A Bounded Random Matrix Approach for Stochastic Upscaling

In this paper, a novel random matrix approach is utilized to characterize the uncertainty propagation in the articulated robotic systems. This method permits the probabilistic formulation to take into account the complexity and structural inter-dependencies of the kinematic manipulators. However, conventional random variable/vector based approaches are unable to address this issue. The kinematic Jacobian matrix of the manipulator is probabilistically formulated so that it adopts a random symmetric positive definite perturbation matrix. The density function of the perturbation matrix is constructed using maximum entropy principle that yields a Wishart distribution with specific parameters. The only information used in the construction of the Jacobian matrix probability model are mean and a scalar value dispersion parameter. We propose to use this framework to study the propagation of the uncertainties in both serial and parallel manipulators. Parallel architecture manipulators offer superior manipulation accuracy and structural rigidity compared to serial chain manipulators but feature more complex physical layout and limited workspace. Therefore, for the design purposes, a trade off between two different architectures is required. This necessitates the mathematical characterization of their capability with respect to the uncertainty propagation. As a case study, the developed random matrix method is implemented on a planar RRR serial and a planar 3-RRR parallel manipulator, that are performing an identical task. A Monte Carlo analysis is conducted and the statistics of the response of both systems provides appropriate measures to compare their dexterity dealing with the uncertainties. The final results verify superiority of the parallel architecture.

The symmetric positive definite (SPD) matrices have been widely used in image and vision problems. Recently there are growing interests in studying sparse representation (SR) of SPD matrices, motivated by the great success of SR for vector data. Though the space of SPD matrices is well-known to form a Lie group that is a Riemannian manifold, existing work fails to take full advantage of its geometric structure. This paper attempts to tackle this problem by proposing a kernel based method for SR and dictionary learning (DL) of SPD matrices. We disclose that the space of SPD matrices, with the operations of logarithmic multiplication and scalar logarithmic multiplication defined in the Log-Euclidean framework, is a complete inner product space. We can thus develop a broad family of kernels that satisfies Mercer's condition. These kernels characterize the geodesic distance and can be computed efficiently. We also consider the geometric structure in the DL process by updating atom matrices in the Riemannian space instead of in the Euclidean space. The proposed method is evaluated with various vision problems and shows notable performance gains over state-of-the-arts.

The structure of this survey is the following: at first we repeat in Part I the first 20 years old proof of the strong Elliptic law for random matrices Then in Part II we prove the strong Elliptic law for random matrices Ξ n of the general form, i.e. when their diagonal entries have nonzero expectations, and when we do not require the existence of the probability densities of the entries of random matrices, but instead of Lyapunov condition we require the boundedness of the absolute moments of the entries of the order 4 + δ , δ &gt; 0. In this case the Elliptic law means that the support of the accompanying spectral density of eigenvalues looks like the picture of several galaxies made by telescope. If the distances between the centers of these galaxies are large enough we have several almost elliptical galaxies. We also present the sand clock law density for the sum of diagonal complex matrix and the symmetric random matrix belonging the domain of attraction of the Semicircular law. After this we prove in the Part III the weak Elliptic law, when all entries can have nonzero expectations. At the end of survey we give the proof of the Elliptic law for the Unitary random matrices from the so called Class N 12 of random Unitary matrices and give the over turned stool law density for the sum of diagonal complex matrix A n and where Ξ n belongs to the domain of attraction of the Circular law. We also establish in Part III for Elliptic Law at the first time the convergency rate and the Central Limit Theorem. These statements are based on the VICTORIA -transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Random Independent Arrays. We follow the main strategy of the theory of limit theorems of the probability theory, i.e. we try to solve the problem of description of all limits of normalized spectral functions where λ k ( A n Ξ n B n + C n ) are eigenvalues of the matrix A n Ξ n B n + C n , A n , B n , and C n are nonrandom matrices, under general (as only possible) conditions on the entries of random matrices Ξ n , χ is the indicator function. We emphasize that the spectral theory of Hermitian random matrices is rather profound. For example, in 1975 in [3] V. Girko proved the general stochastic canonical equation for ACE (Asymptotically Constant Entries)-symmetric matrices: Assume that for any n , the random entries of a symmetric matrix are independent and they are asymptotically constant entries (ACE), i.e., for any ε &gt; 0, and τ &gt; 0 is an arbitrary constant, and that, for every 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1, where the symbol denotes the weak convergence of distribution when n → ∞, is a nondecreasing function with bounded variation in z and continuous in u and v in the domain 0 ≤ u , v ≤ 1. Then, with probability one, for almost all x , where λ k (Ξ n × n ) are eigenvalues, F ( x ) is a distribution function whose Stieltjes transform satisfies the relation G α ( x , y , t ), as a function of x , is a distribution function satisfying the regularized stochastic canonical equation K 3 [3, 23] at the points x of continuity, is a random real functional whose Laplace transform of one-dimensional distribution is equal to The integrand <jats:inli

