition):
Han, Z. (2015). Reflected diffusions and applications to finance and operations management. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/21698
Chicago Manual of Style (16th Edition):
Han, Zheng. “Reflected diffusions and applications to finance and operations management.” 2015. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/21698.
MLA Handbook (7th Edition):
Han, Zheng. “Reflected diffusions and applications to finance and operations management.” 2015. Web. 25 Jan 2021.
Vancouver:
Han Z. Reflected diffusions and applications to finance and operations management. [Internet] [Doctoral dissertation]. University of Kansas; 2015. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/21698.
Council of Science Editors:
Han Z. Reflected diffusions and applications to finance and operations management. [Doctoral Dissertation]. University of Kansas; 2015. Available from: http://hdl.handle.net/1808/21698
University of Kansas
6.
Li, Chaozheng.
Learning, Observability and Time-varying Macroeconomic Volatility.
Degree: PhD, Economics, 2018, University of Kansas
URL: http://hdl.handle.net/1808/27889
► Our paper provides a theoretical explanation for the time-varying macroeconomic volatility by introducing the unobservability of regime switching and learning. With the unobservability of regime…
(more)
▼ Our paper provides a theoretical explanation for the time-varying macroeconomic volatility by introducing the unobservability of regime switching and learning. With the unobservability of regime switching, agents must endogenously form their expectations using best-performed forecasting models. We find that if the regime switching is observable to agents, agents do not shift their expectation frequently and so will not generate a larger macroeconomic volatility. However, with the unobservability of regime switching where no agents can know which regime is dominant, allowing endogenous expectation formation would give rise to larger macro fluctuations (first-layer amplification mechanism), which is made through agents frequently shifting their expectations. Furthermore, we consider the policy implication under the zero lower bound. Our simulations show that in the unobservable regime switching, the economy is more likely to fall into a deflationary trap. To avoid the deflation risk, the policy maker should set a higher expected inflation based threshold. If the expected inflation is under the threshold, an aggressive policy rate will be implemented; otherwise, the normal Taylor-rule monetary policy will be used. Furthermore, to reduce the deflation risk, the strategy for the policy maker is to raise the threshold, and this will generate larger macroeconomic fluctuations (second-layer amplification mechanism) due to more frequent policy strategy switching. We argue that sometimes only with unobservability, the policy maker faces a dilemma between avoiding deflation risk and maintaining macroeconomic stability, and huge macroeconomic fluctuations do not necessarily result from bad luck or bad policy but from the two-layer amplification mechanism caused by the unobservability.
Advisors/Committee Members: Iwata, Shigeru (advisor), Zhang, Jianbo (cmtemember), Sabarwal, Tarun (cmtemember), Comolli, Paul (cmtemember), Tu, Xuemin (cmtemember).
Subjects/Keywords: Economics; Learning; Macroeconomic Volatility; Observability; Regime Switching; Time-varying
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APA ·
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APA (6th Edition):
Li, C. (2018). Learning, Observability and Time-varying Macroeconomic Volatility. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/27889
Chicago Manual of Style (16th Edition):
Li, Chaozheng. “Learning, Observability and Time-varying Macroeconomic Volatility.” 2018. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/27889.
MLA Handbook (7th Edition):
Li, Chaozheng. “Learning, Observability and Time-varying Macroeconomic Volatility.” 2018. Web. 25 Jan 2021.
Vancouver:
Li C. Learning, Observability and Time-varying Macroeconomic Volatility. [Internet] [Doctoral dissertation]. University of Kansas; 2018. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/27889.
Council of Science Editors:
Li C. Learning, Observability and Time-varying Macroeconomic Volatility. [Doctoral Dissertation]. University of Kansas; 2018. Available from: http://hdl.handle.net/1808/27889
University of Kansas
7.
Jia, Bao.
Experimental and numerical investigations of shale gas permeability.
Degree: PhD, Chemical & Petroleum Engineering, 2018, University of Kansas
URL: http://hdl.handle.net/1808/29537
► Shale gas has become an essential source of energy supply in the United States and the world. It is important to analyze shale gas flow…
(more)
▼ Shale gas has become an essential source of energy supply in the United States and the world. It is important to analyze shale gas flow behavior and the associated production behavior from experimental and simulation perspectives. The specific objectives in this study are: (1) analyze shale gas flow behavior in the laboratory using pulse-decay experiments in both un-fractured and fractured cores, which will be accomplished in chapter 2 and chapter 3, respectively (2) propose a workflow to estimate shale gas permeability evolution taking into account multiple physics in the organic-rich matrix, which will be accomplished in chapter 4 (3) perform shale gas production simulations to justify the importance of these physics for shale gas production volume, which will be accomplished in chapter 5. Pulse-decay experiments using adsorptive gas (nitrogen and carbon dioxide) and non-adsorptive gas relate complex relationships between gas apparent porosity and adsorption (in forms of Gibbs/excess and absolute), and gas apparent permeability and adsorption. Among the three types of gas, helium apparent permeability is the highest. The ratio between apparent porosity and intron porosity of tight porous media is the summation of unity and the density ratio between the adsorbed phase and the free gas phase. A coefficient different from the density ratio is involved in the flow governing equation of adsorptive gas, which is in accordance with the adsorption contribution to the gas apparent permeability. For nitrogen, the coefficient is positive; and for carbon dioxide, the coefficient can be positive or negative below and above the phase change pressure, respectively. Virtual experiment simulations by discrete-fracture modeling of the pulse-decay systems reveal that the effective permeability of the tight porous media with complex configurations during the transient flow process is different from that during the steady-state flow process. History matchings of pulse-decay experiments on one naturally fractured core under different pore pressure and effective stress reveal that, matrix permeability is more sensitive to effective stress than matrix porosity, fracture porosity, and fracture permeability. It also implies that gas flow under a higher pressure is more likely in the homogeneous porous media as the matching errors show a decreasing trend with the pore pressure. The workflow of estimating the shale gas permeability evolution curve reveals that, slip flow/pore diffusion and surface diffusion in shale matrix largely mitigate the permeability decrease by the effective stress during the gas depletion process. Shale gas production simulations considering dynamic properties of both the matrix and fractures indicates that, the natural fracture spacing is the most important to shale gas recovery. By contrast, slip flow and pore diffusion are less important. Surface diffusion might be essential to shale gas recovery depending on the surface diffusivity value: the 5-year forecast error of the gas recovery factor without considering…
Advisors/Committee Members: Barati, Reza (advisor), Tsau, Jyun-Syung (cmtemember), Willhite, Paul G (cmtemember), Vossoughi, Shapour (cmtemember), Tu, Xuemin (cmtemember).
