Parameter Manifold Research Articles

REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS

In this paper we present reduced basis (RB) approximations and associated rigorous a posteriori error bounds for the parametrized unsteady Boussinesq equations. The essential ingredients are Galerkin projection onto a low-dimensional space associated with a smooth parametric manifold — to provide dimension reduction; an efficient proper orthogonal decomposition–Greedy sampling method for identification of optimal and numerically stable approximations — to yield rapid convergence; accurate (online) calculation of the solution-dependent stability factor by the successive constraint method — to quantify the growth of perturbations/residuals in time; rigorous a posteriori bounds for the errors in the RB approximation and associated outputs — to provide certainty in our predictions; and an offline–online computational decomposition strategy for our RB approximation and associated error bound — to minimize marginal cost and hence achieve high performance in the real-time and many-query contexts. The method is applied to a transient natural convection problem in a two-dimensional "complex" enclosure — a square with a small rectangle cutout — parametrized by Grashof number and orientation with respect to gravity. Numerical results indicate that the RB approximation converges rapidly and that furthermore the (inexpensive) rigorous a posteriori error bounds remain practicable for parameter domains and final times of physical interest.

The geometry of (non)-Abelian adiabatic pumping

We give a gauge description of the adiabatic charge pumping in closed systems, both in Abelian and non-Abelian processes, and by means of asymptotic Wilson loops in a suitable parameter manifold. Our geometric formulation provides new insights into this issue, and a very simple algorithm for numerical computations. Indeed, as we show first, discretized Berry–Wilczek–Zee holonomies are easy to implement. Finally, we study non-Abelian pumping in a solvable four-state model, already used in several contexts, to demonstrate the relevance of our approach.

Reduced Basis Approximation and Error Bounds for Potential Flows in Parametrized Geometries

AbstractIn this paper we consider (hierarchical, Lagrange) reduced basis approximation anda posteriorierror estimation for potential flows in affinely parametrized geometries. We review the essential ingredients: i) a Galerkin projection onto a low-dimensional space associated with a smooth “parametric manifold” in order to get a dimension reduction; ii) an efficient and effective greedy sampling method for identification of optimal and numerically stable approximations to have a rapid convergence; iii) ana posteriorierror estimation procedure: rigorous and sharp bounds for the linear-functional outputs of interest and over the potential solution or related quantities of interest like velocity and / or pressure; iv) an Offline-Online computational decomposition strategies to achieve a minimummarginal computational costfor high performance in the real-time and many-query (e.g., design and optimization) contexts. We present three illustrative results for inviscid potential flows in parametrized geometries representing a Venturi channel, a circular bend and an added mass problem.

Preface to the special issue, CMAM 2011, no. 3.

Abstract In the recent years, multidimensional numerical simulations with tensor-structured data formats have been recognized as the basic concept for breaking the "curse of dimensionality". Modern applications of tensor methods include the challenging high-dimensional problems of material sciences, bio-science, stochastic modeling, signal processing, machine learning, and data mining, financial mathematics, etc. The guiding principle of the tensor methods is an approximation of multivariate functions and operators with some separation of variables to keep the computational process in a low parametric tensor-structured manifold. Tensors structures had been wildly used as models of data and discussed in the contexts of differential geometry, mechanics, algebraic geometry, data analysis etc. before tensor methods recently have penetrated into numerical computations. On the one hand, the existing tensor representation formats remained to be of a limited use in many high-dimensional problems because of lack of sufficiently reliable and fast software. On the other hand, for moderate dimensional problems (e.g. in "ab-initio" quantum chemistry) as well as for selected model problems of very high dimensions, the application of traditional canonical and Tucker formats in combination with the ideas of multilevel methods has led to the new efficient algorithms. The recent progress in tensor numerical methods is achieved with new representation formats now known as "tensor-train representations" and "hierarchical Tucker representations". Note that the formats themselves could have been picked up earlier in the literature on the modeling of quantum systems. Until 2009 they lived in a closed world of those quantum theory publications and never trespassed the territory of numerical analysis. The tremendous progress during the very recent years shows the new tensor tools in various applications and in the development of these tools and study of their approximation and algebraic properties. This special issue treats tensors as a base for efficient numerical algorithms in various modern applications and with special emphases on the new representation formats.

