Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces

Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. In this paper, we consider phaseless sampling and reconstruction of real-valued signals in a high-dimensional shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. The determination of a signal in a shift-invariant space, up to a sign, by its magnitude measurements on the whole Euclidean space has been shown in the literature to be equivalent to its nonseparability. In this paper, we introduce an undirected graph associated with the signal in a shift-invariant space and use connectivity of the graph to characterize nonseparability of the signal. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that nonseparable signals in the shift-invariant space can be reconstructed in a stable way from their phaseless samples taken on that set. In this paper, we also propose a reconstruction algorithm which provides an approximation to the original signal when its noisy phaseless samples are available only. Finally, numerical simulations are performed to demonstrate the robustness of the proposed algorithm to reconstruct box spline signals from their noisy phaseless samples.

     

Numerical solution of a PDE model for a ratchet-cap pricing with BGM interest rate dynamics

In this paper we present a new numerical method to price an interest rate derivative. The financial product consists of a particular ratchet cap contract which contains a set of ratchet caplets. For this purpose, we first pose the PDE pricing model for each ratchet caplet by means of Feynman-Kac theorem. The underlying interest rates are the forward LIBOR rates, the dynamics of which are assumed to follow the recently introduced BGM (LMM) market model. For the set of PDEs associated to the ratchet caplets pricing problems, we propose a second order Crank-Nicolson characteristics time discretization scheme combined with a finite element discretization in the interest rate variables. In order to illustrate the performance of the numerical methods, we present an academic test and a real example of a particular ratchet cap pricing. In the second case, a comparison between the results obtained by Monte Carlo simulation and the proposed method is presented.     

Noniterative Reconstruction Method for an Inverse Potential Problem Modeled by a Modified Helmholtz Equation

This paper deals with an inverse potential problem posed in two dimensional space whose forward problem is governed by a modified Helmholtz equation. The inverse problem consists in the reconstruction of a set of anomalies embedded into a geometrical domain from partial measurements of the associated potential. Since the inverse problem, we are dealing with, is written in the form of an ill-posed boundary value problem, the idea is to rewrite it as a topology optimization problem. In particular, a shape functional is defined to measure the misfit of the solution obtained from the model and the data taken from the partial measurements. This shape functional is minimized with respect to a set of ball-shaped anomalies using the concept of topological derivatives. It means that the shape functional is expanded asymptotically and then truncated up to the desired order term. The resulting expression is trivially minimized with respect to the parameters under consideration which leads to a noniterative second-order reconstruction algorithm. As a result, the reconstruction process becomes very robust with respect to noisy data and independent of any initial guess. Finally, some numerical experiments are presented to show the effectiveness of the proposed reconstruction algorithm.     

The Krein Formula for Generalized Resolvents in Degenerated Inner Product Spaces

 Let S be a symmetric operator with defect index (1,1) in a Pontryagin space ℋ. The Krein formula establishes a bijective correspondence between the generalized resolvents of S and the set of Nevanlinna functions as parameters. We give an analogue of the Krein formula in the case that ℋ is a degenerated inner product space. The set of parameters is determined by a kernel condition. These results are applied to some classical interpolation problems with singular data. Received 3 February 1997; in revised form 9 June 1997     

Space/time convergence analysis of a class of conservative schemes for linear wave equations

This paper concerns the space/time convergence analysis of conservative two-step time discretizations for linear wave equations. Explicit and implicit, second- and fourth-order schemes are considered, while the space discretization is given and satisfies minimal hypotheses. Convergence analysis is done using energy techniques and holds if the time step is upper-bounded by a quantity depending on space discretization parameters. In addition to showing the convergence for recently introduced fourth-order schemes, the novelty of this work consists in the independency of the convergence estimates with respect to the difference between the time step and its greatest admissible value.     

