Sensitivity analysis of finite element-based equilibrium problems using Padé approximants

Finite element analysis is commonly used to analyze structures. It is often desirable to use the solution of finite element problems as part of the objective function for structural optimization;Since finite element solutions can be quite demanding numerically, it is a common practice to approximate the finite element solution in the neighborhood of an original design for the purpose of simplifying the calculation of the objective function. This procedure is called sensitivity analysis. The range of validity of the approximation is very important. The range usually depends on whether high order terms are included in the expansion and, if they are included, on the convergence of the expansion;This thesis presents the sensitivity analysis of finite element-based equilibrium problems using Pade approximants. The goal is to provide an approximation valid for a large design change. The thesis lays the foundation for Pade approximants in a finite element context and illustrates their use through several examples

Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation

[EN] A new way to compute the Taylor polynomial of a matrix exponential is presented which reduces the number of matrix multiplications in comparison with the de-facto standard Paterson-Stockmeyer method for polynomial evaluation. Combined with the scaling and squaring procedure, this reduction is sufficient to make the Taylor method superior in performance to Pade approximants over a range of values of the matrix norms. An efficient adjustment to make the method robust against overscaling is also introduced. Numerical experiments show the superior performance of our method to have a similar accuracy in comparison with state-of-the-art implementations, and thus, it is especially recommended to be used in conjunction with Lie-group and exponential integrators where preservation of geometric properties is at issue.This work was funded by Ministerio de Economia, Industria y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE). P.B. was additionally supported by a contract within the Program Juan de la Cierva Formacion (Spain).Bader, P.; Blanes Zamora, S.; Casas, F. (2019). Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation. Mathematics. 7(12):1-19. https://doi.org/10.3390/math7121174S119712Iserles, A., Munthe-Kaas, H. Z., Nørsett, S. P., & Zanna, A. (2000). Lie-group methods. Acta Numerica, 9, 215-365. doi:10.1017/s0962492900002154Blanes, S., Casas, F., Oteo, J. A., & Ros, J. (2009). The Magnus expansion and some of its applications. Physics Reports, 470(5-6), 151-238. doi:10.1016/j.physrep.2008.11.001Casas, F., & Iserles, A. (2006). Explicit Magnus expansions for nonlinear equations. Journal of Physics A: Mathematical and General, 39(19), 5445-5461. doi:10.1088/0305-4470/39/19/s07Celledoni, E., Marthinsen, A., & Owren, B. (2003). Commutator-free Lie group methods. Future Generation Computer Systems, 19(3), 341-352. doi:10.1016/s0167-739x(02)00161-9Crouch, P. E., & Grossman, R. (1993). Numerical integration of ordinary differential equations on manifolds. Journal of Nonlinear Science, 3(1), 1-33. doi:10.1007/bf02429858Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286. doi:10.1017/s0962492910000048Najfeld, I., & Havel, T. F. (1995). Derivatives of the Matrix Exponential and Their Computation. Advances in Applied Mathematics, 16(3), 321-375. doi:10.1006/aama.1995.1017Sidje, R. B. (1998). Expokit. ACM Transactions on Mathematical Software, 24(1), 130-156. doi:10.1145/285861.285868Higham, N. J., & Al-Mohy, A. H. (2010). Computing matrix functions. Acta Numerica, 19, 159-208. doi:10.1017/s0962492910000036Paterson, M. S., & Stockmeyer, L. J. (1973). On the Number of Nonscalar Multiplications Necessary to Evaluate Polynomials. SIAM Journal on Computing, 2(1), 60-66. doi:10.1137/0202007Ruiz, P., Sastre, J., Ibáñez, J., & Defez, E. (2016). High performance computing of the matrix exponential. Journal of Computational and Applied Mathematics, 291, 370-379. doi:10.1016/j.cam.2015.04.001Sastre, J., Ibán͂ez, J., Defez, E., & Ruiz, P. (2015). New Scaling-Squaring Taylor Algorithms for Computing the Matrix Exponential. SIAM Journal on Scientific Computing, 37(1), A439-A455. doi:10.1137/090763202Sastre, J. (2018). Efficient evaluation of matrix polynomials. Linear Algebra and its Applications, 539, 229-250. doi:10.1016/j.laa.2017.11.010Westreich, D. (1989). Evaluating the matrix polynomial I+A+. . .+A/sup N-1/. IEEE Transactions on Circuits and Systems, 36(1), 162-164. doi:10.1109/31.16591An Efficient Alternative to the Function Expm of Matlab for the Computation of the Exponential of a Matrix http://www.gicas.uji.es/Research/MatrixExp.htmlKenney, C. S., & Laub, A. J. (1998). A Schur--Fréchet Algorithm for Computing the Logarithm and Exponential of a Matrix. SIAM Journal on Matrix Analysis and Applications, 19(3), 640-663. doi:10.1137/s0895479896300334Dieci, L., & Papini, A. (2000). Padé approximation for the exponential of a block triangular matrix. Linear Algebra and its Applications, 308(1-3), 183-202. doi:10.1016/s0024-3795(00)00042-2Higham, N. J., & Tisseur, F. (2000). A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra. SIAM Journal on Matrix Analysis and Applications, 21(4), 1185-1201. doi:10.1137/s0895479899356080Celledoni, E., & Iserles, A. (2000). Approximating the exponential from a Lie algebra to a Lie group. Mathematics of Computation, 69(232), 1457-1481. doi:10.1090/s0025-5718-00-01223-