Visualization of Riemannian-manifold-valued elements by multidimensional scaling

The present contribution suggests to utilize a multidimensional scaling algorithm as a visualization tool for high-dimensional smoothly constrained learnable-system's patterns that lie on Riemannian manifolds. Such visualization tool proves useful in machine learning whenever learning/adaptation algorithms insist on high-dimensional Riemannian parameter manifolds. In particular, the manuscript describes the cases of interest in the recent scientific literature that the parameter space is the set of special orthogonal matrices, the unit hypersphere and the manifold of symmetric positive-definite matrices. The paper also recalls the notion of multidimensional scaling and discusses its algorithmic implementation. Some numerical experiments performed on toy problems help the readers to get acquainted with the problem at hand, while experiments performed on independent component analysis data as well as averaging data show the usefulness of the proposed visualization tool.

Topological Excitations in Spinor Bose-Einstein Condensates

A rich variety of order parameter manifolds of multicomponent Bose-Einstein condensates (BECs) admit various kinds of topological excitations, such as fractional vortices, monopoles, skyrmions, and knots. In this paper, we discuss two topological excitations in spinor BECs: non-Abelian vortices and knots. Unlike conventional vortices, non-Abelian vortices neither reconnect themselves nor pass through each other, but create a rung between them in a topologically stable manner. We discuss the collision dynamics of non-Abelian vortices in the cyclic phase of a spin-2 BEC. In the latter part, we show that a knot, which is a unique topological object characterized by a linking number or a Hopf invariant [$\pi_3 (S^2)=Z$], can be created using a conventional quadrupole magnetic field in a cold atomic system.

Intrinsic geometry of quantum adiabatic evolution and quantum phase transitions

We elucidate the geometry of quantum adiabatic evolution. By minimizing the deviation from adiabaticity we find a Riemannian metric tensor underlying adiabatic evolution. Equipped with this tensor, we identify a unified geometric description of quantum adiabatic evolution and quantum phase transitions, which generalizes previous treatments to allow for degeneracy. The same structure is relevant for applications in quantum information processing, including adiabatic and holonomic quantum computing, where geodesics over the manifold of control parameters correspond to paths which minimize errors. We illustrate this geometric structure with examples, for which we explicitly find adiabatic geodesics. By solving the geodesic equations in the vicinity of a quantum critical point, we identify universal characteristics of optimal adiabatic passage through a quantum phase transition. In particular, we show that in the vicinity of a critical point describing a second order quantum phase transition, the geodesic exhibits power-law scaling with an exponent given by twice the inverse of the product of the spatial and scaling dimensions.

Ground-State Degeneracies Leave Recognizable Topological Scars in the Electronic Density

In Kohn-Sham density functional theory (KS DFT) a fictitious system of noninteracting particles is constructed having the same ground-state (GS) density as the physical system of interest. A fundamental open question in DFT concerns the ability of an exact KS calculation to spot and characterize the GS degeneracies in the physical system. In this Letter we provide theoretical evidence suggesting that the GS density, as a function of position on a 2D manifold of parameters affecting the external potential, is "topologically scarred" in a distinct way by degeneracies. These scars are sufficiently detailed to enable determination of the positions of degeneracies and even the associated Berry phases. We conclude that an exact KS calculation can spot and characterize the degeneracies of the physical system.

Geometric phases in adiabatic Floquet theory, Abelian gerbes and Cheon's anholonomy

We study the geometric phase phenomenon in the context of the adiabatic Floquet theory (so-called (t, t′) Floquet theory). A double integration appears in the geometric phase formula because of the presence of two time variables within the theory. We show that the geometric phases are then identified with horizontal lifts of surfaces in an Abelian gerbe with connection, rather than with horizontal lifts of curves in an Abelian principal bundle. This higher degree in the geometric phase gauge theory is related to the appearance of changes in the Floquet blocks at the transitions between two local charts of the parameter manifold. We present the physical example of a kicked two-level system where the changes are involved by a Cheon's anholonomy. In this context, the analogy between the usual geometric phase theory and the classical field theory also provides an analogy with the classical string theory.