Subjects/Keywords: Petroleum engineering; numerical modeling; pulse-decay; shale gas; slip flow; surface diffusion
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APA ·
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APA (6th Edition):
Jia, B. (2018). Experimental and numerical investigations of shale gas permeability. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/29537
Chicago Manual of Style (16th Edition):
Jia, Bao. “Experimental and numerical investigations of shale gas permeability.” 2018. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/29537.
MLA Handbook (7th Edition):
Jia, Bao. “Experimental and numerical investigations of shale gas permeability.” 2018. Web. 25 Jan 2021.
Vancouver:
Jia B. Experimental and numerical investigations of shale gas permeability. [Internet] [Doctoral dissertation]. University of Kansas; 2018. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/29537.
Council of Science Editors:
Jia B. Experimental and numerical investigations of shale gas permeability. [Doctoral Dissertation]. University of Kansas; 2018. Available from: http://hdl.handle.net/1808/29537
University of Kansas
8.
Yu, Yufei.
An Adaptive Moving Mesh Finite Element Method and Its Application to Mathematical Models from Physical Sciences and Image Processing.
Degree: PhD, Mathematics, 2019, University of Kansas
URL: http://hdl.handle.net/1808/30071
► Moving sharp fronts are an important feature of many mathematical models from physical sciences and cause challenges in numerical computation. In order to obtain accurate…
(more)
▼ Moving sharp fronts are an important feature of many mathematical models from physical sciences and cause challenges in numerical computation. In order to obtain accurate solutions, a high resolution of mesh is necessary, which results in high computational cost if a fixed mesh is used. As a solution to this issue, an adaptive mesh method, which is called the moving mesh partial differential equation (MMPDE) method, is described in this work. The MMPDE method has the advantage of adaptively relocating the mesh points to increase the densities around sharp layers of the solutions, without increasing the mesh size. Moreover, this strategy can generate a nonsingular mesh even on non-convex and non-simply connected domains, given that the initial mesh is nonsingular. The focus of this thesis is on the application of the MMPDE method to mathematical models from physical sciences and image segmentation. In particular, this thesis includes the selection of the regularization parameter for the Ambrosio-Tortorelli functional, a simulation of the contact sets in the evolution of the micro-electro mechanical systems, and a numerical study of the flux selectivity in the Poisson-Nernst-Planck model. Sharp interfaces take place in all these three models, bringing interesting features and rich phenomena to study.
Advisors/Committee Members: Huang, Weizhang (advisor), Huang, Weizhang (cmtemember), Keshmiri, Shawn (cmtemember), Liu, Weishi (cmtemember), Tu, Xuemin (cmtemember), Van Vleck, Erik (cmtemember).
Subjects/Keywords: Applied mathematics
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APA ·
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APA (6th Edition):
Yu, Y. (2019). An Adaptive Moving Mesh Finite Element Method and Its Application to Mathematical Models from Physical Sciences and Image Processing. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/30071
Chicago Manual of Style (16th Edition):
Yu, Yufei. “An Adaptive Moving Mesh Finite Element Method and Its Application to Mathematical Models from Physical Sciences and Image Processing.” 2019. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/30071.
MLA Handbook (7th Edition):
Yu, Yufei. “An Adaptive Moving Mesh Finite Element Method and Its Application to Mathematical Models from Physical Sciences and Image Processing.” 2019. Web. 25 Jan 2021.
Vancouver:
Yu Y. An Adaptive Moving Mesh Finite Element Method and Its Application to Mathematical Models from Physical Sciences and Image Processing. [Internet] [Doctoral dissertation]. University of Kansas; 2019. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/30071.
Council of Science Editors:
Yu Y. An Adaptive Moving Mesh Finite Element Method and Its Application to Mathematical Models from Physical Sciences and Image Processing. [Doctoral Dissertation]. University of Kansas; 2019. Available from: http://hdl.handle.net/1808/30071
University of Kansas
9.
Duan, Zhaowen.
A High Order Overset Flux Reconstruction Method for Dynamic Moving Grids.
Degree: PhD, Aerospace Engineering, 2019, University of Kansas
URL: http://hdl.handle.net/1808/30483
► Overset meshes have a unique advantage in handling moving boundary problems as remeshing is often unnecessary. Recently, overset Cartesian and strand meshes were used successfully…
(more)
▼ Overset meshes have a unique advantage in handling moving boundary problems as remeshing is often unnecessary. Recently, overset Cartesian and strand meshes were used successfully to compute complex flow over rotorcraft. Although it is quite straightforward to deploy high-order finite difference method on the Cartesian mesh, the near-body solver for the strand mesh is often limited to second order accuracy. In the present work of this dissertation, we develop a high-order FR/CPR solver, hpMusic, on both the near-body and background grids, and extend it to handle moving boundary problems. The solver is also extended to sliding meshes, which can be considered a special case of overset meshes. The use of sliding meshes can often simplify the treatment of moving boundary problems with simple translational and rotational motions. Two different approaches to handle the overset interfaces are evaluated for accuracy, efficiency and robustness. Accuracy studies are carried out and the designed order of accuracy is obtained for both inviscid and viscous flows. Steady and unsteady flow problems are solved on stationary overset meshes. The results agree well with those in the literature and from experiments. A turbine blade under the wake of moving cylinders is simulated using sliding meshes. The flow structures are compared with those without moving cylinders. The solver is then tested for moving overset meshes with a benchmark dynamic airfoil problem from the 4th International Workshop on High-Order CFD Methods. Hp-convergent results are obtained and compared with those from other groups. Finally flow over a hovering rotor is simulated to compare with experimental data. In this case, the present high-order solver is capable of generating and propagating tip vortices with high resolution. Good agreement is achieved with experimental data in tip vortex core size, location, and the swirl velocity at 3rd order accuracy.
Advisors/Committee Members: Wang, ZJ (advisor), Wang, ZJ (cmtemember), Farokhi, Saeed (cmtemember), Taghavi, Ray (cmtemember), Wu, Huixuan (cmtemember), Tu, Xuemin (cmtemember).
Subjects/Keywords: Aerospace engineering; high order; large eddy simulation; moving grids; overset meshes; rotor simulation; sliding meshes
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APA ·
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APA (6th Edition):
Duan, Z. (2019). A High Order Overset Flux Reconstruction Method for Dynamic Moving Grids. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/30483
Chicago Manual of Style (16th Edition):
Duan, Zhaowen. “A High Order Overset Flux Reconstruction Method for Dynamic Moving Grids.” 2019. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/30483.