In the 70’s, Robert May demonstrated that complexity creates instability in generic models of ecological networks having random interaction matrices A. Similar random matrix models have since been applied in many disciplines. Central to assessing stability is the “circular law” since it describes the eigenvalue distribution for an important class of random matrices A. However, despite widespread adoption, the “circular law” does not apply for ecological systems in which density-dependence operates (i.e., where a species growth is determined by its density). Instead one needs to study the far more complicated eigenvalue distribution of the community matrix S = DA, where D is a diagonal matrix of population equilibrium values. Here we obtain this eigenvalue distribution. We show that if the random matrix A is locally stable, the community matrix S = DA will also be locally stable, providing the system is feasible (i.e., all species have positive equilibria D > 0). This helps explain why, unusually, nearly all feasible systems studied here are locally stable. Large complex systems may thus be even more fragile than May predicted, given the difficulty of assembling a feasible system. It was also found that the degree of stability, or resilience of a system, depended on the minimum equilibrium population.

With the displacement field taken as the only fundamental unknown field in a mixed-boundary-value problem for linear elastostatics, the principle of minimum potential energy asserts that a potential energy functional, which is defined as the difference between the free energy of the body and the work done by the prescribed surface tractions and the body forces --- assumes a smaller value for the actual solution of the mixed problem than for any other kinematically admissible displacement field which satisfies the displacement boundary condition. This principle provides a weak or variational method for solving mixed boundary-value-problems of elastostatics. In particular, instead of solving the governing Navier form of the partial differential equations of equilibrium, one can search for a displacement field such that the first variation of the potential energy functional vanishes. A similar principle of minimum complementary energy, which is phrased in terms of statically admissible stress fields which satisfy the equilibrium equation and the traction boundary condition, is also discussed. The principles of minimum potential energy and minimum complementary energy can also be applied to derive specialized principles which are particularly well-suited to solving structural problems; in this context the celebrated theorems of Castigliano are discussed.

Log-Cholesky filtering of diffusion tensor fields: Impact on noise reduction

Diffusion Tensor Imaging (DTI) is a noise-sensitive method, where a low Signal-to-Noise Ratio (SNR) results in significant errors in the estimated tensor field. Post-reconstruction tensor field smoothing is a simple and effective solution for alleviating this problem. Diffusion tensors can be represented by Symmetric Positive-Definite (SPD) matrices which can be viewed as a Riemannian manifold after defining a suitable metric on the space of SPD matrices. The Log-Cholesky metric is a recently developed concept with several advantages over previously defined metrics, e.g., Frobenius, Log-Euclidean, and affine-invariant metrics. In this work, we implemented a smoothing method based on the Log-Cholesky metric and show its effectiveness as a simple solution to filtering noisy diffusion tensor fields.

Fading is the time-dependent variations in signal strength measured at a receiver, due to temporally evolving multipath scattering and interference. In our previous work we introduced a statistical fading model for the time-reversal invariant case by combining the predictions of random matrix theory with the random coupling model that includes system-speci c properties such as the radiation impedance of the ports and short-orbit e ects. In the high-loss limit this random matrix theory model reduced to the most common fading models in the wireless communication eld. In this paper we discuss the theoretical model in more detail and extend it to the case of broken time-reversal invariance.

