Echet Derivative Research Articles

Optimal computational parameters for maximum accuracy and minimum cost of Arnoldi-based time-stepping methods for flow global stability analysis

Global instability analysis of flows is often performed via time-stepping methods, based on the Arnoldi algorithm. When setting up these methods, several computational parameters must be chosen, which affect intrinsic errors of the procedure, such as the truncation errors, the discretization error of the flow solver, the error associated with the nonlinear terms of the Navier-Stokes equations and the error associated with the limited size of the approximation of the Jacobian matrix. This paper develops theoretical equations for the estimation of optimal balance between accuracy and cost for each case. The 2D open cavity flow is used both for explaining the effect of the parameters on the accuracy and the cost of the solution, and for verifying the quality of the predictions. The equations demonstrate the impact of each parameter on the quality of the solution. For example, if higher order methods are used for approaching a Fr\'echet derivative in the procedure, it is shown that the solution deteriorates more rapidly for larger grids or less accurate flow solvers. On the other hand, lower order approximations are more sensitive to the initial disturbance magnitude. Nevertheless, for accurate flow solvers and moderate grid dimensions, first order Fr\'echet derivative approximation with optimal computational parameters can provide 5 decimal place accurate eigenvalues. It is further shown that optimal parameters based on accuracy tend to also lead to the most cost-effective solution. The predictive equations, guidelines and conclusions are general, and, in principle, applicable to any flow, including 3D ones.

Combinatory Adjoints and Differentiation

We develop a compositional approach for automatic and symbolic differentiation based on categorical constructions in functional analysis where derivatives are linear functions on abstract vectors rather than being limited to scalars, vectors, matrices or tensors represented as multi-dimensional arrays. We show that both symbolic and automatic differentiation can be performed using a differential calculus for generating linear functions representing Fr\'echet derivatives based on rules for primitive, constant, linear and bilinear functions as well as their sequential and parallel composition. Linear functions are represented in a combinatory domain-specific language. Finally, we provide a calculus for symbolically computing the adjoint of a derivative without using matrices, which are too inefficient to use on high-dimensional spaces. The resulting symbolic representation of a derivative retains the data-parallel operations from the input program. The combination of combinatory differentiation and computing formal adjoints turns out to be behaviorally equivalent to reverse-mode automatic differentiation. In particular, it provides opportunities for optimizations where matrices are too inefficient to represent linear functions.

Stability of solutions to abstract evolution equations in Banach spaces under nonclassical assumptions

The stability of the solution to the equation $(*)\dot{u} = F(t,u)+f(t)$, $t\ge 0$, $u(0)=u_0$ is studied. Here $F(t,u)$ is a nonlinear operator in a Banach space $\mathcal{X}$ for any fixed $t\ge 0$ and $F(t,0)=0$, $\forall t\ge 0$. We assume that the Fr\'echet derivative of $F(t,u)$ is H\"{o}lder continuous of order $q>0$ with respect to $u$ for any fixed $t\ge 0$, i.e., $\|F'_u(t,w) - F'_u(t,v)\|\le \alpha(t)\|v - w\|^{q}$, $q>0$. We proved that the equilibrium solution $v=0$ to the equation $\dot{v} = F(t,v)$ is Lyapunov stable under persistently acting perturbation $f(t)$ if $\sup_{t\ge 0}\int_0^t \alpha(\xi)\|U(t,\xi)\|\, d\xi<\infty$ and $\sup_{t\ge 0}\|U(t)\|<\infty$. Here, $U(t):=U(t,0)$ and $U(t,\xi)$ is the solution to the equation $\frac{d}{dt}{U}(t,\xi) = F'_u(t,0)U(t,\xi)$, $t\ge \xi$, $U(\xi,\xi)=I$, where $I$ is the identity operator in $\mathcal{X}$. Sufficient conditions for the solution $u(t)$ to equation (*) to be bounded and for $\lim_{t\to\infty}u(t) = 0$ are proposed and justified. Stability of solutions to equations with unbounded operators in Hilbert spaces is also studied.

Quantum storage in quantum ferromagnets

We must protect inherently fragile quantum data to unlock the potential of quantum technologies. A pertinent concern in schemes for quantum storage is their potential for near-term implementation. Since Heisenberg ferromagnets are readily available, we investigate their potential for robust quantum storage. We propose to use permutation-invariant quantum codes to store quantum data in Heisenberg ferromagnets, because the ground space of any Heisenberg ferromagnet must be symmetric under any permutation of the underlying qubits. By exploiting an area law on the expected energy of Pauli errors, we show that increasing the effective dimension of Heisenberg ferromagnets can improve the storage lifetime. When the effective dimension of Heisenberg ferromagnets is maximal, we also obtain an upper bound for the storage error. This result relies on perturbation theory, where we use Davis' divided difference representation for Fr\'echet derivatives along with the recursive structure of these divided differences. Our numerical bounds allow us to better understand how quantum memory lifetimes can be enhanced in Heisenberg ferromagnets.