MLA Handbook (7th Edition):
Duan, Zhaowen. “A High Order Overset Flux Reconstruction Method for Dynamic Moving Grids.” 2019. Web. 25 Jan 2021.
Vancouver:
Duan Z. A High Order Overset Flux Reconstruction Method for Dynamic Moving Grids. [Internet] [Doctoral dissertation]. University of Kansas; 2019. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/30483.
Council of Science Editors:
Duan Z. A High Order Overset Flux Reconstruction Method for Dynamic Moving Grids. [Doctoral Dissertation]. University of Kansas; 2019. Available from: http://hdl.handle.net/1808/30483
University of Kansas
10.
Anwar, Rajib.
Fluid dynamics of non-Newtonian thin films: squeezing flow, film rupture, and contact line instability.
Degree: PhD, Bioengineering, 2016, University of Kansas
URL: http://hdl.handle.net/1808/22338
► HIV is a global pandemic that affects millions of lives every year. Vaginal gel formulation has been touted as an effective strategy to fight sexually…
(more)
▼ HIV is a global pandemic that affects millions of lives every year. Vaginal gel formulation has been touted as an effective strategy to fight sexually transmitted diseases (STD) by acting as a physical barrier, and also by delivering active pharmaceutical agents at the intended location. Microbicidal gels can act as a low cost, self-administered method to prevent HIV and other STD’s. The overall long term goal of our research group is to rationally design these delivery vehicles by optimizing their rheological properties for their specific applications. In that regard, there has been an ongoing effort in our group to mathematically model the flow of thin films in order to understand the dynamics and stability of these films, so that we can understand/predict the behavior of the gel during their applications. In my dissertation, I have used thin film lubrication theory to develop mathematical models of (a) boundary elasticity-driven squeezing flow of yield stress fluids, (b) rupture/dewetting of Ellis-type shear-thinning liquid films, and (c) contact line instability of Ellis-type shear-thinning liquid films. Our investigations showed that yield stress can improve the control over the spreading characteristics of liquids, and also possibly aid in the retention of liquid in place. We have demonstrated an optimization framework that can be used to estimate the rheological properties of microbicidal gels for different combinations of tissue elasticity, initial bolus volume, and deployment time. Numerical models of the rupture and dewetting of Ellis-type thin films showed that the shear-thinning rheology could accelerate the rupture process which could create dry spots earlier than Newtonian fluids. Dry spots are usually unwanted for both functional and aesthetic purposes. Numerical models of contact line instability have shown that the shear-thinning rheology increases the finger growth rate for vertical inclines but shows a more complex behavior for flatter inclines. For flatter inclines, the most unstable mode shifts to larger wavelength but the finger growth rate is dependent on the degree of shear-thinning. Fluids with low degree of shear-thinning suppress the fingering but fluids with the highest degree of shear-thinning show an increased growth rate despite the shift in the most unstable mode to larger wavelength. This behavior possibly occurs due to the decrease of apparent viscosity far beyond contact line, especially for fluids with a narrow range for the Newtonian plateau. Lubrication theory can be applied in the modeling of numerous biological and industrial applications. The models developed here are not limited to any specific application, and can easily be modified to study other applications that involves non-Newtonian fluids.
Advisors/Committee Members: Kieweg, Sarah L. (advisor), Camarda, Kyle V. (advisor), Wilson, Sara (cmtemember), Hefty, P. Scott (cmtemember), Tu, Xuemin (cmtemember).
Subjects/Keywords: Mechanical engineering; Biomedical engineering; Applied mathematics
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APA (6th Edition):
Anwar, R. (2016). Fluid dynamics of non-Newtonian thin films: squeezing flow, film rupture, and contact line instability. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/22338
Chicago Manual of Style (16th Edition):
Anwar, Rajib. “Fluid dynamics of non-Newtonian thin films: squeezing flow, film rupture, and contact line instability.” 2016. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/22338.
MLA Handbook (7th Edition):
Anwar, Rajib. “Fluid dynamics of non-Newtonian thin films: squeezing flow, film rupture, and contact line instability.” 2016. Web. 25 Jan 2021.
Vancouver:
Anwar R. Fluid dynamics of non-Newtonian thin films: squeezing flow, film rupture, and contact line instability. [Internet] [Doctoral dissertation]. University of Kansas; 2016. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/22338.
Council of Science Editors:
Anwar R. Fluid dynamics of non-Newtonian thin films: squeezing flow, film rupture, and contact line instability. [Doctoral Dissertation]. University of Kansas; 2016. Available from: http://hdl.handle.net/1808/22338
University of Kansas
11.
Shi, Lei.
Adaptive High-Order Differential Formulation for the Compressible Navier-Stokes Equations.
Degree: PhD, Aerospace Engineering, 2014, University of Kansas
URL: http://hdl.handle.net/1808/27533
► High-order methods have the potential to achieve higher accuracy at lower cost than lower order methods. This potential has been demonstrated conclusively for smooth problems…
(more)
▼ High-order methods have the potential to achieve higher accuracy at lower cost than lower order methods. This potential has been demonstrated conclusively for smooth problems in the 1st International Workshop on High-Order Methods. For non-smooth problems, solution based hp-adaptations offer the best promise. Adjoint-based adaptive methods have the capability of dynamically distributing computing resources to areas which are important for predicting engineering performance parameters, such as lift or drag. This thesis presents a robust and efficient adjoint-based adaptive high-order differential formulation for the compressible Navier-Stokes equations, which can rapidly determine an accurate estimate of an engineering output within a prescribed error threshold. The flux reconstruction (FR) or the correction procedure via reconstruction (CPR) method used in this work is a high-order differential formulation. We develop a parallel adjoint-based adaptive CPR solver which can work with any element-based error estimate and handle arbitrary discretization orders for mixed elements. First, a dual-consistent discrete form of the CPR method is derived. Then, an efficient and accurate adjoint-based error estimation method for the CPR method is developed and its accuracy and effectiveness are verified for the linear and non-linear partial differential equations (PDE). For anisotropic h-adaptations, we use a local output error sampling procedure to find the optimal refinement option. The current method has been applied to aerodynamic problems. Numerical tests show that significant savings in the number of DOFs can be achieved through the adjoint-based adaptation.
Advisors/Committee Members: Wang, Z.J. (advisor), Farokhi, Saeed (cmtemember), Huang, Weizhang (cmtemember), Taghavi, Ray (cmtemember), Tu, Xuemin (cmtemember), Zheng, Zhongquan (cmtemember).
