Let Φ(ω), ω ∈ Ω, be a family of n × N random matrices whose entries φ i,j are independent realizations of a symmetric, real random variable η with expectation IEη = 0 and variance IEη 2 = 1/n. Such matrices are used in compressed sensing to encode a vector x ∈ IR N by y = Φx. The information y holds about x is extracted by using a decoder ∆ : IR n → IR N.… (More)
The reduced basis method was introduced for the accurate online evaluation of solutions to a parameter dependent family of elliptic partial differential equations. Abstractly, it can be viewed as determining a " good " n dimensional space H n to be used in approximating the elements of a compact set F in a Hilbert space H. One, by now popular, computational… (More)
We study the approximation of a function class F in L p by choosing first a basis B and then using n-term approximation with the elements of B. Into the competition for best bases we enter all greedy (i.e. democratic and unconditional ) bases for L p. We show that if the function class F is well oriented with respect to a particular basis B then, in a… (More)
We discover that the choice of a piecewise polynomial reconstruction is crucial in computing solutions of nonconvex hyperbolic (systems of) conservation laws. Using semidiscrete central-upwind schemes, we illustrate that the obtained numerical approximations may fail to converge to the unique entropy solution or the convergence may be so slow that achieving… (More)
We derive a second-order semi-discrete central-upwind scheme for one-and two-dimensional systems of two-layer shallow water equations. We prove that the presented scheme is well-balanced in the sense that stationary steady-state solutions are exactly preserved by the scheme, and positivity preserving, that is, the depth of each fluid layer is guaranteed to… (More)
We give a bivariate analog of the Micchelli-Rivlin quadrature for computing the integral of a function over the unit disk using its Radon projections.
We introduce a new second-order central-upwind scheme for the Saint-Venant system of shallow water equations on triangular grids. We prove that the scheme both preserves " lake at rest " steady states and guarantees the positivity of the computed fluid depth. Moreover, it can be applied to models with discontinuous bottom topography and irregular channel… (More)