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We propose a new numerical algorithm to construct a structured numerical elliptic grid of a doubly connected domain. Our method is applicable to domains with boundaries defined by two contour lines of a two-dimensional function. The…
This book draft gives the theory of a new method for p dimensional adaptive integration by repeated 2^p subdivision of simplexes and hypercubes. A new method of constructing high order integration routines for these geometries permits…
Hamiltonian simulation, i.e., simulating the real time evolution of a target quantum system, is a natural application of quantum computing. Trotter-Suzuki splitting methods can generate corresponding quantum circuits; however, a faithful…
This article presents a systematic methodology for modeling a class of flexible multidimensional mechanical structures defined by linear elastic relations that directly allows to obtain their infinite-dimensional port-Hamiltonian…
Various Hamiltonian simulation algorithms have been proposed to efficiently study the dynamics of quantum systems on a quantum computer. The existing algorithms generally approximate the time evolution operators, which may need a deep…
Many hard graph problems, such as Hamiltonian Cycle, become FPT when parameterized by treewidth, a parameter that is bounded only on sparse graphs. When parameterized by the more general parameter clique-width, Hamiltonian Cycle becomes…
The capacity to custom tailor the properties of quantum matter and materials is a central requirement for enlarging their range of possible functionalities. A particularly promising route is the use of driving protocols to engineer specific…
Randomization has been applied to Hamiltonian simulation in a number of ways to improve the accuracy or efficiency of product formulas. Deterministic product formulas are often constructed in a symmetric way to provide accuracy of even…
Long-term stability studies of nonlinear Hamiltonian systems require symplectic integration algorithms which are both fast and accurate. In this paper, we study a symplectic integration method wherein the symplectic map representing the…
This paper studies the parameter tuning problem of positive linear systems for optimizing their stability properties. We specifically show that, under certain regularity assumptions on the parametrization, the problem of finding the…
In this paper we present a procedure to compute reducible invariant tori and their stable and unstable manifolds in stroboscopic Poincar\'e maps. The method has two steps. In the first step we compute, by means of a quadratically convergent…
In this thesis, I explore the possibilities of conducting Bayesian optimization techniques in high dimensional domains. Although high dimensional domains can be defined to be between hundreds and thousands of dimensions, we will primarily…
The low-rank approximation is a complexity reduction technique to approximate a tensor or a matrix with a reduced rank, which has been applied to the simulation of high dimensional problems to reduce the memory required and computational…
This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). Searching this approximation in a data-driven approach is formalised as attempting to solve a low-rank…
This paper introduces a novel variational Bayesian method that integrates Tucker decomposition for efficient high-dimensional inverse problem solving. The method reduces computational complexity by transforming variational inference from a…
We describe a renormalization group transformation that is related to the breakup of golden invariant tori in Hamiltonian systems with two degrees of freedom. This transformation applies to a large class of Hamiltonians, is conceptually…
The extended flow equations of the multi-component Toda hierarchy are constructed. We give the Hirota bilinear equations and tau function of this new extended multi-component Toda hierarchy(EMTH). Because of logarithmic terms, some extended…
In recent years, a number of fast algorithms for computing the determinant of a Toeplitz matrix were developed. The fastest algorithm we know so far is of order $k^2\log{n}+k^3$, where $n$ is the number of rows of the Toeplitz matrix and…
We explore the utilization of higher-order discretization techniques in optimizing the gate count needed for quantum computer based solutions of partial differential equations. To accomplish this, we present an efficient approach for…
In this paper, we discuss adaptive approximations of an elliptic eigenvalue optimization problem in a phase-field setting by a conforming finite element method. An adaptive algorithm is proposed and implemented in several two dimensional…