Subjects/Keywords: Aerospace engineering; Adaptive Method; Adjoint-based Adaptation; Computational Fluid Dynamics; High-order Method; Navier-Stokes Equations
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APA (6th Edition):
Shi, L. (2014). Adaptive High-Order Differential Formulation for the Compressible Navier-Stokes Equations. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/27533
Chicago Manual of Style (16th Edition):
Shi, Lei. “Adaptive High-Order Differential Formulation for the Compressible Navier-Stokes Equations.” 2014. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/27533.
MLA Handbook (7th Edition):
Shi, Lei. “Adaptive High-Order Differential Formulation for the Compressible Navier-Stokes Equations.” 2014. Web. 25 Jan 2021.
Vancouver:
Shi L. Adaptive High-Order Differential Formulation for the Compressible Navier-Stokes Equations. [Internet] [Doctoral dissertation]. University of Kansas; 2014. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/27533.
Council of Science Editors:
Shi L. Adaptive High-Order Differential Formulation for the Compressible Navier-Stokes Equations. [Doctoral Dissertation]. University of Kansas; 2014. Available from: http://hdl.handle.net/1808/27533
University of Kansas
12.
Clifton, Cody Edward.
A Stochastic System Model for PageRank: Parameter Estimation and Adaptive Control.
Degree: PhD, Mathematics, 2015, University of Kansas
URL: http://hdl.handle.net/1808/19025
► A key feature of modern web search engines is the ability to display relevant and reputable pages near the top of the list of query…
(more)
▼ A key feature of modern web search engines is the ability to display relevant and reputable pages near the top of the list of query results. The PageRank algorithm provides one way of achieving such a useful hierarchical indexing by assigning a measure of relative importance, called the PageRank value, to each webpage. PageRank is motivated by the inherently hypertextual structure of the World Wide Web; specifically, the idea that pages with more incoming hyperlinks should be considered more popular and that popular pages should rank highly in search results, all other factors being equal. We begin by overviewing the original PageRank algorithm and discussing subsequent developments in the mathematical theory of PageRank. We focus on important contributions to improving the quality of rankings via topic-dependent or "personalized" PageRank, as well as techniques for improving the efficiency of PageRank computation based on Monte Carlo methods, extrapolation and adaptive methods, and aggregation methods We next present a model for PageRank whose dynamics are described by a controlled stochastic system that depends on an unknown parameter. The fact that the value of the parameter is unknown implies that the system is unknown. We establish strong consistency of a least squares estimator for the parameter. Furthermore, motivated by recent work on distributed randomized methods for PageRank computation, we show that the least squares estimator remains strongly consistent within a distributed framework. Finally, we consider the problem of controlling the stochastic system model for PageRank. Under various cost criteria, we use the least squares estimates of the unknown parameter to iteratively construct an adaptive control policy whose performance, according to the long-run average cost, is equivalent to the optimal stationary control that would be used if we had knowledge of the true value of the parameter. This research lays a foundation for future work in a number of areas, including testing the estimation and control procedures on real data or larger scale simulation models, considering more general parameter estimation methods such as weighted least squares, and introducing other types of control policies.
Advisors/Committee Members: Pasik-Duncan, Bozenna (advisor), Duncan, Tyrone E. (cmtemember), Evans, Joseph B. (cmtemember), Talata, Zsolt (cmtemember), Tu, Xuemin (cmtemember).
Subjects/Keywords: Mathematics; Applied mathematics; Adaptive Control; PageRank; Parameter Estimation; Stochastic System
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APA (6th Edition):
Clifton, C. E. (2015). A Stochastic System Model for PageRank: Parameter Estimation and Adaptive Control. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/19025
Chicago Manual of Style (16th Edition):
Clifton, Cody Edward. “A Stochastic System Model for PageRank: Parameter Estimation and Adaptive Control.” 2015. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/19025.
MLA Handbook (7th Edition):
Clifton, Cody Edward. “A Stochastic System Model for PageRank: Parameter Estimation and Adaptive Control.” 2015. Web. 25 Jan 2021.
Vancouver:
Clifton CE. A Stochastic System Model for PageRank: Parameter Estimation and Adaptive Control. [Internet] [Doctoral dissertation]. University of Kansas; 2015. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/19025.
Council of Science Editors:
Clifton CE. A Stochastic System Model for PageRank: Parameter Estimation and Adaptive Control. [Doctoral Dissertation]. University of Kansas; 2015. Available from: http://hdl.handle.net/1808/19025
University of Kansas
13.
Liu, Yanghui.
Numerical solutions of rough differential equations and stochastic differential equations.
Degree: PhD, Mathematics, 2016, University of Kansas
URL: http://hdl.handle.net/1808/21866
► In this dissertation, we investigate time-discrete numerical approximation schemes for rough differential equations and stochastic differential equations (SDE) driven by fractional Brownian motions (fBm). The…
(more)
▼ In this dissertation, we investigate time-discrete numerical approximation schemes for rough differential equations and stochastic differential equations (SDE) driven by fractional Brownian motions (fBm). The dissertation is organized as follows. In Chapter 1, we introduce the basic settings and define time-discrete numerical approximation schemes. In Chapter 2, we consider the Euler scheme for SDEs driven by fBms. For a SDE driven by a fBm with Hurst parameter H> ½ it is known that the existing (naive) Euler scheme has the rate of convergence n
1-2H. Since the limit H → ½ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It\
o SDEs for H=½, the convergence rate of the naive Euler scheme deteriorates for H → ½. The new (modified Euler) approximation scheme we are introducing in this chapter is closer to the classical Euler scheme for Stratonovich SDEs for H=½ and it has the rate of convergence γ
n-1, where γ
n=n
2H-½ when H ½ it is known that the existing (naive) Euler scheme has the rate of convergence n
1-2H. Since the limit H → ½ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It\
o SDEs for H=½, the convergence rate of the naive Euler scheme deteriorates for H → ½. The new (modified Euler) approximation scheme we are introducing in this chapter is closer to the classical Euler scheme for Stratonovich SDEs for H=½ and it has the rate of convergence γ
n-1, where γ
n=n
2H-½ when H \frac34. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if {X
t, 0 ≤ t ≤ T} is the solution of a SDE driven by a fBm and if {X
tn, 0 ≤ t ≤ T} is its approximation obtained by the new modified Euler scheme, then we prove that γ
n (X
n-X) converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when H∈ ( ½, \frac34]. In the case H \frac 34, we show the L
p convergence of n(X
nt-Xt) and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme. In Chapter 3, we consider the Crank-Nicolson method for a SDE driven by a m-dimensional fBm. We consider the Crank-Nicolson method in three cases: (i) m1; (ii) m=1 and and the drift term is equal to non-zero; and (iii) m=1 and the drift term is equal to zero. We will show that the convergence rate of the Crank-Nicolson method is n
1/2-2H, n
-1/2-H and n
-2H, respectively, in these three cases, and these convergence rates are exact in the sense that the error process for the Crank-Nicolson method…
Advisors/Committee Members: Nualart, David (advisor), Hu, Yaozhong (advisor), Nualart, David (cmtemember), Hu, Yaozhong (cmtemember), Feng, Jin (cmtemember), Tu, Xuemin (cmtemember), Zhang, Jianbo (cmtemember), Soo, Terry (cmtemember).