Quantum Adiabatic Brachistochrone

We formulate a time-optimal approach to adiabatic quantum computation (AQC). A corresponding natural Riemannian metric is also derived, through which AQC can be understood as the problem of finding a geodesic on the manifold of control parameters. This geometrization of AQC is demonstrated through two examples, where we show that it leads to improved performance of AQC, and sheds light on the roles of entanglement and curvature of the control manifold in algorithmic performance.

Group Comparison of Eigenvalues and Eigenvectors of Diffusion Tensors

Diffusion tensor imaging (DTI) data differ from most medical images in that values at each voxel are not scalars, but 3×3 symmetric positive definite matrices called diffusion tensors (DTs). The anatomic characteristics of the tissue at each voxel are reflected by the DT eigenvalues and eigenvectors. In this article we consider the problem of testing whether the means of two groups of DT images are equal at each voxel in terms of the DT’s eigenvalues, eigenvectors, or both. Because eigendecompositions are highly nonlinear, existing likelihood ratio statistics (LRTs) for testing differences in eigenvalues or eigenvectors of means of Gaussian symmetric matrices assume an orthogonally invariant covariance structure between the matrix entries. While retaining the form of the LRTs, we derive new approximations to their true distributions when the covariance between the DT entries is arbitrary and possibly different between the two groups. The approximate distributions are those of similar LRT statistics computed on the tangent space to the parameter manifold at the true value of the parameter, but plugging in an estimate for the point of application of the tangent space. The resulting distributions, which are weighted sums of chi-squared distributions, are further approximated by scaled chi-squared distributions by matching the first two moments. For validity of the Gaussian model, the positive definite constraints on the DT are removed via a matrix log transformation, although this is not crucial asymptotically. Voxelwise application of the test statistics leads to a multiple-testing problem, which is solved by false discovery rate inference. The foregoing methods are illustrated in a DTI group comparison of boys versus girls.

FAMILIES OF HOLOMORPHIC BUNDLES

The first goal of the article is to solve several fundamental problems in the theory of holomorphic bundles over non-algebraic manifolds. For instance, we prove that stability and semi-stability are Zariski open properties in families when the Gauduchon degree map is a topological invariant, or when the parameter manifold is compact. Second, we show that, for a generically stable family of bundles over a Kähler manifold, the Petersson–Weil form extends as a closed positive current on the whole parameter space of the family. This extension theorem uses classical tools from Yang–Mills theory (e.g., the Donaldson functional on the space of Hermitian metrics and its properties). We apply these results to study families of bundles over a Kählerian manifold Y parametrized by a non-Kählerian surface X, proving that such families must satisfy very restrictive conditions. These results play an important role in our program to prove existence of curves on class VII surfaces [22–24].

Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations

In this paper we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations. The essential ingredients are (primal-dual) Galerkin projection onto a low-dimensional space associated with a smooth “parametric manifold”—dimension reduction; efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations—rapid convergence; a posteriori error estimation procedures—rigorous and sharp bounds for the linear-functional outputs of interest; and Offline-Online computational decomposition strategies—minimum marginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control) and many-query (e.g., design optimization, multi-model/scale) contexts. We present illustrative results for heat conduction and convection-diffusion, inviscid flow, and linear elasticity; outputs include transport rates, added mass, and stress intensity factors.

A non-conventional neutron polariser concept

Even in an idealised case conventional methods of neutron polarisation inevitably loose at least 50% of the intensity by removing one spin state from the incident beam. We demonstrate, however, that it should be possible to polarise a pulsed beam of slow neutrons without loosing any particle. This so-called “Dynamical Neutron Polarisation” (DNP) method is based upon acceleration and deceleration, respectively, of the two neutron spin states during passage through an NMR-like arrangement of crossed static and radio-frequency magnetic fields, followed by spin-dependent spin rotation in a homogeneous precession field. Precise ramping of the strength of this precession field synchronously with a time modulated π / 2 -spin turn device at the exit position allows to stop precession for all neutron wavelengths exactly at the moment where both spin states are aligned parallel, albeit on the cost of a tiny energy difference between them. We present a theoretical description of DNP which is essentially based upon a semi-classical spin-rotation formalism and present a typical result of a thorough investigation of the performance of such a non-conventional polariser with respect to a manifold of instrumental parameters.

