Ruben D. Spies

Arbitrary divergence speed of the least-squares method in infinite-dimensional inverse ill-posed problems

A standard engineering procedure for approximating the solutions of an infinite-dimensional inverse problem of the form Ax = y, where A is a given compact linear operator on a Hilbert space X and y is the given data, is to find a sequence {XN} of finite-dimensional approximating subspaces of X whose union is dense in X and to construct the sequence {xN} of least-squares solutions of the problem in XN. In 1980, Seidman showed that if the problem is ill-posed, then, without any additional assumptions on the exact solution or on the sequence of approximating subspaces XN, it cannot be guaranteed that the sequence {xN} will converge to the exact solution. In this paper, this result is extended in the following sense: it is shown that if X is separable, then for any y X, y ≠ 0 and for any arbitrarily given function there exists an injective, compact linear operator A and an increasing sequence of finite-dimensional subspaces XN ⊂ X such that for all , where xN is the least-squares solution of Ax = y in XN.

Mixed spatially varying L2-BV regularization of inverse ill-posed problems

Abstract Several generalizations of the traditional Tikhonov–Phillips regularization method have been proposed during the last two decades. Many of these generalizations are based upon inducing stability throughout the use of different penalizers which allow the capturing of diverse properties of the exact solution (e.g. edges, discontinuities, borders, etc.). However, in some problems in which it is known that the regularity of the exact solution is heterogeneous and/or anisotropic, it is reasonable to think that a much better option could be the simultaneous use of two or more penalizers of different nature. Such is the case, for instance, in some image restoration problems in which preservation of edges, borders or discontinuities is an important matter. In this work we present some results on the simultaneous use of penalizers of L2 and of bounded variation (BV) type. For particular cases, existence and uniqueness results are proved. Open problems are discussed and results to signal restoration problems are presented.

Polynomial stability of a joint-leg-beam system with local damping

Recent advances in the design and construction of large inflatable/rigidizable space structures and potential new applications of such structures have produced a demand for better analysis and computational tools to deal with the new class of structures. Understanding stability and damping properties of truss systems composed of these materials is central to the successful operation of future systems. In this paper, we consider a mathematical model for an assembly of two elastic beams connected to a joint through legs. The dynamic joint model is composed of two rigid bodies (the joint-legs) with an internal moment. In an ideal design all struts and joints will have identical material and geometric properties. In this case we previously established exponential stability of the beam-joint system. However, in order to apply theoretical stability estimates to realistic systems one must deal with the case where the individual truss components are not identical and still be able to analyze damping. We consider a problem of this type where one beam is assumed to have a small Kelvin-Voigt damping parameter and the second beam has no damping. In this case, we prove that the component system is only polynomially damped even if additional rotational damping is assumed in the joint.

Modelling for control of shape memory alloys

The problem of modeling a one-dimensional SMA (shape memory alloy) body is considered. Based on experimental results, new stress-strain relationships are introduced that are able to capture the dynamics of these materials.<<ETX>>

Local Discriminant Wavelet Packet Basis for Signal Classification in Brain Computer Interface

A Brain-Computer Interface (BCI) is a system that provides direct communication between the brain of a person and the outside world. In the present work we use a BCI based on Event Related Potentials (ERP). The aim of this paper is to efficiently solve the classification problem consisting on labeling electroencephalogram records as target (with ERP) or non-target records (without ERP).We evaluate the performance of a BCI by using the Wavelet Packet Transform with the Local Discriminant Basis (LDB) method to find an orthogonal basis that maximizes the difference between the two classes involved. The performance of the LDB patterns and the temporal data (without post-processing) are analyzed with the Fisher Linear Classifier. It is shown that the bets results are obtained with LDB patterns calculated by Daubechies 4 as filter, Sum of Squares as discriminant function and the first 18 more discriminant basis vectors.