Subjects/Keywords: Mathematics; fourth moment theorem; fractional Brownian motions; Numerical solutions; rough differential equations; stochastic differential equations
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APA ·
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APA (6th Edition):
Liu, Y. (2016). Numerical solutions of rough differential equations and stochastic differential equations. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/21866
Chicago Manual of Style (16th Edition):
Liu, Yanghui. “Numerical solutions of rough differential equations and stochastic differential equations.” 2016. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/21866.
MLA Handbook (7th Edition):
Liu, Yanghui. “Numerical solutions of rough differential equations and stochastic differential equations.” 2016. Web. 25 Jan 2021.
Vancouver:
Liu Y. Numerical solutions of rough differential equations and stochastic differential equations. [Internet] [Doctoral dissertation]. University of Kansas; 2016. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/21866.
Council of Science Editors:
Liu Y. Numerical solutions of rough differential equations and stochastic differential equations. [Doctoral Dissertation]. University of Kansas; 2016. Available from: http://hdl.handle.net/1808/21866
14.
Ngo, Cuong.
Moving mesh methods for numerical solution of porous medium equations.
Degree: PhD, Mathematics, 2017, University of Kansas
URL: http://hdl.handle.net/1808/26343
► Porous medium equation (PME) has been found in many applications of the physical sciences. The equation is nonlinear, degenerate, and in many situations has a…
(more)
▼ Porous medium equation (PME) has been found in many applications of the physical sciences. The equation is nonlinear, degenerate, and in many situations has a free boundary, which altogether pose great challenges for mathematical and numerical analyses. In contrast with the mathematical development of PME, which began in the 1950s and has since had much success, studies of numerical solution did not appear until the 1980s. Though a significant progress has been made since then for the 1D setting, only limited success has been observed for 2D cases. In this dissertation, we will propose several moving mesh methods which improve the accuracy and convergence order of the PME numerical solution.
Advisors/Committee Members: Huang, Weizhang (advisor), Van Vleck, Erik (cmtemember), Tu, Xuemin (cmtemember), Xu, Hongguo (cmtemember), Zheng, Charlie (cmtemember).
Subjects/Keywords: Mathematics; Adaptive moving mesh method; Finite element method; Free boundary; Hessian-based adaptivity; MMPDE method; Porous medium equation
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APA (6th Edition):
Ngo, C. (2017). Moving mesh methods for numerical solution of porous medium equations. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/26343
Chicago Manual of Style (16th Edition):
Ngo, Cuong. “Moving mesh methods for numerical solution of porous medium equations.” 2017. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/26343.
MLA Handbook (7th Edition):
Ngo, Cuong. “Moving mesh methods for numerical solution of porous medium equations.” 2017. Web. 25 Jan 2021.
Vancouver:
Ngo C. Moving mesh methods for numerical solution of porous medium equations. [Internet] [Doctoral dissertation]. University of Kansas; 2017. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/26343.
Council of Science Editors:
Ngo C. Moving mesh methods for numerical solution of porous medium equations. [Doctoral Dissertation]. University of Kansas; 2017. Available from: http://hdl.handle.net/1808/26343
15.
Li, Yanan.
Large-eddy simulations of turbulent flows using the high-order FR/CPR method.
Degree: D.Eng., Aerospace Engineering, 2016, University of Kansas
URL: http://hdl.handle.net/1808/24138
► Large eddy simulation (LES) was originally proposed for simulating atmospheric flows and then has become one of the most successful methodologies for turbulence simulation for…
(more)
▼ Large eddy simulation (LES) was originally proposed for simulating atmospheric flows and then has become one of the most successful methodologies for turbulence simulation for its good balance between accuracy and cost. In LES, energetic scales are resolved while the small equilibrium scales are modeled by the sub-grid scale(SGS) stress models. The resolution of the wide spectrum of the energetic scales is a big challenge for numerical methods. High-order methods are very promising in LES for its low dissipation and dispersion errors. For smooth turbulent flow, high-order methods have the potential to achieve high accuracy at lower cost than lower order methods. This thesis presents the investigation of the performance of different LES sub-grid scale stress (SGS) models with the high-order flux reconstruction or the correction procedure via reconstruction( FR/CPR) method. A mathematical analysis of scale similarity is conducted and presented as well. In addition, numerical schemes’ behavior in nonlinear wave propagation is studied and presented. The computationa of discontinuities, such as shocks, is another challenge to the numerical methods. In the simulation of shocks, non-physical oscillations can occur at the discontinuities and lead to divergence. The situation is worse for high-order methods. This thesis also presents a new flux limiter for the FR/CPR method. The new technique shows good properties, convergence for steady problems and accuracy preserving for vortex dominated flows. It is very promising in handling shock and turbulence interaction problems.
Advisors/Committee Members: Wang, Z. J. (advisor), Farokhi, Saeed (cmtemember), Taghavi, Ray (cmtemember), Tu, Xuemin (cmtemember), Zheng, Zhongquan (cmtemember).
Subjects/Keywords: Aerospace engineering; computational fluid dynamics; high-order numerical schemes; large-eddy simulation; shock capturing; turbulence
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APA ·
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APA (6th Edition):
Li, Y. (2016). Large-eddy simulations of turbulent flows using the high-order FR/CPR method. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/24138
Chicago Manual of Style (16th Edition):
Li, Yanan. “Large-eddy simulations of turbulent flows using the high-order FR/CPR method.” 2016. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/24138.
MLA Handbook (7th Edition):
Li, Yanan. “Large-eddy simulations of turbulent flows using the high-order FR/CPR method.” 2016. Web. 25 Jan 2021.
Vancouver:
Li Y. Large-eddy simulations of turbulent flows using the high-order FR/CPR method. [Internet] [Doctoral dissertation]. University of Kansas; 2016. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/24138.
Council of Science Editors:
Li Y. Large-eddy simulations of turbulent flows using the high-order FR/CPR method. [Doctoral Dissertation]. University of Kansas; 2016. Available from: http://hdl.handle.net/1808/24138
16.
Wang, Bin.
Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations.