Environmental noise reduction for holonomic quantum gates

We study the performance of holonomic quantum gates, driven by lasers, under the effect of a dissipative environment modeled as a thermal bath of oscillators. We show how to enhance the performance of the gates by a suitable choice of the loop in the manifold of the controllable parameters of the laser. For a simplified, albeit realistic model, we find the surprising result that for a long time evolution the performance of the gate (properly estimated in terms of average fidelity) increases. On the basis of this result, we compare holonomic gates with the so-called stimulated raman adiabatic passage (STIRAP) gates.

A Short- Time Beltrami Kernel for Smoothing Images and Manifolds

We introduce a short-time kernel for the Beltrami image enhancing flow. The flow is implemented by "convolving" the image with a space dependent kernel in a similar fashion to the solution of the heat equation by a convolution with a Gaussian kernel. The kernel is appropriate for smoothing regular (flat) 2-D images, for smoothing images painted on manifolds, and for simultaneously smoothing images and the manifolds they are painted on. The kernel combines the geometry of the image and that of the manifold into one metric tensor, thus enabling a natural unified approach for the manipulation of both. Additionally, the derivation of the kernel gives a better geometrical understanding of the Beltrami flow and shows that the bilateral filter is a Euclidean approximation of it. On a practical level, the use of the kernel allows arbitrarily large time steps as opposed to the existing explicit numerical schemes for the Beltrami flow. In addition, the kernel works with equal ease on regular 2-D images and on images painted on parametric or triangulated manifolds. We demonstrate the denoising properties of the kernel by applying it to various types of images and manifolds.

Learning independent components on the orthogonal group of matrices by retractions

Neural independent component learning algorithms based on optimization on manifolds have attracted interest in the neural network community. In the past years, we have developed learning algorithms specialized for the orthogonal group of matrices as parameters manifold. Here, we sketch a view of these algorithms by the help of `retractions' on manifolds.

Weighted distance maps computation on parametric three-dimensional manifolds

We propose an efficient computational solver for eikonal equations on parametric three-dimensional manifolds. Our approach is based on the fast marching method for solving the eikonal equation in O ( n log n ) steps on n grid points by numerically simulating wavefront propagation. The obtuse angle splitting problem is reformulated as a set of small integer linear programs, that can be solved in O ( n ) . Numerical simulations demonstrate the accuracy of the proposed algorithm.

Continuous Medial Representation for Anatomical Structures

The m-rep approach pioneered by Pizer et al. (2003) is a powerful morphological tool that makes it possible to employ features derived from medial loci (skeletons) in shape analysis. This paper extends the medial representation paradigm into the continuous realm, modeling skeletons and boundaries of three-dimensional objects as continuous parametric manifolds, while also maintaining the proper geometric relationship between these manifolds. The parametric representation of the boundary-medial relationship makes it possible to fit shape-based coordinate systems to the interiors of objects, providing a framework for combined statistical analysis of shape and appearance. Our approach leverages the idea of inverse skeletonization, where the skeleton of an object is defined first and the object's boundary is derived analytically from the skeleton. This paper derives a set of sufficient conditions ensuring that inverse skeletonization is well-posed for single-manifold skeletons and formulates a partial differential equation whose solutions satisfy the sufficient conditions. An efficient variational algorithm for deformable template modeling using the continuous medial representation is described and used to fit a template to the hippocampus in 87 subjects from a schizophrenia study with sub-voxel accuracy and 95% mean overlap.

Geometric curve flows on parametric manifolds

Planar geometric curve evolution equations are the basis for many image processing and computer vision algorithms. In order to extend the use of these algorithms to images painted on manifolds it is necessary to devise numerical schemes for the implementation of the geodesic generalizations of these equations. We present efficient numerical schemes for the implementation of the classical geodesic curve evolution equations on parametric manifolds. The efficiency of the schemes is due to their implementation on the parameterization plane rather than on the manifold itself. We demonstrate these flows on various manifolds and use them to implement two applications: scale space of images painted on manifolds and segmentation by an active contour model.