Free boundary problems can be reformulated as shape optimization problems with partial differential equation (PDE) constraints, making them solvable through numerical shape optimization algorithms. The Smooth Extension Embedding Method (SEEM), a novel approach for solving PDEs, leverages straightforward discretization of complex domains and achieves a high degree of convergence for problems with smooth solutions. This makes SEEM a viable alternative to the finite element methods used in traditional numerical shape optimization algorithms, effectively overcoming key challenges such as mesh degeneration and low-order convergence. To enhance the stability of SEEM while maintaining its high-order accuracy for boundary value problems on non-smooth numerical representations of inherently smooth boundaries, we employ a regularized level set boundary approximation technique. This enhancement broadens the applicability of SEEM to the class of free boundary problems with a shape optimization approach. Through theoretical and practical examples, experiments show that our improved SEEM-based algorithm maintains high-order accuracy in shape gradient approximation and circumvents computational pitfalls like mesh degeneration that can arise during shape evolution. This makes the direct use of a simple shape gradient formula and an explicit gradient descent flow possible. The resulting algorithm employs straightforward domain grids throughout the shape optimization process, resulting in significant computational savings and ease of implementation.

This thesis presents SEEM (Smooth Extension Embedding Method), a novel approach to the

solution of boundary value problems within the framework of the fictitious domain method

philosophy. The salient feature of the novel method is that it reduces the whole boundary

value problem to a linear constraint for an appropriate optimization problem formulated in

a larger, simpler set which contains the domain on which the boundary value problem is

posed and which allows for the use of straightforward discretizations. It can also be viewed

as a fully discrete meshfree method which uses a novel class of basis functions, thus building

a bridge between fictitious domain and meshfree methods.

SEEM in essence computes a (discrete) extension of the solution to the boundary value

problem by selecting it as a smooth element of the complete affine family of solutions of the

original equations, which now yield an underdetermined problem for an unknown defined in

the whole fictitious domain. The actual regularity of this extension is determined by that

of the analytic solution and by the choice of objective functional. Numerical experiments

are presented which demonstrate that the method can be stably used to efficiently solve

boundary value problems on general geometries, and that it produces solutions of tunable

(and high) accuracy. Divergence-free and time-dependent problems are considered as well.

Asymptotic spectral techniques have become a powerful tool in estimating statistical properties of systems that can be well approximated by rectangular matrices with i.i.d. or highly structured entries. Starting in the mid 2000s, results from Random Matrix Theory were used along these lines to investigate problems related to financial data, particularly the out-of sample risk underestimation of portfolios constructed through mean-variance optimization. If the returns of assets to be held in a portfolio are assumed independent and stationary, then these results are universal in that they do not depend on the precise distribution of

returns. This universality has been somewhat misrepresented in the literature, however, as asymptotic results require that an arbitrarily long time horizon be available before such predictions necessarily become accurate. This makes these methods ill-suited when moving to high frequency data, for example, where the number of data-points increases but the overall time horizon remains the same or even decreases. In order to reconcile these models with the highly non-Gaussian returns observed in financial data, a new ensemble of random rectangular matrices are introduced, modeled on the observations of independent Levy processes over a fixed time horizon. The key mathematical results describe the eigenvalues of these models' sample covariance matrices, which exhibit remarkably similar scaling behavior to what is seen when working with daily and intraday data on the S&P 500 and Nikkei 225.

In this paper we present a novel point-based rendering approach based on object-space point interpolation of densely sampled surfaces. We introduce the concept of a transformation-invariant covariance matrix of a set of points which can efficiently be used to determine splat sizes in a multiresolution point hierarchy. We also analyze continuous point interpolation in object-space, and we define a new class of parametrized blending kernels as well as a normalization procedure to achieve smooth blending. Furthermore, we present a hardware accelerated rendering algorithm based on texture mapping and a-blending as well as programmable vertex- and pixel-shaders. Experimental results show the high quality and rendering efficiency that is achieved by our approach.

In this thesis, we explore Markov chains with random transition matrices. Such chains are a development on classic Markov chains where the transition matrix is taken to be random. The intuition for this is that we may be interested in modeling phenomenas where the homogeneity assumption of classic Markov chains is invalid. We first use such chains to model credit risk. The randomness of the transition matrix is used to represent the randomness of the economy that underlies credit risk. With this, we model a portfolio of loans and the risk due to having a shared economy. We then proceed to explore theoretical properties of such chains with a focus on their asymptotic behavior. In the case of absorbing chains, we show that the the infinite product of independent and identically distributed random matrices must converge almost surely. We also introduce perturbed Markov chains as a special form of Markov chains with random transition matrices. Perturbed Markov chains are Markov chains with transition matrices of the form P+εQ where P is taken to be a constant matrix, 0<ε<1 is a constant and Q random with rows summing to 0. For perturbed Markov chains with P irreducible, we approximate the long-run fluctuation of such chains. As for perturbed Markov chains with P having absorbing states, we approximate the variance of the fundamental matrix and mean time to absorption. We also explore some applications. In the Moran process application, we study the impact of temporal randomness on the Moran process and derive an analytic result to calculate the probability of fixation.