Series Representations and Approximation of some Quantile Functions appearing in Finance

It has long been agreed by academics that the inversion method is the method of choice for generating random variates, given the availability of a cheap but accurate approximation of the quantile function. However for several probability distributions arising in practice a satisfactory method of approximating these functions is not available. The main focus of this thesis will be to develop Taylor and asymptotic series representations for quantile functions of the following probability distributions; Variance Gamma, Generalized Inverse Gaussian, Hyperbolic, -Stable and Snedecor’s F distributions. As a secondary matter we briefly investigate the problem of approximating the entire quantile function. Indeed with the availability of these new analytic expressions a whole host of possibilities become available. We outline several algorithms and in particular provide a C++ implementation for the variance gamma case. To our knowledge this is the fastest available algorithm of its sort

Localized magnetic field in the O(N) model

We consider the critical O(N) model in the presence of an external magnetic field localized in space. This setup can potentially be realized in quantum simulators and in some liquid mixtures. The external field can be understood as a relevant perturbation of the trivial line defect, and thus triggers a defect Renormalization Group (RG) flow. In agreement with the g-theorem, the external localized field leads at long distances to a stable nontrivial defect CFT (DCFT) with g < 1. We obtain several predictions for the corresponding DCFT data in the epsilon expansion and in the large N limit. The analysis of the large N limit involves a new saddle point and, remarkably, the study of fluctuations around it is enabled by recent progress in AdS loop diagrams. Our results are compatible with results from Monte Carlo simulations and we make several predictions that can be tested in the future

The approximation of functions with branch points

In recent years Pade approximants have proved to be one of the most useful computational tools in many areas of theoretical physics, most notably in statistical mechanics and strong interaction physics. The underlying reason for this is that very often the equations describing a physical process are so complicated that the simplest (if not the only) way of obtaining their solution is to perform a power series expansion in some parameters of the problem. Furthermore, the physical values of the para­meters are often such that this perturbation expansion does not converge and is therefore only a formal solution to the problem; as such it cannot be used quantitatively. However, the relevant information is contained in the coefficients of the perturbation series and the Fade approximants provide a convenient mathematical technique for extracting this information in a convergent way. A major difficulty with these approximants is that their convergence is restricted to regions of the complex plane free from any branch cuts; for example, the (N/N+j) Pade approximants to a series of Stieltjes converge to an analytic function in the complex plane cut along the negative real axis. The central idea of the present work is to obtain convergence along these branch cuts by using approximants which themselves have branch points. The ideas presented in this thesis are expected to be only the beginning of a large investigation into the use of multi-valued approximants as a practical method of approximation. In Chapter 1 we shall see that such approximants arise as natural generalisations of Pade approximants and possess many of the properties of Pade approximants; in particular, the very important property of homographic covariance. We term these approximants ’algebraic' approximants (since they satisfy an algebraic equation) and we are mainly concerned with the 'simplest' of these approximants, the quadratic approximants of Shafer. Chapter 2 considers some of the known convergence results for Pade approximants to indicate the type of results we nay reasonably expect to hold (and to be able to prove) for quadratic (and higher order) approximants. A discussion of various numerical examples is then given to illustrate the possible practical usefulness of these latter approximants. A major application of all these approximants is discussed in Chapter 3, where the problem of evaluating Feynman matrix elements in the physical region is considered; in this case, the physical region is along branch cuts. Several simple Feynman diagrams are considered to illustrate (a) the potential usefulness of the calculational scheme presented and (b) the relative merits of rational (Pade), quadratic and cubic approximation schemes. The success of these general approximation schemes in one variable (as exhibited by the results of Chapters 2 and 3) leads, in Chapter 4 to a consideration of the corresponding approximants in two variables. We shall see that the two variable scheme developed for rational approximants can be extended in a very natural way to define two variable "t-power" approximants. Numerical results are presented to indicate the usefulness of these schemes in practice. A final application to strong interaction physics is given in Chapter 5, where the analytic continuation of Legendre series is considered. Such series arise in partial wave expansions of the scattering amplitude. We shall see that the Pade Legendre approximants of Fleischer and Common can be generalised to produce corresponding quadratic Legendre approximants: various examples are considered to illustrate the relative merits of these schemes