In this talk we suggest a new formula for the residual of Galerkin projection onto rational Krylov spaces applied to a Sylvester equation, and establish a relation to three different underlying extremal problems for rational functions. These extremal problems enable us to compare the size of the residual for the above method with that obtained by ADI. In addition, we may deduce several new a pr...

It is well known that the block Krylov subspace solvers work efficiently for some cases of the solution of differential equations with multiple right-hand sides. In lattice QCD calculation of physical quantities on a given configuration demands us to solve the Dirac equation with multiple sources. We show that a new block Krylov subspace algorithm recently proposed by the authors reduces the co...

Many problems in engineering and physics require the solution of a large sequence of linear systems. We can reduce the cost of solving subsequent systems in the sequence by recycling information from previous systems. We consider two di erent approaches. For several model problems, we demonstrate that we can reduce the iteration count required to solve a linear system by a factor of two. We con...

When solving the linear system Ax = b with a Krylov method, the smallest eigenvalues of the matrix A often slow down the convergence. This is usually still the case even after the system has been preconditioned. Consequently if the smallest eigenvalues of A could be somehow “removed” the convergence would be improved. Several techniques have been proposed in the past few years that attempt to t...

We develop a class of lters based upon the numerical solution of high-order elliptic problems in l R d that allow for independent determination of order and cut-oo wave number and that default to classical Fourier-based lters in homogeneous domains. However, because they are based on the solution of a PDE, the present lters are not restricted to applications in tensor-product-based geometries a...

Two recent methods, Neighborhood Components Analysis (NCA) and Informative Discriminant Analysis (IDA), search for a class-discriminative subspace or discriminative components of data, equivalent to learning of distance metrics invariant to changes perpendicular to the subspace. Constraining metrics to a subspace is useful for regularizing the metrics, and for dimensionality reduction. We intro...

Over the last two decades, the canonical variate analysis method for subspace system identification has been widely applied. A number of these applications have demonstrated near maximum likelihood accuracy of the adaptx CVA subspace algorithm in large samples with unknown feedback. The critical step in the algorithm is the use of an ARX model estimated by conditional maximum likelihood to remo...

Subspace clustering groups data into several lowrank subspaces. In this paper, we propose a theoretical framework to analyze a popular optimization-based algorithm, Sparse Subspace Clustering (SSC), when the data dimension is compressed via some random projection algorithms. We show SSC provably succeeds if the random projection is a subspace embedding, which includes random Gaussian projection...