Numerical analysis of least squares and perceptron learning for classification problems

This work presents study on regularized and non-regularized versions of perceptron learning and least squares algorithms for classification problems. Fr'echet derivatives for regularized least squares and perceptron learning algorithms are derived. Different Tikhonov's regularization techniques for choosing the regularization parameter are discussed. Decision boundaries obtained by non-regularized algorithms to classify simulated and experimental data sets are analyzed.

On Regularity of the Logarithmic Forward Map of Electrical Impedance Tomography

This work considers properties of the logarithm of the Neumann-to-Dirichlet boundary map for the conductivity equation in a Lipschitz domain. It is shown that the mapping from the (logarithm of) the conductivity, i.e. the (logarithm of) the coefficient in the divergence term of the studied elliptic partial differential equation, to the logarithm of the Neumann-to-Dirichlet map is continuously Fr\'echet differentiable between natural topologies. Moreover, for any essentially bounded perturbation of the conductivity, the Fr\'echet derivative defines a bounded linear operator on the space of square integrable functions living on the domain boundary, although the logarithm of the Neumann-to-Dirichlet map itself is unbounded in that topology. In particular, it follows from the fundamental theorem of calculus that the difference between the logarithms of any two Neumann-to-Dirichlet maps is always bounded on the space of square integrable functions. All aforementioned results also hold if the Neumann-to-Dirichlet boundary map is replaced by its inverse, i.e. the Dirichlet-to-Neumann map.

On the non-linear integral equation approach for an inverse boundary value problem for the heat equation

We consider the inverse problem of reconstructing the interior boundary curve of a doubly connected domain from the knowledge of the temperature and the thermal flux on the exterior boundary curve. The use of the Laguerre transform in time leads to a sequence of stationary inverse problems. Then, the application of the modified single-layer ansatz, reduces the problem to a sequence of systems of non-linear boundary integral equations. An iterative algorithm is developed for the numerical solution of the obtained integral equations. We find the Fr\'echet derivative of the corresponding integral operator and we show the unique solvability of the linearized equation. Full discretization is realized by a trigonometric quadrature method. Due to the inherited ill-possedness of the derived system of linear equations we apply the Tikhonov regularization. The numerical results show that the proposed method produces accurate and stable reconstructions.

Bouligand-Levenberg-Marquardt iteration for a non-smooth ill-posed inverse problem

In this paper, we consider a modified Levenberg--Marquardt method for solving an ill-posed inverse problem where the forward mapping is not G\^ateaux differentiable. By relaxing the standard assumptions for the classical smooth setting, we derive asymptotic stability estimates that are then used to prove the convergence of the proposed method. This method can be applied to an inverse source problem for a non-smooth semilinear elliptic PDE where a Bouligand subdifferential can be used in place of the non-existing Fr\'echet derivative, and we show that the corresponding Bouligand-Levenberg-Marquardt iteration is an iterative regularization scheme. Numerical examples illustrate the advantage over the corresponding Bouligand-Landweber iteration.

Fréchet differentiability of molecular distribution functions II: the Ursell function

For a grand canonical ensemble of classical point-like particles at equilibrium in continuous space we investigate the functional relationship between a stable and regular pair potential describing the interaction of the particles and the thermodynamical limit of the Ursell or pair correlation function. For certain admissible perturbations of the pair potential and sufficiently small activity we rigorously establish Frechet differentiability of the Ursell function in the $L^1$ norm. Furthermore, concerning the thermodynamical limit of the pair distribution function we explicitly compute its Fr\'echet derivative as a sum of a multiplication operator and an integral operator.

SKEW-SYMMETRIC SOLVENT FOR SOLVING A POLYNOMIAL EIGENVALUE PROBLEM

In this paper a nonlinear matrix equation is considered which has the form <TEX>$$P(X)=A_0X^m+A_1X^{m-1}+{\cdots}+A_{m-1}X+A_m=0$$</TEX> where X is an <TEX>$n{\times}n$</TEX> unknown real matrix and <TEX>$A_m$</TEX>, <TEX>$A_{m-1}$</TEX>, <TEX>${\cdots}$</TEX>, <TEX>$A_0$</TEX> are <TEX>$n{\times}n$</TEX> matrices with real elements. Newtons method is applied to find the skew-symmetric solvent of the matrix polynomial P(X). We also suggest an algorithm which converges the skew-symmetric solvent even if the Fr<TEX>$\acute{e}$</TEX>echet derivative of P(X) is singular.

Simultaneous Reconstruction of Outer Boundary Shape and Admittivity Distribution in Electrical Impedance Tomography

The aim of electrical impedance tomography is to reconstruct the admittivity distribution inside a physical body from boundary measurements of current and voltage. Due to the severe ill-posedness of the underlying inverse problem, the functionality of impedance tomography relies heavily on accurate modelling of the measurement geometry. In particular, almost all reconstruction algorithms require the precise shape of the imaged body as an input. In this work, the need for prior geometric information is relaxed by introducing a Newton-type output least squares algorithm that reconstructs the admittivity distribution and the object shape simultaneously. The method is built in the framework of the complete electrode model and it is based on the Fr\'echet derivative of the corresponding current-to-voltage map with respect to the object boundary shape. The functionality of the technique is demonstrated via numerical experiments with simulated measurement data.

