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Max-Balancing Hungarian Scalings
Get PDFA Hungarian scaling is a diagonal scaling of a matrix that is typically applied along with a permutation to a sparse symmetric or nonsymmetric indefinite linear system before
calling a direct or iterative solver.
A Hungarian scaled and reordered matrix has all its entries of modulus less than or equal to 1 and entries of modulus 1
on the diagonal. We use max-plus algebra to characterize the set of all Hungarian scalings for a given matrix and show that max-balancing a Hungarian scaled matrix yields the most ``diagonally dominant" Hungarian scaled matrix
possible with respect to some ordering. We also propose a new scaling, called centre of mass scaling, which can be seen as an approximate max-balancing Hungarian scaling and whose computation is embarrassingly parallel.
Numerical experiments illustrate the increased diagonal dominance produced by max-balancing and centre of mass scaling of Hungarian scaled matrices as well as the reduced
need for pivoting in Gaussian elimination with partial pivoting and the improved stability of LU factorizations without pivoting
When do the iterative reconstruction methods become worth the effort?
Get PDFA driving force for the development of new reconstruction algorithms is to achieve better quality images using less information (lower dose, fewer projections, in less time), but under what circumstances do iterative methods become worth the effort? In this paper we propose a framework that enables the performance of reconstruction algorithms to be mapped. Such a framework allows fair comparisons to be made, providing insights into experimental acquisition strategies and methods of quantifying the quality of reconstructions, and identifying the sweet spot for different algorithms
Analysis of optimal liquidation in limit order books for portfolios of correlated assets with stochastic volatility
Get PDFIn this paper we study optimal liquidation under two settings: the first being for a basket of correlated assets, the second being for a portfolio of a single asset but under a stochastic volatility model. Under both frameworks we use a combined approach of accurate numerical methods and asymptotic analysis to investigate and gain insight into the solution, with each approach informing and confirming the other. We are able to make a significant improvement in efficiency in both problems, reducing the resulting Hamiliton-Jacobi-Bellman (HJB) partial differential equations (PDEs) to classical non-linear PDEs, as well as reducing the number of variables and input parameters, the latter through non-dimensionalisation. We present numerical solutions to both problems, before further investigating the solution topology through the use of asymptotic analysis in various limits. In some cases we are able to find analytic approximations for both the value function and indeed the optimal liquidation strategies. Furthermore, the solutions we present are comparable with those of Markowitz Portfolio Theory (MPT) for the multiple-asset case, and to those of option pricing theory under stochastic volatility for the stochastic volatility model. For the former we find the trader trades in a way that results in a diversified portfolio, while for the latter we find that more noise in the volatility can be beneficial for the trader in certain regimes, despite being risk-averse
The effects of tomographic scans with fewer radiographs on the image reconstruction
Get PDFReconstructing a 2D slice or a 3D volume from a set of insufficient tomographic data is a difficult problem, and it is often tackled with analytical reconstruction algorithms. However, these types of methods fall short on delivering a quality image due to the severe artefacts introduced by the insufficiency of the data. The presented work shows the effects of taking tomographic scans with fewer radiographs on the quality of the reconstructed images. The aim here is to show the advantages of using iterative reconstruction methods over analytical methods, which are demonstrated by a quantitative comparison
A new strain energy function for modelling ligaments and tendons whose fascicles have a helical arrangement of fibrils
Get PDFA new strain energy function for the hyperelastic modelling of ligaments and tendons whose fascicles have a helical arrangement of fibrils is derived. The stress-strain response of a single fascicle whose fibrils exhibit varying levels of crimp throughout its radius is calculated and used to determine the form of the strain energy function. The new constitutive law is used to model uniaxial extension test data for human patellar tendon and is shown to provide an excellent fit, with the average relative error being 9.8%. It is then used to model shear and predicts that the stresses required to shear a tendon are much smaller than those required to uniaxially stretch it to the same strain level. Finally, the strain energy function is used to model ligaments and tendons whose fascicles are helical, and the relative effects of the fibril helix angle, the fascicle helix angle and the fibril crimp variable are compared. It is shown that they all have a significant effect; the fibril crimp variable governs the non-linearity of the stress-strain curve, whereas the helix angles primarily affect its stiffness. Smaller values of the helix angles lead to stiffer tendons; therefore, the model predicts that one would expect to see fewer helical sub-structures in stiff positional tendons, and more in those that are required to be more flexible
Max-plus LU
No full textWe present a new method for the a priori approximation of the order of magnitude of the entries in the LU factors of a matrix A �¢���� Cn��â��n. We are also able to predict which permutation matrices will be chosen by partial pivoting or complete pivoting Gaussian elimination. Our method uses max- plus algebra and is based purely on the moduli of the entries in the matrix. This approximation can be used in the construction of ILU preconditioners, where the max-plus LU approximation can be used to quickly determine the positions of the largest entries in the LU factors. These positions can subsequently be used as the sparsity pattern for an ILU preconditioner
Estimating the Largest Elements of a Matrix
Get PDFWe derive an algorithm for estimating the largest p �¢â�°�¥ 1 values a ij or |a ij | for an
m ��â�� n matrix A, along with their locations in the matrix. The matrix is accessed using only matrix�¢â�¬â��
vector or matrix�¢â�¬â��matrix products. For p = 1 the algorithm estimates the norm A M := max i,j |a ij |
or max i,j a ij . The algorithm is based on a power method for mixed subordinate matrix norms and
iterates on n ��â�� t matrices, where t �¢â�°�¥ p is a parameter. For p = t = 1 we show that the algorithm
is essentially equivalent to rook pivoting in Gaussian elimination; we also obtain a bound for the
expected number of matrix�¢â�¬â��vector products for random matrices and give a class of counter-examples.
Our numerical experiments show that for p = 1 the algorithm usually converges in just two iterations,
requiring the equivalent of 4t matrix�¢â�¬â��vector products, and for t = 2 the algorithm already provides
excellent estimates that are usually within a factor 2 of the largest element and frequently exact.
For p > 1 we incorporate deflation to improve the performance of the algorithm. Experiments on
real-life datasets show that the algorithm is highly effective in practice
Linearizations of Matrix Polynomials in Bernstein Bases
Get PDFWe discuss matrix polynomials expressed
in a Bernstein basis,
and the associated polynomial eigenvalue problems.
Using Mobius transformations of matrix polynomials,
large new families of strong
linearizations are generated.
Matrix polynomials that are structured
with respect to a Bernstein basis,
together with their associated spectral symmetries,
are also investigated.
The results in this paper apply equally well
to scalar polynomials,
and include the development of new companion pencils
for polynomials expressed in a Bernstein basis
Energy norm a posteriori error estimation for parametric operator equations
Get PDFStochastic Galerkin approximation is an increasingly popular approach for the solution of elliptic PDE problems with correlated random data. A typical strategy is to combine conventional (-) finite element approximation on the spatial domain with spectral (-) approximation on a finite-dimensional manifold in the (stochastic) parameter domain. The issues involved in a posteriori error analysis of computed solutions are outlined in this paper. A novel energy error estimator that uses a parameter-free part of the underlying differential operator is introduced which effectively exploits the tensor product structure of the approximation space. We prove that our error estimator is reliable and efficient. We also discuss different strategies for enriching the approximation space and prove two-sided estimates of the error reduction for the corresponding enhanced approximations. These give computable estimates of the error reduction that depend only on the problem data and the original approximation