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Numerical analysis and simulation of partial differential equation in various fields

  • Sangkwon Kim (김상권, 대학원 수학과 응용수학전공)

초록/요약

The primary purpose of this dissertation is to explore the numerical methods for the phase-field model and financial model. The dissertation consists of published and working papers conducted as a graduate student. First, we consider the phase-field model. This dissertation is describes numerical analysis and solving of vector-valued Allen--Cahn (vAC) equation that model multi-phase separation by extending the Allen--Cahn equation with high order polynomial free energy (ACh) that models phase separation in two components. When solving the vAC equation numerically, this model can lead to the generation of additional spurious phases at interfaces at multiple junctions. The ACh equation ($4$th order or higher) has the advantage of better representing the interfacial phenomenon of the interface, which can solve the problem of generating additional phases at the interface. However, as the order of polynomials increases, the time step restriction is very strict and the equation is very stiff. Implicit and explicit Euler schemes cannot be used. Therefore, we proposed interpolation-based numerical solution scheme with the operator splitting method (OSM) to solve this problem. We apply the developed method in this dissertation. A numerically stable solution to the vAC equation must consider a large number of phase field variables to capture the limitations and exact dynamics of using small time steps. This leads to huge computational costs and makes computations very inefficient. To overcome this problem, OSM and variable Lagrange multipliers can be used to convert multi-phase system problems into a series of binary system problems, reducing the complexity of the problem, computational cost, and memory required. We proved the efficiency and accuracy of the algorithm through various numerical experiments. In addition, we applied an algorithm to the multi-phase image segmentation example to verify its efficiency. Second, we propose a numerical algorithm to reconstruct a local volatility surface function inherent in the European option price at strike prices and maturity times of the market data using the Black--Scholes (BS) equation. The proposed numerical algorithm is defined that induces function in the underlying asset price based on time-dependent volatility. The BS equation is a partial differential equation and has been applied to model financial option pricing. The classical BS equation assumes constant volatility. Unfortunately, these assumption do not reflect the behaviour of real financial market data. The proposed algorithm consists of the three steps. First, we calibrate time-dependent volatility function. Second, Using Monte Carlo simulations, we find effective regions that strongly affect option prices. Finally, we present a guide function for the underlying asset price and use fully implicit finite difference scheme and the conjugate gradient optimization algorithms to calibrate local volatility surface functions.

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목차

Chapter 1. Introduction 1
Chapter 2. Allen–Cahn type equation for data classification 5
2.1. Allen–Cahn equation 6
2.2. Modified Allen–Cahn equation 8
2.3. Numerical solution algorithm 9
2.4. Numerical experiments 10
Chapter 3. Allen–Cahn with high order polynomial free energy 28
3.1. Interpolation-based numerical scheme 29
3.2. Numerical experiments 34
Chapter 4. Vector-valued Allen-Cahn with high-order polynomial potential 42
4.1. Numerical solution algorithm 45
4.2. Numerical experiments 47
Chapter 5. Reconstruct local volatility using Black–Scholes 54
5.1. Numerical solution algorithm 56
5.2. Numerical experiment 61
Chapter 6. Conclusion 73

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