Consider an underdetermined system of linear equations y = Ax with known d*n matrix A and known y. We seek the sparsest nonnegative solution, i.e. the nonnegative x with fewest nonzeros satisfying y = Ax. In general this problem is NP-hard. However, for many matrices A there is a threshold phenomenon: if the sparsest solution is sufficiently sparse, it can be found by linear programming. In classical convex polytope theory, a polytope P is called k-neighborly if every set of k vertices of P span a face of P. Let aj denote the j-th column of A, 1<=_j<=_n, let a0 = 0 and let P denote the convex hull of the aj . We say P is outwardly k-neighborly if every subset of k vertices not including 0 spans a face of P. We show that outward k-neighborliness is completely equivalent to the statement that, whenever y = Ax has a nonnegative solution with at most k nonzeros, it is the nonnegative solution to y = Ax having minimal sum. Using this and classical results on polytope neighborliness we obtain two types of corollaries. First, because many [d/2]-neighborly polytopes are known, there are many systems where the sparsest solution is available by convex optimization rather than combinatorial optimization - provided the answer has fewer nonzeros than half the number of equations. We mention examples involving incompletely-observed Fourier transforms and Laplace transforms. Second, results on classical neighborliness of high-dimensional randomly-projected simplices imply that, if A is a typical uniformly-distributed random orthoprojector with n = 2d and n large, the sparsest nonnegative solution to y = Ax can be found by linear programming provided it has fewer nonzeros than 1/8 the number of equations. We also consider a notion of weak neighborliness, in which the overwhelming majority of k-sets of aj's not containing 0 span a face. This implies that most nonnegative vectors x with k nonzeros are uniquely determined by y = Ax. As a corollary of recent work counting faces of random simplices, it is known that most polytopes P generated by large n by 2n uniformly-distributed orthoprojectors A are weakly k-neighborly with k _~.558n. We infer that for most n by 2n underdetermined systems having a sparse solution with fewer nonzeros than roughly half the number of equations, the sparsest solution can be found by linear programming.

We propose a linear-time line segment detector that gives accurate results, a controlled number of false detections, and requires no parameter tuning. This algorithm is tested and compared to state-of-the-art algorithms on a wide set of natural images.

A common approach in model reduction is balanced truncation, which is based on gramian matrices classifiying certain attributes of states or parameters of a given dynamic system. Initially restricted to linear systems, the empirical gramians not only extended this concept to nonlinear systems, but also provide a uniform computational method. This work introduces a unified software framework supplying routines for six types of empirical gramians. The gramian types will be discussed and applied in a model reduction framework for multiple-input-multiple-output (MIMO) systems.

An accepted model reduction technique is balanced truncation, by which negligible states of a linear system of ODEs are determined by balancing the systems controllability and observability gramian matrices. To be applicable for nonlinear system this method was enhanced through the empirical gramians, while the cross gramian conjoined both gramians into one gramian matrix. This work introduces the empirical cross gramian for square Multiple-Input-Multiple-Output systems as well as the (empirical) joint gramian. Based on the cross gramian, the joint gramian determines, in addition to the Hankel singular values, the parameter identifiability allowing a combined model reduction, concurrently reducing state and parameter spaces. Furthermore, a controllability and an observability based combined reduction method are presented and the usage of empirical gramians is extended to parameter reduction in (Bayesian) inverse problems. All methods presented are evaluated by numerical experiments.

We consider the problem of finding optimally stable polynomial approximations to the exponential for application to one-step integration of initial value ordinary and partial differential equations. The objective is to find the largest stable step size and corresponding method for a given problem when the spectrum of the initial value problem is known. The problem is expressed in terms of a general least deviation feasibility problem. Its solution is obtained by a new fast, accurate, and robust algorithm based on convex optimization techniques. Global convergence of the algorithm is proven in the case that the order of approximation is one and in the case that the spectrum encloses a starlike region. Examples demonstrate the effectiveness of the proposed algorithm even when these conditions are not satisfied.

A common approach in model reduction is balanced truncation, which is based on gramian matrices classifiying certain attributes of states or parameters of a given dynamic system. Initially restricted to linear systems, the empirical gramians not only extended this concept to nonlinear systems, but also provide a uniform computational method. This work introduces a unified software framework supplying routines for six types of empirical gramians. The gramian types will be discussed and applied in a model reduction framework for multiple-input-multiple-output (MIMO) systems.

In compressed sensing, one takes n < N samples of an N -dimensional vector x0 using an n × N matrix A, obtaining un-dersampled measurements y = Ax0 . For random matrices with Gaussian i.i.d entries, it is known that, when x0 is k-sparse, there is a precisely determined phase transition: for a certain region in the (k/n, n/N )-phase diagram, convex optimization min ||x||_1 subject to y = Ax, x ∈ X^N typically finds the sparsest solution, while outside that region, it typically fails. It has been shown empirically that the same property – with the same phase transition location – holds for a wide range of non-Gaussian random matrix ensembles. We consider specific deterministic matrices including Spikes and Sines, Spikes and Noiselets, Paley Frames, Delsarte-Goethals Frames, Chirp Sensing Matrices, and Grassmannian Frames. Extensive experiments show that for a typical k-sparse object, convex optimization is successful over a region of the phase diagram that coincides with the region known for Gaussian matrices. In our experiments, we considered coefficients constrained to X^N for four different sets X ∈ {[0, 1], R_+ , R, C}. We establish this finding for each of the associated four phase transitions.