Degree: PhD, Mathematics, 2017, University of Kansas
URL: http://hdl.handle.net/1808/27005
► Hybridizable Discontinuous Galerkin (HDG) is an important family of methods, which combine the advantages of both Discontinuous Galerkin in terms of flexibility and standard finite…
(more)
▼ Hybridizable Discontinuous Galerkin (HDG) is an important family of methods, which combine the advantages of both Discontinuous Galerkin in terms of flexibility and standard finite elements in terms of accuracy and efficiency. The impact of this method is partly evidenced by the prolificacy of research work in this area. Weak Galerkin (WG) is a relatively newly proposed method by introducing weak functions and generalizing the differential operator for them. This method has also drawn remarkable interests from both numerical practitioners and analysts recently. HDG and WG are different but closely related. BDDC algorithms are developed for numerical solution of elliptic problems with both methods. We prove that the optimal condition number estimate for BDDC operators with standard finite element methods can be extended to the counterparts arising from the HDG and WG methods, which are nonconforming finite element methods. Numerical experiments are conducted to verify the theoretical analysis. Further, we propose BDDC algorithms for the saddle point system arising from the Stokes equations using both HDG and WG methods. By design of the preconditioner, the iterations are restricted to a benign subspace, which makes the BDDC operator effectively positive definite thus solvable by the conjugate gradient method. We prove that the algorithm is scalable in the number of subdomains with convergence rate only dependent on subdomain problem size. The condition number bound for the BDDC preconditioned Stokes system is the same as the optimal bound for the elliptic case. Numerical results confirm the theoretical analysis.
Advisors/Committee Members: Tu, Xuemin (advisor), Huang, Weizhang (cmtemember), Van Vleck, Erik (cmtemember), Xu, Hongguo (cmtemember), Wang, Z.J. (cmtemember).
Subjects/Keywords: Mathematics; BDDC; domain decomposition; hybridizable discontinuous Galerkin; saddle point problems; Stokes; weak Galerkin
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APA ·
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Export
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APA (6th Edition):
Wang, B. (2017). Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/27005
Chicago Manual of Style (16th Edition):
Wang, Bin. “Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations.” 2017. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/27005.
MLA Handbook (7th Edition):
Wang, Bin. “Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations.” 2017. Web. 25 Jan 2021.
Vancouver:
Wang B. Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations. [Internet] [Doctoral dissertation]. University of Kansas; 2017. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/27005.
Council of Science Editors:
Wang B. Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations. [Doctoral Dissertation]. University of Kansas; 2017. Available from: http://hdl.handle.net/1808/27005
17.
Su, Chen.
Some Studies on Parameter Estimations.
Degree: PhD, Mathematics, 2016, University of Kansas
URL: http://hdl.handle.net/1808/21898
► Parameter estimation has wide applications in such fields as finance, biological science, weather prediction, oil deposit detection, etc. Researchers are particularly interested in reconstructing some…
(more)
▼ Parameter estimation has wide applications in such fields as finance, biological science, weather prediction, oil deposit detection, etc. Researchers are particularly interested in reconstructing some unknown parameters from the observed data set which may be sparse and noisy. This is a typical inverse problem which tends to be ill-conditioned in many cases. A plethora of literature has been devoted to this area and there has been a concrete progress recently in the design of more efficient estimation techniques. Depending on the model (usually an equation or a system of equations) we choose, we divide the estimation into two categories: stochastic and deterministic parameter estimations. The former involves a stochastic system, usually a stochastic differential equation (SDE) or system of SDEs where unknown parameters are present. A standard estimator to use for stochastic parameter estimation problems is the maximum likelihood estimator (MLE), since it, in many situations, enjoys desirable properties such as consistency and asymptotic normality. One major obstacle in obtaining the MLE is that the transition density of SDE, which is essential for deriving MLE, is often not available. To address this issue, various approximations of transition density was introduced in the past decades. In chapter 1, we will present several popular density approximation schemes including Euler-Maruyama methods and Hermite expansions. We will also introduce the parametrix approximation in which we derive a point-wise approximation of the transition density that is uniform in the parameter. As a consequence, the approximated MLE from parametrix method will eventually converge to the true MLE so that those desired properties of MLE can be preserved. We will see some applications of the parametrix approximation and some necessary preliminaries regarding the ergodicity of SDEs and the consistency of the estimators will also be presented. The deterministic parameter estimation involves a partial differential equation (PDE) or a system of PDEs. A key feature for this type of estimation is the high level of uncertainty for recovering the parameters, i.e. different choices of parameters may all yield reasonable explanation of the data. This is a typical feature for many ill-conditioned inverse problems. The Bayesian inference formulation provides a systematic way to characterize this uncertainty. It incorporates a prior, which is from the historical data before any experiment is done, and a likelihood, which measures how likely the data will be provided that certain parameter value is chosen, to form a posterior density. It generates a neat solution which takes the form of a posterior probability density. However, how to interpret this posterior density is a non-trivial task since the forward model may be very expensive and the discretized parameter field may result in a high dimensional density. As a consequence, efficient sampling techniques are called for to better characterize the posterior. In chapter 2, we will introduce some…
Advisors/Committee Members: Hu, Yaozhong (advisor), Tu, Xuemin (advisor), Nualart, David (cmtemember), Soo, Terry (cmtemember), Zhang, Jianbo (cmtemember).
Subjects/Keywords: Mathematics; Bayesian methods; implicit sampling; inverse problems; maximum likelihood estimator; stochastic differential equations
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APA ·
Chicago ·
MLA ·
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Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Su, C. (2016). Some Studies on Parameter Estimations. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/21898
Chicago Manual of Style (16th Edition):
Su, Chen. “Some Studies on Parameter Estimations.” 2016. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/21898.
MLA Handbook (7th Edition):
Su, Chen. “Some Studies on Parameter Estimations.” 2016. Web. 25 Jan 2021.
Vancouver:
Su C. Some Studies on Parameter Estimations. [Internet] [Doctoral dissertation]. University of Kansas; 2016. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/21898.
Council of Science Editors:
Su C. Some Studies on Parameter Estimations. [Doctoral Dissertation]. University of Kansas; 2016. Available from: http://hdl.handle.net/1808/21898
18.
Zhou, Cheng.
Adaptive High-Order Discretization of the Reynolds-Averaged Navier-Stokes(RANS) Equations.