Cone Fields and Topological Sampling in Manifolds with Bounded Curvature

A standard reconstruction problem is how to discover a compact set from a noisy point cloud that approximates it. A finite point cloud is a compact set. This paper proves a reconstruction theorem which gives a sufficient condition, as a bound on the Hausdorff distance between two compact sets, for when certain offsets of these two sets are homotopic in terms of the absence of μ-critical points in an annular region. We reduce the problem of reconstructing a subset from a point cloud to the existence of a deformation retraction from the offset of the subset to the subset itself. The ambient space can be any Riemannian manifold but we focus on ambient manifolds which have nowhere negative curvature (this includes Euclidean space). We get an improvement on previous bounds for the case where the ambient space is Euclidean whenever μ≤0.945 (μ∈(0,1) by definition). In the process, we prove stability theorems for μ-critical points when the ambient space is a manifold.     

The Cauchy–Schwarz inequality and the induced metrics on real vector spaces mainly on the real line

It is very well known that the Cauchy–Schwarz inequality is an important property shared by all inner product spaces and the inner product induces a norm on the space. A proof of the Cauchy–Schwarz inequality for real inner product spaces exists, which does not employ the homogeneous property of the inner product. However, it is shown that a real vector space with a product satisfying properties of an inner product except the homogeneous property induces a metric but not a norm. It is remarkable to see that the metric induced on the real line by such a product has highly contrasting properties relative to the absolute value metric. In particular, such a product on the real line is given so that the induced metric is not complete and the set of rational numbers is not dense in the real line.     

Inversion of a generalized Radon transform by algebraic iterative methods

Inversion of a generalized Radon transform (GRT) of seismic type is investigated by employing the algebraic iterative methods ART and SIRT, which require discrete formulations of the reconstruction problem. To this aim, two discretization procedures are proposed: a direct discretization and a discretization via the relation of GRT with the regular Radon transform (RT). The feasibility and the semi-convergence behavior of the proposed methods are analyzed and compared by discussing the effect of noisy data.     

A noniterative reconstruction method for the inverse potential problem with partial boundary measurements

In this paper, a noniterative reconstruction method for solving the inverse potential problem is proposed. The forward problem is governed by a modified Helmholtz equation. The inverse problem consists in the reconstruction of a set of anomalies embedded into a geometrical domain from partial or total boundary measurements of the associated potential. Since the inverse problem is written in the form of an ill‐posed boundary value problem, the idea is to rewrite it as a topology optimization problem. In particular, a shape functional measuring the misfit between the solution obtained from the model and the data taken from the boundary measurements is minimized with respect to a set of ball‐shaped anomalies by using the concept of topological derivatives. It means that the shape functional is expanded asymptotically and then truncated up to the desired order term. The resulting truncated expansion is trivially minimized with respect to the parameters under consideration that leads to a noniterative second order reconstruction algorithm. As a result, the reconstruction process becomes very robust with respect to the noisy data and independent of any initial guess. Finally, some numerical experiments are presented showing the capability of the proposed method in reconstructing multiple anomalies of different sizes and shapes by taking into account complete or partial boundary measurements.     

Sufficiency theorems for target capture

A dynamical system is assumed to be governed by a set of ordinary differential equations subject to control. The set of points in state space from which there exist permissible controls that can transfer these points to a prescribed target set in a finite time interval is called a capture set. The task of determining the capture set is studied in two contexts. first, in the case of the system subject to a single control vector; and second, in the case of the system subject to two control vectors each operated independently. In the latter case, it is assumed that one controller's aim is to cause the system to attain the target, and the other's is to prevent that from occurring.Sufficient conditions are developed that, when satisfied everywhere on the interior of some subset of the state space, ensure that this subset is truly a capture set. A candidate capture set is assumed to have already been predetermined by independent methods. The sufficient conditions developed herein require the use of an auxiliary scalar function of the state, similar to a Lyapunov function.To ensure capture, five conditions must be satisfied. Four of these constrain the auxiliary state function. Basically, these four conditions require that the boundary of the controllable set be an envelope of the auxiliary state function and that that function be positive inside the capture set, approaching zero value as the target set is approached. The final condition tests the inner product of the gradient of the auxiliary state function with the system state velocity vector. If the sign of that inner product can be made negative everywhere within the test subset, then that subset is a capture set.Dedicated to Professor A. BusemannThe authors are indebted to Professors G. Leitmann and J. M. Skowronskii for their useful comments and discussion.     