Regularization methods for ill-posed problems in multiple Hilbert scales

Several convergence results in Hilbert scales under different source conditions are proved and orders of convergence and optimal orders of convergence are derived. Also, relations between those source conditions are proved. The concept of a multiple Hilbert scale on a product space is introduced, and regularization methods on these scales are defined, both for the case of a single observation and for the case of multiple observations. In the latter case, it is shown how vector-valued regularization functions in these multiple Hilbert scales can be used. In all cases, convergence is proved and orders and optimal orders of convergence are shown. Finally, some potential applications and open problems are discussed.

Well-posedness and exponential stability of a thermoelastic Joint-Leg-Beam system with Robin boundary conditions

An important class of proposed large space structures features a triangular truss backbone. In this paper we study thermomechanical behavior of a truss component; namely, a triangular frame consisting of two thin-walled circular beams connected through a joint. Transverse and axial mechanical motions of the beams are coupled though a mechanical joint. The nature of the external solar load suggests a decomposition of the temperature fields in the beams leading to two heat equations for each beam. One of these fields models the circumferential average temperature and is coupled to axial motions of the beam, while the second field accounts for a temperature gradient across the beam and is coupled to beam bending. The resulting system of partial and ordinary differential equations formally describes the coupled thermomechanical behavior of the joint-beam system. The main work is in developing an appropriate state-space form and then using semigroup theory to establish well-posedness and exponential stability.

A quasilinearization approach for parameter identification in a nonlinear model of shape memory alloys

The nonlinear partial differential equations considered here arise from the conservation laws of linear momentum and energy, and describe structural phase transitions (martensitic transformations) in one-dimensional shape memory alloys (SMA) with non-convex Landau-Ginzburg free energy potentials. This system is formally written as a nonlinear abstract Cauchy problem in an appropriate Hilbert space. A quasilinearization-based algorithm for parameter identification in this type of Cauchy problem is proposed. Sufficient conditions for the convergence of the algorithm are derived in terms of the regularity of the solutions with respect to the parameters. Numerical examples are presented in which the algorithm is applied to recover the non-physical parameters describing the free energy potential in SMA, from both exact and noisy data.

Generalized sparse discriminant analysis for event-related potential classification

Abstract A brain–computer interface (BCI) is a system which provides direct communication between the mind of a person and the outside world by using only brain activity (EEG). The event-related potential (ERP)-based BCI problem consists of a binary pattern recognition. Linear discriminant analysis (LDA) is widely used to solve this type of classification problems, but it fails when the number of features is large relative to the number of observations. In this work we propose a penalized version of the sparse discriminant analysis (SDA), called generalized sparse discriminant analysis (GSDA), for binary classification. This method inherits both the discriminative feature selection and classification properties of SDA and it also improves SDA performance through the addition of Kullback–Leibler class discrepancy information. The GSDA method is designed to automatically select the optimal regularization parameters. Numerical experiments with two real ERP-EEG datasets show that, on one hand, GSDA outperforms standard SDA in the sense of classification performance, sparsity and required computing time, and, on the other hand, it also yields better overall performances, compared to well-known ERP classification algorithms, for single-trial ERP classification when insufficient training samples are available. Hence, GSDA constitute a potential useful method for reducing the calibration times in ERP-based BCI systems.

Image restoration with a half-quadratic approach to mixed weighted smooth and anisotropic bounded variation regularization

The problem of restoring a signal or image is often tantamount to approximating the solution of a linear inverse ill-posed problem. In order to find such an approximation one might regularize the problem by penalizing variations on the estimated solution. Among the wide variety of methods available to perform this penalization, the most commonly used is the Tikhonov-Phillips regularization, which is appropriate when the sought signal or image is expected to be smooth, but it results unsuitable whenever preservation of discontinuities and edges is an important matter. Nonetheless, there are other methods with edge preserving properties, such as bounded variation (BV ) regularization. However, these methods tend to produce piecewise constant solutions showing the so called “staircasing effect” and their numerical implementations entail great computational effort and cost. In order to overcome these obstacles, we consider a mixed weighted Tikhonov and anisotropic BV regularization method to obtain improved restorations and we use a half-quadratic approach to construct highly efficient numerical algorithms. Several numerical results in signal and image restoration problems are presented.