Degree: PhD, Aerospace Engineering, 2016, University of Kansas
URL: http://hdl.handle.net/1808/24139
► The use of high-order methods to compute turbulent flows governed by the Reynolds- averaged Navier-Stokes (RANS) equations is an active research topic in the compu-…
(more)
▼ The use of high-order methods to compute turbulent flows governed by the Reynolds- averaged Navier-Stokes (RANS) equations is an active research topic in the compu- tational fluid dynamics (CFD) community. However, it is well known that high-order methods for the non-smooth turbulence modeling equations are difficult to converge to the steady-state because of the numerical stiffness. The objective of this work is to de- velop a robust and efficient high-order discretization that can simulate turbulent flows governed by the Reynolds-Averaged Navier-Stokes equations, which involves the de- velopment of high-order space discretization of robust turbulence modeling equations, the improvement of time integration strategy, and the application of effective mesh adaptation methods. In the present study, correction procedure via reconstruction (CPR) high-order dis- cretization is developed to solve the Reynolds-averaged Navier-Stokes (RANS) equa- tions with the modified Spalart and Allmaras (SA) model. In this model, the non- dimensional length scale depends on the distance to the nearest wall. To compute the distance of each solution point in the domain to the nearest curved polynomial wall boundaries, the CPR high-order discretization is extended to solve the Eikonal equa- tion. On the other hand, to improve time integration strategy for the simulation of turbulent flows, the present work carried out a comparative study of several implicit time integration schemes to determine which is the most efficient, robust and general scheme. Additionally, an adjoint-based adaptive mesh refinement method is utilized to minimize the output error. Numerical results show that, to achieve a certain level of accuracy, the adaptive CPR discretization of the RANS equations with the SA model saves orders of magnitude in terms of number of degrees of freedom comparing to the numerical results of uniform mesh refinement, when applied to the simulations of turbulent flows.
Advisors/Committee Members: Wang, Z.J. (advisor), Farokhi, Saeed (cmtemember), Taghavi, Ray (cmtemember), Tu, Xuemin (cmtemember), Zheng, Zhongquan Charlie (cmtemember).
Subjects/Keywords: Aerospace engineering; Mathematics; Adaptive; CFD; CPR; High-order method; RANS
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APA (6th Edition):
Zhou, C. (2016). Adaptive High-Order Discretization of the Reynolds-Averaged Navier-Stokes(RANS) Equations. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/24139
Chicago Manual of Style (16th Edition):
Zhou, Cheng. “Adaptive High-Order Discretization of the Reynolds-Averaged Navier-Stokes(RANS) Equations.” 2016. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/24139.
MLA Handbook (7th Edition):
Zhou, Cheng. “Adaptive High-Order Discretization of the Reynolds-Averaged Navier-Stokes(RANS) Equations.” 2016. Web. 25 Jan 2021.
Vancouver:
Zhou C. Adaptive High-Order Discretization of the Reynolds-Averaged Navier-Stokes(RANS) Equations. [Internet] [Doctoral dissertation]. University of Kansas; 2016. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/24139.
Council of Science Editors:
Zhou C. Adaptive High-Order Discretization of the Reynolds-Averaged Navier-Stokes(RANS) Equations. [Doctoral Dissertation]. University of Kansas; 2016. Available from: http://hdl.handle.net/1808/24139
19.
Gu, Peidi.
Finding Eigenvalues of Unitary Matrices.
Degree: MA, Mathematics, 2013, University of Kansas
URL: http://hdl.handle.net/1808/12977
► The study introduces methods of finding eigenvalues for unitary matrices and pencils. Bunse-Gerstner and Elsner ([2]) proposed an algorithm of using the Schur parameter pencil…
(more)
▼ The study introduces methods of finding eigenvalues for unitary matrices and pencils. Bunse-Gerstner and Elsner ([2]) proposed an algorithm of using the Schur parameter pencil to solve eigenproblems for unitary matrices and pencils. This thesis reviews the Schur parameter pencil algorithm. The method is divided into two phases: Reducing a unitary pencil to a Schur parameter form and QR-type shifted iteration. The algorithm is proved to be backward stable and more efficient than the standard QR/QZ algorithm. However, during the process of reduction, norms of vectors are frequently compared for numerical stability, which causes a lot of extra work for computations. Based on the idea in [8], we introduce a modified Schur parameter algorithm to avoid such frequent comparison. The modified algorithm is still divided into two phases similar to the one in [2]. A detailed reduction process and shifted iteration are described in this thesis.
Advisors/Committee Members: Xu, Hongguo (advisor), Xu, Hongguo (cmtemember), Tu, Xuemin (cmtemember), Van Vleck, Erik (cmtemember).
Subjects/Keywords: Mathematics; Eigenvalue; Givens matrix; Householder reflector; Iteration; Schur parameter form; Unitary matrix
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APA (6th Edition):
Gu, P. (2013). Finding Eigenvalues of Unitary Matrices. (Masters Thesis). University of Kansas. Retrieved from http://hdl.handle.net/1808/12977
Chicago Manual of Style (16th Edition):
Gu, Peidi. “Finding Eigenvalues of Unitary Matrices.” 2013. Masters Thesis, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/12977.
MLA Handbook (7th Edition):
Gu, Peidi. “Finding Eigenvalues of Unitary Matrices.” 2013. Web. 25 Jan 2021.
Vancouver:
Gu P. Finding Eigenvalues of Unitary Matrices. [Internet] [Masters thesis]. University of Kansas; 2013. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/12977.
Council of Science Editors:
Gu P. Finding Eigenvalues of Unitary Matrices. [Masters Thesis]. University of Kansas; 2013. Available from: http://hdl.handle.net/1808/12977
20.
Lu, Fei.
Some application of Malliavin calculus to SPDE and convergence of densities.
Degree: PhD, Mathematics, 2013, University of Kansas
URL: http://hdl.handle.net/1808/12309
► Some applications of Malliavin calculus to stochastic partial differential equations (SPDEs) and to normal approximation theory are studied in this dissertation. In Chapter 3, a…
(more)
▼ Some applications of Malliavin calculus to stochastic partial differential equations (SPDEs) and to normal approximation theory are studied in this dissertation. In Chapter 3, a Feynman-Kac formula is established for a stochastic heat equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst parameter H<1/2. To establish such a formula, we introduce and study a nonlinear stochastic integral of the Gaussian noise. The existence of the Feynman-Kac integral then follows from the exponential integrability of this nonlinear stochastic integral. Then, techniques from Malliavin calculus is used to show that the Feynman-Kac integral is the weak solution to the stochastic heat equation. In Chapter 4, the density formula in Malliavin calculus is used to study the joint H"{o}lder continuity of the solution to a nonlinear SPDE arising from the study of one dimensional super-processes. Dawson, Vaillancourt and Wang [Ann. Inst. Henri. Poincaré Probab. Stat., 36 (2000) 167-180] proved that the solution of this SPDE gives the density of the branching particles in a random environment. The time-space joint continuity of the density process was left as an open problem. Li, Wang, Xiong and Zhou [Probab. Theory Related Fields 153 (2012), no. 3-4, 441 – 469] proved that this solution is joint H"{o}lder continuous with exponent up to 1/10 in time and up to 1/2 in space. Using our new method of Malliavin calculus, we improve their result and obtain the optimal exponent 1/4 in time. In Chapter 5, we study the convergence of densities of a sequence of random variables to a normal density. The random variables considered are nonlinear functional of a Gaussian process, in particular, the multiple integrals. They are assumed to be non-degenerate so that their probability densities exist. The tool we use is the Malliavin calculus, in particular, the density formula, the integration by parts formula and the Stein's method. Applications to the convergence of densities of the least square estimator for the drift parameter in Ornstein-Ulenbeck is also considered. In Chapter 6, we apply an upper bound estimate from small deviation theory to prove the non-degeneracy of some functional of fractional Brownian motion.
