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Title: MONTE CARLO SIMULATION OF INFINITE-DIMENSIONAL INTEGRALS
Creator: Niu, Ben
Date: 2011-04-13, 2011-05
Description: This thesis is motivated by pricing a path-dependent financial derivative, such as an Asian option, which requires the computation of the...
Show moreThis thesis is motivated by pricing a path-dependent financial derivative, such as an Asian option, which requires the computation of the expectation of a payoff function, which depends on a Brownian motion. Employing a standard series expansion of the Brownian motion, the latter problem is equivalent to the computation of the expectation of a function of the corresponding i.i.d. sequence of random coefficients. This motivates the construction and the analysis of algorithms for numerical integration with respect to a product probability measure on the infinite-dimensional sequence. The class of integrands studied in this thesis resides in the unit ball in a reproducing kernel Hilbert space obtained by superposition of weighted tensor product spaces of functions of finitely many variables. Combining tractability results for high-dimensional integration with the multi-level technique we obtain new algorithms for infinite-dimensional integration. These deterministic multi-level algorithms use variable subspace sampling and they are superior to any deterministic algorithm based on fixed subspace sampling with respect to the respective worst case error. Numerical experiment results are presented at the end.
Ph.D. in Applied Mathematics, May 2011
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Title: A NUMERICAL AND ANALYTICAL STUDY OF THE GROWTH OF SECOND PHASE PARTICLES USING A SHARP INTERFACE APPROACH
Creator: Barua, Amlan K.
Date: 2012-08-15, 2012-12
Description: Two phase alloys are quite important in materials science and metallurgy. Some common examples include nickel-aluminum system, iron-carbon...
Show moreTwo phase alloys are quite important in materials science and metallurgy. Some common examples include nickel-aluminum system, iron-carbon system etc. The most important macroscopic properties of these alloys depend on size, orientation and concentration of the second-phase precipitates. It is necessary to understand the details of formation, growth and equilibrium conditions of these micro-structures for better material production. In this dissertation we investigate the growth of the precipitates within the matrix using a sharp interface approach. We consider the effects of elastic fields on the evolution of the precipitates. The elastic fields can either be applied at the far field or can simply arise as a result of crystallographic difference between matrix and precipitate phase. The precipitates exhibit complicated morphology because of the Mullins-Sekerka instability. Our investigation is based on both analytical and numerical techniques. We use linear analysis to understand the qualitative behavior of the problem, at least for short time. To simulate the long time dynamics of the problem and to understand the effects of nonlinearity, we use highly accurate boundary integral methods. Our main contribution in this thesis is threefold. First, starting from linear analysis, we focus on the conditions under which stable growth, in presence of elastic field, is possible for a single precipitate. Finding such conditions are important in material production and simple conditions like constant material flux and constant elastic fields produce precipitates with complicated shapes. Second, we propose a space-time rescaling of the original boundary integral equations of the problem. The rescaling enables us to accurately simulate very long time behavior of the system comprising of multiple precipitates growing under different mass flux and elasticity. It also helps us to understand the long time interaction of precipitates. Third, we xiii implement an adaptive treecode to reduce the computational complexity of the iterative solver from O(N2) to O(N logN) where N is the dimension of the discrete problem. The efficiency of the treecode is demonstrated by performing simulations. Also a parallelization strategy for the treecode is discussed. The speed-up from the parallelization is demonstrated using moderate number of cores. xiv
PH.D in Applied Mathematics, December 2012
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Title: RELIABLE QUASI-MONTE CARLO WITH CONTROL VARIATES
Creator: Li, Da
Date: 2016, 2016-07
Description: Recently, Quasi-Monte Carlo (QMC) methods have been implemented in a reliable adaptive algorithm. This raises the possibility of combining...
Show moreRecently, Quasi-Monte Carlo (QMC) methods have been implemented in a reliable adaptive algorithm. This raises the possibility of combining adaptive QMC with efficiency improvement techniques for independent and identically distributed (IID) Monte Carlo (MC) such as control variates (CV). The challenge for adding CV to QMC is that the optimal CV coefficient for QMC is generally not the same as that for MC. Here we propose a method for imple- menting CV in a reliable adaptive QMC algorithm. One merit of using CV with MC is that theoretically the efficiency is always no worse than vanilla MC. Our method is implemented in an efficient way so that the extra cost for CV is tolerable, and the overall time savings can be substantial. We test our algorithm on various problems including option pricing and mul- tivariate normal probability estimation for dimensions from 4 to 64. The same tests are performed on adaptive QMC algorithm without CV as a comparison. Our results show that with good CV, the cost of adaptive QMC is greatly reduced compared to vanilla QMC.
M.S. in Applied Mathematics, July 2016
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Title: FAST AUTOMATIC BAYESIAN CUBATURE USING MATCHING KERNELS AND DESIGNS
Creator: Rathinavel, Jagadeeswaran
Date: 2019, 2019-12-20
Publisher: Chicago
Description: Automatic cubatures approximate multidimensional integrals to user-specified...
Show moreAutomatic cubatures approximate multidimensional integrals to user-specified error tolerances. In many real-world integration problems, the analytical solution is either unavailable or difficult to compute. To overcome this, one can use numerical algorithms that approximately estimate the value of the integral. For high dimensional integrals, quasi-Monte Carlo (QMC) methods are very popular. QMC methods are equal-weight quadrature rules where the quadrature points are chosen deterministically, unlike Monte Carlo (MC) methods where the points are chosen randomly. The families of integration lattice nodes and digital nets are the most popular quadrature points used. These methods consider the integrand to be a deterministic function. An alternate approach, called Bayesian cubature, postulates the integrand to be an instance of a Gaussian stochastic process.
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Title: AN ACCELERATING COUETTE FLOW IN NEK5000: APPLICATIONS IN OCEANOGRAPHY AND MAGNETOHYDRODYNAMICS
Creator: Miksis, Zachary M.
Date: 2017, 2017-05
Description: Nek5000 is a highly scalable spectral element code used in a broad array of problems in computational fluid dynamics. In this thesis, we focus...
Show moreNek5000 is a highly scalable spectral element code used in a broad array of problems in computational fluid dynamics. In this thesis, we focus on applying the code to a model problem of an accelerating Couette flow, or a hydrodynamic flow between two plates, of which the top plate is accelerating and the bottom plate is stationary, and verifying the numerical methods as applied to this problem. We obtain an analytical solution to the hydrodynamic flow problem, and use this to analyze the effects of changing time step length, the size of the computational mesh, and the computational polynomial order on the accuracy and stability of Nek5000. Additionally, we discuss the addition of an applied magnetic field to the hydrodynamic Couette flow, and provide a formulation for an exact solution to this magnetohydrodynamic problem that can be used to further verify Nek5000 in a similar fashion to the hydrodynamic problem.
M.S. In Applied Mathematics, May 2017
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Title: RUNTIME FOR PERFORMING EXACT TESTS ON THE PI STATISTICAL MODEL FOR RANDOM GRAPHS
Creator: Dillon, Martin
Date: 2016, 2016-05
Description: In statistics, we ask whether some statistical model ts observed data. We use a Markov chain proposed by Gross, Petrovi c, and Stasi to...
Show moreIn statistics, we ask whether some statistical model ts observed data. We use a Markov chain proposed by Gross, Petrovi c, and Stasi to perform exact testing for the p1 random graph model. By comparing it to the simple switch Markov chain, we prove that it mixes rapidly on many classes of degree sequences, and we discuss why it is sometimes better suited than the simple switch chain, and try to easily introduce the concepts from the general theory along the way.
M.S. in Applied Mathematics, May 2016
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