Forward and adjoint simulations of seismic wave propagation on fully unstructured hexahedral meshes

We present forward and adjoint spectral-element simulations of coupled acoustic and (an)elastic seismic wave propagation on fully unstructured hexahedral meshes. Simulations benefit from recent advances in hexahedral meshing, load balancing and software optimization. Meshing may be accomplished using a mesh generation tool kit such as CUBIT, and load balancing is facilitated by graph partitioning based on the SCOTCH library. Coupling between fluid and solid regions is incorporated in a straightforward fashion using domain decomposition. Topography, bathymetry and Moho undulations may be readily included in the mesh, and physical dispersion and attenuation associated with anelasticity are accounted for using a series of standard linear solids. Finite-frequency Fr'echet derivatives are calculated using adjoint methods in both fluid and solid domains. The software is benchmarked for a layercake model. We present various examples of fully unstructured meshes, snapshots of wavefields and finite-frequency kernels generated by Version 2.0 'Sesame' of our widely used open source spectral-element package SPECFEM3D.

Rapid perturbational calculations for the Helmholtz equation in two dimensions

Existing approaches to the solution of the inverse scattering problems in two and three dimensions rely on linearization of the Helmholtz equation, which requires the knowledge of the Fr\'echet derivative of the far field with respect to the index of refraction. We present an efficient algorithm for this perturbational calculation in two dimensions. Our method is based on the merging and splitting procedures already established for the solution of the Lippmann-Schwinger equation [2], [3], [4]. For an $m$-by-$m$ wavelength problem, the algorithm obtains perturbations to scattered waves for $m$ distinct incident waves in $O(m^3)$ steps.

Perturbation analysis of singular subspaces and deflating subspaces

Perturbation expansions for singular subspaces of a matrix and for deflating subspaces of a regular matrix pair are derived by using a technique previously described by the author. The perturbation expansions are then used to derive Fr\'echet derivatives, condition numbers, and\(r\) th-order perturbation bounds for the subspaces. Vaccaro's result on second-order perturbation expansions for a special class of singular subspaces can be obtained from a general result of this paper. Besides, new perturbation bounds for singular subspaces and deflating subspaces are derived by applying a general theorem on solution of a system of nonlinear equations. The results of this paper reveal an important fact: Each singular subspace and each deflating subspace have individual perturbation bounds and individual condition numbers.

Solutions of a Nonlinear Integral Equation for High-Energy Scattering. II. Numerical Solutions

Solutions of the Ball-Zachariasen equation for high-energy scattering are calculated unmerically. For small values of the parameter $c$, which measures the strength of particle production, the solutions coincide with those obtained in the existence proof of paper I. When $c{\ensuremath{\sigma}}_{\mathrm{el}}$ (${\ensuremath{\sigma}}_{\mathrm{el}}=\mathrm{elastic}\mathrm{cross}\mathrm{section}$) is much smaller than the physical value, the solutions are obtained successfully by either plain iteration or Newton-Kantorovich iteration, but they disagree qualitatively with experiment. A continuation to larger $c$ is attempted by using the result of a successful iteration as the starting point for an iteration with slightly larger $c$. The continuation proceeds only a short distance before coming to a halt, because of a singularity of the Fr\'echet derivative of the nonlinear integral operator. The singularity is circumvented by an excursion into the complex $c$ plane. On return to the real axis the solutions are complex, however, which is not allowed physically. A proposed approximate solution of Ball and Zachariasen, quite different from those of paper I, is also investigated. Plain or Newton-Kantorovich iteration starting with this function fails to converge. The failure is explained again in terms of a nearby singularity of the Fr\'echet derivative. The singularity makes it difficult to decide whether the Ball-Zachariasen function is close to a true solution.

Solutions of a Nonlinear Integral Equation for High-Energy Scattering. I. An Existence Theorem

Ball and Zachariasen have proposed a model of high-energy diffraction scattering, which entails nonshrinking diffraction peaks. The model has some resemblance to the multiperipheral model, in that it emphasizes $s$-channel unitarity with a simple factored form of production amplitudes. It leads to an integral equation for the elastic amplitude which is relatively simple in appearance. The equation is analyzed here with the help of methods which may also be useful in the study of more realistic high-energy models. The Hankel transform of the original equation is studied as a nonlinear equation in a certain Banach space. Existence of an infinite class of solutions is proved by means of the contraction mapping principle. These solutions are constructed by iteration for small values of a parameter $c$, which measures the strength of particle production. The range of allowed values for the product of $c$ and the elastic cross section does not include the physical value. One can try, however, to continue the solutions to the physical value, since they are analytic in $c$. In paper II, the solutions are calculated numerically, and the continuation to large $c$ is attempted. The continuation stops short of its goal, because of a singularity of the Fr\'echet derivative of the nonlinear operator. This derivative becomes a linear integral operator of the "third kind," which has no inverse in a space of continuous functions. No physically acceptable solutions of the Ball-Zachariasen equation have been found. A proposed approximate solution appears to have some difficulties, as is explained in paper II.