Advisors/Committee Members: Hu, Yaozhong (advisor), Nualart, David (advisor), Duncan, Tyrone E. (cmtemember), Feng, Jin (cmtemember), Juhl, Ted (cmtemember), Tu, Xuemin (cmtemember).
Subjects/Keywords: Mathematics; Central limit theorems on wiener chaos; Convergence of densities; Feynman-kac formula; Holder continuity of solutions to spdes; Malliavin calculus; Stochastic partial differential equatons
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APA ·
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MLA ·
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CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Lu, F. (2013). Some application of Malliavin calculus to SPDE and convergence of densities. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/12309
Chicago Manual of Style (16th Edition):
Lu, Fei. “Some application of Malliavin calculus to SPDE and convergence of densities.” 2013. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/12309.
MLA Handbook (7th Edition):
Lu, Fei. “Some application of Malliavin calculus to SPDE and convergence of densities.” 2013. Web. 25 Jan 2021.
Vancouver:
Lu F. Some application of Malliavin calculus to SPDE and convergence of densities. [Internet] [Doctoral dissertation]. University of Kansas; 2013. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/12309.
Council of Science Editors:
Lu F. Some application of Malliavin calculus to SPDE and convergence of densities. [Doctoral Dissertation]. University of Kansas; 2013. Available from: http://hdl.handle.net/1808/12309
21.
Zhang, Mingji.
Dynamics of Poisson-Nernst-Planck systems and applications to ionic channels.
Degree: PhD, Mathematics, 2013, University of Kansas
URL: http://hdl.handle.net/1808/12250
► Dynamics of Poisson-Nernst-Planck systems and its applications to ion channels are studied in this dissertation. The Poisson-Nernst-Planck systems serve as basic electro-diffusion equations modeling, for…
(more)
▼ Dynamics of Poisson-Nernst-Planck systems and its applications to ion channels are studied in this dissertation. The Poisson-Nernst-Planck systems serve as basic electro-diffusion equations modeling, for example, ion flow through membrane channels and transport of holes and electrons in semiconductors. The model can be derived from the more fundamental models of the Langevin-Poisson system and the Maxwell-Boltzmann equations, and from the energy variational analysis EnVarA. A brief description of the model is given in Chapter 2 including the physical meaning of each equation involved. Ion channels are cylindrical, hollow proteins that regulate the movement of ions ( mainly Na+, K+, Ca++ and Cl-;) through nearly all the membrane channels. When an initial potential is applied at one end of the channel, it will drive the ions through the channel, and the movement of these ions will produce the current which can be measured. Different initial potentials will result in different currents, and the collection of all those data will provide a relation, the so-called I-V (current-voltage) relation, which is an important characterization of two most relevant properties of a channel: permeation and selectivity. In Chapter 3, a classical Poisson-Nernst-Planck system is studied both analytically and numerically to investigate the cubic-like feature of the I-V relation. For the case of zero permanent charge, under electroneutrality boundary conditions at both ends of the channel, our result concerning the I-V relation for two oppositely charged ion speciesis that the third order correction is cubic in the potential V , and furthermore, up to the third order, the cubic I-V relation has three distinct real roots (except for a very degenerate case) which corresponds to the bi-stable structure in the FitzHugh-Nagumo simplification of the Hodgkin-Huxley model. Numerical simulations are performed and and they are consistent with our analytical results. In Chapter 4, we consider a one-dimensional steady-state Poisson-Nernst-Planck type model for ionic flow through membrane channels including ionic interaction modeled by a nonlocal hard-sphere potential from the Density Functional Theory. The resulting problem is a singularly perturbed boundary value problem of an integro-differential system. Ion size effect on the I-V relations is investigated numerically. Two numerical tasks are conducted. The first one is a numerical approach of solving the boundary value problem and obtaining I-V curves. This is accomplished through a numerical implement of the analytical strategy introduced in [46]. The second task is to numerically detect two critical potential values Vc and Vc. Our numerical detections are based on the defining properties of Vc and V c and are designed to use the numerical I-V curves directly. For the setting in the above mentioned reference, our numerical results agree well with the analytical predictions. In Chapter 5, a one-dimensional steady-state Poisson-Nernst-Planck type model for ionic flow through a membrane channel is…
Advisors/Committee Members: Liu, Weishi (advisor), Liu, Weishi (cmtemember), Stanislavova, Milena (cmtemember), Tu, Xuemin (cmtemember), Van Vleck, Erik (cmtemember), Shi, Jack (cmtemember).
Subjects/Keywords: Mathematics; Biochemistry
…China), Dr. Xuemin Tu ( from University
of Kansas), Dr. Yingfei Yi (from…
Record Details
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❌
APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Zhang, M. (2013). Dynamics of Poisson-Nernst-Planck systems and applications to ionic channels. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/12250
Chicago Manual of Style (16th Edition):
Zhang, Mingji. “Dynamics of Poisson-Nernst-Planck systems and applications to ionic channels.” 2013. Doctoral Dissertation, University of Kansas. Accessed January 25, 2021.
http://hdl.handle.net/1808/12250.
MLA Handbook (7th Edition):
Zhang, Mingji. “Dynamics of Poisson-Nernst-Planck systems and applications to ionic channels.” 2013. Web. 25 Jan 2021.
Vancouver:
Zhang M. Dynamics of Poisson-Nernst-Planck systems and applications to ionic channels. [Internet] [Doctoral dissertation]. University of Kansas; 2013. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1808/12250.
Council of Science Editors:
Zhang M. Dynamics of Poisson-Nernst-Planck systems and applications to ionic channels. [Doctoral Dissertation]. University of Kansas; 2013. Available from: http://hdl.handle.net/1808/12250
. 
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