Minimum zone evaluation of conicity error using minimum potential energy algorithms

A distressed material can be considered as a two-phase composite material with the distressed areas (volumes) as one phase and the original material as the other phase. A parallel diffusion model was developed in which the damaged phase and the original phase are arranged parallel to the direction of diffusion. The parallel model is associated with the isogradient principle in that the concentration gradients in the damaged phase and in the original phase are the same. A serial diffusion model was developed in which the two phases are arranged perpendicular to the direction of diffusion. The serial model is associated with the isoflux principle in that the fluxes in the two phases are the same. The two-phase composite models were further extended into general multiphase composite models to evaluate the effect of multilevel damage on the effective diffusivity of distressed materials. A principle of minimum complementary “chemical-flux energy” and a principle of minimum “potential chemical energy” were established similar to the commonly used principles of minimum complementary energy and minimum potential energy. Based on the two new minimum energy principles, it was shown that the parallel model is the lower bound and the serial model is the upper bound for effective diffusivity of distressed materials. Diffusivity of distressed concrete was used as an example. The experimental data agreed quite well with predictions of the theoretical models.

Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle of balanced metrics that relate the affine-invariant metric with the Euclidean and inverse-Euclidean metric, or the Bogoliubov-Kubo-Mori metric with the Euclidean and log-Euclidean metrics. We introduce two new families of balanced metrics, the mixed-power-Euclidean and the mixed-power-affine metrics and we discuss the relation between this new principle of balanced metrics and the concept of dual connections in information geometry.

The geometry of mixed-Euclidean metrics on symmetric positive definite matrices

We present a distributed memory algorithm for the approximate hierarchical of symmetric positive definite (SPD) matrices. Our method is based on the distributed memory GOFMM, an algorithm that appeared in SC18 (doi:10.1109/SC.2018.00018). GOFMM constructs a hierarchical matrix approximation of an arbitrary SPD matrix that compresses the matrix by creating low-rank approximations of the off-diagonal blocks. GOFMM method has no guarantees of success for arbitrary SPD matrices. (This is similar to the SVD; not every matrix admits a good low-rank approximation.) But for many SPD matrices, GOFMM does enable compression that results in fast matrix-vector multiplication that can reach N logN time—as opposed to N2 required for a dense matrix. GOFMM supports shared and distributed memory parallelism. In this paper, we build an approximate based on the Hierarchically Semi-Separable (HSS) compression of the GOFMM. This requires O(N) work (given the compressed matrix) and O(N=p) + O(log p) time on p MPI processes (assuming a hypercube topology). The previous state-of-the-art required O(N logN) work. We present the algorithm, discuss its complexity, and present weak and strong scaling results for the factorization and phases of our algorithm. We also discuss the performance of the inexact ULV as a preconditioner for a few exemplary large dense linear systems. In our largest run, we were able to factorize a 67M-by-67M matrix in less than one second; and solve a system with 64 right-hand sides in less than one-tenth of a second. This run was on 6,144 Intel Skylake cores on the SKX partition of the Stampede2 system at the Texas Advanced Computing Center.

Chapter 8 - Basic Equations and Solution Procedure

We investigate the laws that rule the behavior of the largest Lyapunov exponent (LLE) in many particle systems with long-range interactions. We consider as a representative system the so-called Hamiltonian alpha-XY model where the adjustable parameter alpha controls the range of the interactions of N ferromagnetic spins in a lattice of dimension d. In previous work the dependence of the LLE with the system size N, for sufficiently high energies, was established through numerical simulations. In the thermodynamic limit, the LLE becomes constant for alpha>d whereas it decays as an inverse power law of N for alpha<d. A recent theoretical calculation based on a geometrization of the dynamics is consistent with these numerical results. Here we show that the scaling behavior can also be explained by a random matrix approach, in which the tangent mappings that define the Lyapunov exponents are modeled by random simplectic matrices drawn from a suitable ensemble.