Time discretization via Laplace transformation of an integro-differential equation of parabolic type

We consider the discretization in time of an inhomogeneous parabolic integro-differential equation, with a memory term of convolution type, in a Banach space setting. The method is based on representing the solution as an integral along a smooth curve in the complex plane which is evaluated to high accuracy by quadrature, using the approach in recent work of López-Fernández and Palencia. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. The method is combined with finite element discretization in the spatial variables to yield a fully discrete method. The paper is a further development of earlier work by the authors, which on the one hand treated purely parabolic equations and, on the other, an evolution equation with a positive type memory term. The authors acknowledge the support of the Australian Research Council.     

A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature

We consider the discretization in time of an inhomogeneous parabolicequation in a Banach space setting, using a representation ofthe solution as an integral along a smooth curve in the complexleft half-plane which, after transformation to a finite interval,is then evaluated to high accuracy by a quadrature rule. Thisreduces the problem to a finite set of elliptic equations withcomplex coefficients, which may be solved in parallel. The paperis a further development of earlier work by the authors, wherewe treated the homogeneous equation in a Hilbert space framework.Special attention is given here to the treatment of the forcingterm. The method is combined with finite-element discretizationin spatial variables.     

Reproducing kernel method of solving singular integral equation with cosecant kernel

In the paper, a reproducing kernel method of solving singular integral equations (SIE) with cosecant kernel is proposed. For solving SIE, difficulties lie in its singular term. In order to remove singular term of SIE, an equivalent transformation is made. Compared with known investigations, its advantages are that the representation of exact solution is obtained in a reproducing kernel Hilbert space and accuracy in numerical computation is higher. On the other hand, the representation of reproducing kernel becomes simple by improving the definition of traditional inner product and requirements for image space of operators are weakened comparing with traditional reproducing kernel method. The final numerical experiments illustrate the method is efficient.     

Some remarks on blvarlational bounds and the pointwise estimation of solutions

When a given equation can be realised in some Hilbert space H as an operator equation in the from A[d] then it is known that complementary variational bounds can be obtained for the inner product [d]. Recently it has been shown that complementary bivariational bounds can be obtained for the inner product [d]associated with the equation A[d]= f and an arbitary elementy [d]. These latter bounds are a natural starting point for investigatng pointwise estimatesd of solutions provided thta certain questions relating, on the one hand, to the self-adjointness and the invertibility of associated operators and on the other to the availability of a suitable large Hilbert space can be resolved. We show that if the underlying problem can be given an operator realisation in a suitably equipped Hilbert space then pointwise estimates of solution can be obtained     

POD reduced-order modeling for evolution equations utilizing arbitrary finite element discretizations

The main focus of the present work is the inclusion of spatial adaptivity for the snapshot computation in the offline phase of model order reduction utilizing proper orthogonal decomposition (POD-MOR) for nonlinear parabolic evolution problems. We consider snapshots which live in different finite element spaces, which means in a fully discrete setting that the snapshots are vectors of different length. From a numerical point of view, this leads to the problem that the usual POD procedure which utilizes a singular value decomposition of the snapshot matrix, cannot be carried out. In order to overcome this problem, we here construct the POD model/basis using the eigensystem of the correlation matrix (snapshot Gramian), which is motivated from a continuous perspective and is set up explicitly, e.g., without the necessity of interpolating snapshots into a common finite element space. It is an advantage of this approach that the assembly of the matrix only requires the evaluation of inner products of snapshots in a common Hilbert space. This allows a great flexibility concerning the spatial discretization of the snapshots. The analysis for the error between the resulting POD solution and the true solution reveals that the accuracy of the reduced-order solution can be estimated by the spatial and temporal discretization error as well as the POD error. Finally, to illustrate the feasibility of our approach, we present a test case of the Cahn–Hilliard system utilizing h-adapted hierarchical meshes and two settings of a linear heat equation using nested and non-nested grids.     

