Exploring the mirror TBA

We apply the contour deformation trick to the Thermodynamic Bethe Ansatz equations for the AdS_5 \times S^5 mirror model, and obtain the integral equations determining the energy of two-particle excited states dual to N=4 SYM operators from the sl(2) sector. We show that each state/operator is described by its own set of TBA equations. Moreover, we provide evidence that for each state there are infinitely-many critical values of 't Hooft coupling constant \lambda, and the excited states integral equations have to be modified each time one crosses one of those. In particular, estimation based on the large L asymptotic solution gives \lambda \approx 774 for the first critical value corresponding to the Konishi operator. Our results indicate that the related calculations and conclusions of Gromov, Kazakov and Vieira should be interpreted with caution. The phenomenon we discuss might potentially explain the mismatch between their recent computation of the scaling dimension of the Konishi operator and the one done by Roiban and Tseytlin by using the string theory sigma model.Comment: 69 pages, v2: new "hybrid" equations for YQ-functions, figures and tables are added. Analyticity of Y-system is discussed, v3: published versio

Analytical and numerical techniques for wave scattering

In this thesis, we study the mathematical solution of wave scattering problems which describe the behaviour of waves incident on obstacles and are highly relevant to a raft of applications in the aerospace industry. The techniques considered in the present work can be broadly classed into two categories: analytically based methods which use special transforms and functions to provide a near-complete mathematical description of the scattering process, and numerical techniques which select an approximate solution from a general finite-dimensional space of possible candidates. The first part of this thesis addresses an analytical approach to the scattering of acoustic and vortical waves on an infinite periodic arrangement of finite-length flat blades in parallel mean flow. This geometry serves as an unwrapped model of the fan components in turbo-machinery. Our contributions include a novel semi-analytical solution based on the Wiener–Hopf technique that extends previous work by lifting the restriction that adjacent blades overlap, and a comprehensive study of the composition of the outgoing energy flux for acoustic wave scattering on this array of blades. These results provide an insight into the importance of energy conversion between the unsteady vorticity shed from the trailing edges of the cascade blades and the acoustic field. Furthermore, we show that the balance of incoming and outgoing energy fluxes of the unsteady field provides a convenient tool for understanding several interesting scattering symmetries on this geometry. In the second part of the thesis, we focus on numerical techniques based on the boundary integral method which allows us to write the governing equations for zero mean flow in the form of Fredholm integral equations. We study the solution of these integral equations using collocation methods for two-dimensional scatterers with smooth and Lipschitz boundaries. Our contributions are as follows: Firstly, we explore the extent to which least-squares oversampling can improve collocation. We provide rigorous analysis that proves guaranteed convergence for small amounts of oversampling and shows that superlinear oversampling can ensure faster asymptotic convergence rates of the method. Secondly, we examine the computation of the entries in the discrete linear system representing the continuous integral equation in collocation methods for hybrid numerical-asymptotic basis spaces on simple geometric shapes in the context of high-frequency wave scattering. This requires the computation of singular highly-oscillatory integrals and we develop efficient numerical methods that can compute these integrals at frequency-independent cost. Finally, we provide a general result that allows the construction of recurrences for the efficient computation of quadrature moments in a broad class of Filon quadrature methods, and we show how this framework can also be used to accelerate certain Levin quadrature methods.Supported by EPSRC grant EP/L016516/

On the Application of the Incomplete QR Algorithm to the Analysis of Microstrip Antennas

In this paper, we provide some insight into the usage of fast, iterative, method-of-moments (MoM) solution of integral equations (IE) describing antennas and other metallic structures immersed in a planar multilayered environment. Based on the form of multilayered media Green's functions, we extract free-space terms, associated with direct rays within the analyzed structure, reducing the number of significant interactions required to describe the rest of MoM matrix. Next, we show that it is possible to construct a hybrid algorithm, where the fast multipole method (FMM) is used to the free-space matrix part, while the reduced rank incomplete QR (iQR) decomposition is applied to the remaining portion of the MoM matrix. This HM-iQR (hybrid multipole - incomplete QR) method is applied to a relatively large (in terms o f the number of unknowns) problem of plane wave scattering by a finite array of rectangular microstrip patches printed on a grounded dielectric slab. Computation results from the new algorithm are compared to literature data and to the results of the pure low rank IE-QR method

A Hybrid Boundary Element Method for Elliptic Problems with Singularities

The singularities that arise in elliptic boundary value problems are treated locally by a singular function boundary integral method. This method extracts the leading singular coefficients from a series expansion that describes the local behavior of the singularity. The method is fitted into the framework of the widely used boundary element method (BEM), forming a hybrid technique, with the BEM computing the solution away from the singularity. Results of the hybrid technique are reported for the Motz problem and compared with the results of the standalone BEM and Galerkin/finite element method (GFEM). The comparison is made in terms of the total flux (i.e. the capacitance in the case of electrostatic problems) on the Dirichlet boundary adjacent to the singularity, which is essentially the integral of the normal derivative of the solution. The hybrid method manages to reduce the error in the computed capacitance by a factor of 10, with respect to the BEM and GFEM

Computation of option greeks under hybrid stochastic volatility models via Malliavin calculus

This study introduces computation of option sensitivities (Greeks) using the Malliavin calculus under the assumption that the underlying asset and interest rate both evolve from a stochastic volatility model and a stochastic interest rate model, respectively. Therefore, it integrates the recent developments in the Malliavin calculus for the computation of Greeks: Delta, Vega, and Rho and it extends the method slightly. The main results show that Malliavin calculus allows a running Monte Carlo (MC) algorithm to present numerical implementations and to illustrate its effectiveness. The main advantage of this method is that once the algorithms are constructed, they can be used for numerous types of option, even if their payoff functions are not differentiable.Comment: Published at https://doi.org/10.15559/18-VMSTA100 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/

Scattering and radiation analysis of three-dimensional cavity arrays via a hybrid finite element method

A hybrid numerical technique is presented for a characterization of the scattering and radiation properties of three-dimensional cavity arrays recessed in a ground plane. The technique combines the finite element and boundary integral methods and invokes Floquet's representation to formulate a system of equations for the fields at the apertures and those inside the cavities. The system is solved via the conjugate gradient method in conjunction with the Fast Fourier Transform (FFT) thus achieving an O(N) storage requirement. By virtue of the finite element method, the proposed technique is applicable to periodic arrays comprised of cavities having arbitrary shape and filled with inhomogeneous dielectrics. Several numerical results are presented, along with new measured data, which demonstrate the validity, efficiency, and capability of the technique

A Framework for Worst-Case and Stochastic Safety Verification Using Barrier Certificates

This paper presents a methodology for safety verification of continuous and hybrid systems in the worst-case and stochastic settings. In the worst-case setting, a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region. No explicit computation of reachable sets is required in the construction of barrier certificates, which makes it possible to handle nonlinearity, uncertainty, and constraints directly within this framework. In the stochastic setting, our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set, a bound whose validity is proven by the existence of a barrier certificate. For polynomial systems, barrier certificates can be constructed using convex optimization, and hence the method is computationally tractable. Some examples are provided to illustrate the use of the method