Robust AMLI methods for parabolic Crouzeix-Raviart FEM systems

For the iterative solution of linear systems of equations arising from finite element discretization of elliptic problems there exist well-established techniques to construct numerically efficient and computationally optimal preconditioners. Among those, most often preferred choices are Multigrid methods (geometric or algebraic), Algebraic MultiLevel Iteration (AMLI) methods, Domain Decomposition techniques.In this work, the method in focus is AMLI. We extend its construction and the underlying theory over to systems arising from discretizations of parabolic problems, using non-conforming finite element methods (FEM). The AMLI method is based on an approximated block two-by-two factorization of the original system matrix. A key ingredient for the efficiency of the AMLI preconditioners is the quality of the utilized block two-by-two splitting, quantified by the so-called Cauchy-Bunyakowski-Schwarz (CBS) constant, which measures the abstract angle between the two subspaces, associated with the two-by-two block splitting of the matrix.The particular choice of space discretization for the parabolic equations, used in this paper, is Crouzeix-Raviart non-conforming elements on triangular meshes. We describe a suitable splitting of the so-arising matrices and derive estimates for the associated CBS constant. The estimates are uniform with respect to discretization parameters in space and time as well as with respect to coefficient and mesh anisotropy, thus providing robustness of the method.     

Higher order variational integrators in optimal control theory

Higher order variational integrators are analyzed and applied to optimal control problems posed with mechanical systems. First, we derive two different kinds of high order variational integrators based on different dimensions of the underlying approximation space. While the first well-known integrator is equivalent to a symplectic partitioned Runge-Kutta method, the second integrator, denoted as symplectic Galerkin integrator, yields a method which in general, cannot be written as a standard symplectic Runge-Kutta scheme [1]. Furthermore, we use these integrators for the discretization of optimal control problems. By analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that for these particular integrators optimization and discretization commute [2]. This property guarantees that the accuracy is preserved for the adjoint system which is also referred to as the Covector Mapping Principle. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Operator-splitting methods via the Zassenhaus product formula

In this paper, we contribute an operator-splitting method improved by the Zassenhaus product. Zassenhaus products are of fundamental importance for the theory of Lie groups and Lie algebras. While their applications in physics and physical chemistry are important, novel applications in CFD (computational fluid dynamics) arose based on the fact that their sparse matrices can be seen as generators of an underlying Lie algebra. We apply this to classical splitting and the novel Zassenhaus product formula. The underlying analysis for obtaining higher order operator-splitting methods based on the Zassenhaus product is presented. The benefits of dealing with sparse matrices, given by spatial discretization of the underlying partial differential equations, are due to the fact that the higher order commutators are very quickly computable (their matrix structures thin out and become nilpotent). When applying these methods to convection-diffusion-reaction equations, the benefits of balancing time and spatial scales can be used to accelerate these methods and take into account these sparse matrix structures.The verification of the improved splitting methods is done with numerical examples. Finally, we conclude with higher order operator-splitting methods.     

Nonlinear model reduction based on the finite element method with interpolated coefficients: Semilinear parabolic equations

For nonlinear reduced‐order models (ROMs), especially for those with high‐order polynomial nonlinearities or nonpolynomial nonlinearities, the computational complexity still depends on the dimension of the original dynamical system. To overcome this issue, we develop an efficient finite element (FE) discretization algorithm for nonlinear ROMs. The proposed approach approximates the nonlinear function by its FE interpolant, which makes the inner product evaluations in generating the nonlinear terms computationally cheaper than that in the standard FE discretization. Due to the separation of spatial and temporal variables in the FE interpolation, the discrete empirical interpolation method (DEIM) can be directly applied on the nonlinear functions in the same manner as that in the finite difference setting. Therefore, the main computational hurdles for applying the DEIM in the FE context are conquered. We also establish a rigorous asymptotic error estimation, which shows that the proposed approach achieves the same accuracy as that of the standard FE method under certain smoothness assumptions of the nonlinear functions. Several numerical tests are presented to validate the proposed method and verify the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1713–1741, 2015     

Stability analysis for a fully discrete spectral scheme for Boussinesq systems

In this paper we perform a stability analysis of a fully discrete numerical method for the solution of a family of Boussinesq systems, consisting of a Fourier collocation spectral method for the spatial discretization and a explicit fourth order Runge–Kutta (RK4) scheme for time integration. Our goal is to determine the influence of the parameters, associated to this family of systems, on the efficiency and accuracy of the numerical method. This analysis allows us to identify which regions in the parameter space are most appropriate for obtaining an efficient and accurate numerical solution. We show several numerical examples in order to validate the accuracy, stability and applicability of our MATLAB implementation of the numerical method.