Error analysis of FFT architectures for digital video applications

This paper describes how worst-case error analysis can be applied to solve some of the practical issues in the development and implementation of a low power, high performance radix-4 FFT chip for digital video applications. The chip has been fabricated using a 0.6 µm CMOS technology and can perform a 64 point complex forward or inverse FFT on real-time video at up to 18 Megasamples per second. It comprises 0.5 million transistors in a die area of 7.8×8 mm and dissipates 1 W, leading to a cost-effective silicon solution for high quality video processing applications. The analysis focuses on the effect that different radix-4 architectural configurations and finite wordlengths has on the FFT output dynamic range. These issues are addressed using both mathematical error models and through extensive simulation.

Error analysis of a Monte Carlo algorithm for computing bilinear forms of matrix powers

In this paper we present error analysis for a Monte Carlo algorithm for evaluating bilinear forms of matrix powers. An almost Optimal Monte Carlo (MAO) algorithm for solving this problem is formulated. Results for the structure of the probability error are presented and the construction of robust and interpolation Monte Carlo algorithms are discussed. Results are presented comparing the performance of the Monte Carlo algorithm with that of a corresponding deterministic algorithm. The two algorithms are tested on a well balanced matrix and then the effects of perturbing this matrix, by small and large amounts, is studied.

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived.

On aposteriori error analysis of dG schemes approximating hyperbolic conservation laws

This contribution is concerned with aposteriori error analysis of discontinuous Galerkin (dG) schemes approximating hyperbolic conservation laws. In the scalar case the aposteriori analysis is based on the L1 contraction property and the doubling of variables technique. In the system case the appropriate stability framework is in L2, based on relative entropies. It is only applicable if one of the solutions, which are compared to each other, is Lipschitz. For dG schemes approximating hyperbolic conservation laws neither the entropy solution nor the numerical solution need to be Lipschitz. We explain how this obstacle can be overcome using a reconstruction approach which leads to an aposteriori error estimate.

A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems

We present and analyse a space–time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space–time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al. (2014) and of Monk & Richter (2005). For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree. The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.

A navigation system for percutaneous needle interventions based on PET/CT images: design, workflow and error analysis of soft tissue and bone punctures

Percutaneous needle intervention based on PET/CT images is effective, but exposes the patient to unnecessary radiation due to the increased number of CT scans required. Computer assisted intervention can reduce the number of scans, but requires handling, matching and visualization of two different datasets. While one dataset is used for target definition according to metabolism, the other is used for instrument guidance according to anatomical structures. No navigation systems capable of handling such data and performing PET/CT image-based procedures while following clinically approved protocols for oncologic percutaneous interventions are available. The need for such systems is emphasized in scenarios where the target can be located in different types of tissue such as bone and soft tissue. These two tissues require different clinical protocols for puncturing and may therefore give rise to different problems during the navigated intervention. Studies comparing the performance of navigated needle interventions targeting lesions located in these two types of tissue are not often found in the literature. Hence, this paper presents an optical navigation system for percutaneous needle interventions based on PET/CT images. The system provides viewers for guiding the physician to the target with real-time visualization of PET/CT datasets, and is able to handle targets located in both bone and soft tissue. The navigation system and the required clinical workflow were designed taking into consideration clinical protocols and requirements, and the system is thus operable by a single person, even during transition to the sterile phase. Both the system and the workflow were evaluated in an initial set of experiments simulating 41 lesions (23 located in bone tissue and 18 in soft tissue) in swine cadavers. We also measured and decomposed the overall system error into distinct error sources, which allowed for the identification of particularities involved in the process as well as highlighting the differences between bone and soft tissue punctures. An overall average error of 4.23 mm and 3.07 mm for bone and soft tissue punctures, respectively, demonstrated the feasibility of using this system for such interventions. The proposed system workflow was shown to be effective in separating the preparation from the sterile phase, as well as in keeping the system manageable by a single operator. Among the distinct sources of error, the user error based on the system accuracy (defined as the distance from the planned target to the actual needle tip) appeared to be the most significant. Bone punctures showed higher user error, whereas soft tissue punctures showed higher tissue deformation error.

Methods for automatic error analysis of numerical algorithms /

Vita.

An error analysis of solutions to sparse linear programming problems,

"Supported in part by the Advanced Research Projects Agency ... contract no. US AF 30(602) 4144."

Intelligent execution monitoring and error analysis in planning involving processes

An intelligent agent, operating in an external world which cannot be fully described in its internal world model, must be able to monitor the success of a previously generated plan and to respond to any errors which may have occurred. The process of error analysis requires the ability to reason in an expert fashion about time and about processes occurring in the world. Reasoning about time is needed to deal with causality. Reasoning about processes is needed since the direct effects of a plan action can be completely specified when the plan is generated, but the indirect effects cannot. For example, the action `open tap' leads with certainty to `tap open', whereas whether there will be a fluid flow and how long it might last is more difficult to predict. The majority of existing planning systems cannot handle these kinds of reasoning, thus limiting their usefulness. This thesis argues that both kinds of reasoning require a complex internal representation of the world. The use of Qualitative Process Theory and an interval-based representation of time are proposed as a representation scheme for such a world model. The planning system which was constructed has been tested on a set of realistic planning scenarios. It is shown that even simple planning problems, such as making a cup of coffee, require extensive reasoning if they are to be carried out successfully. The final Chapter concludes that the planning system described does allow the correct solution of planning problems involving complex side effects, which planners up to now have been unable to solve.

A Posteriori Error Analysis of hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic Problems

We develop the a-posteriori error analysis of hp-version interior-penalty discontinuous Galerkin finite element methods for a class of second-order quasilinear elliptic partial differential equations. Computable upper and lower bounds on the error are derived in terms of a natural (mesh-dependent) energy norm. The bounds are explicit in the local mesh size and the local degree of the approximating polynomial. The performance of the proposed estimators within an automatic hp-adaptive refinement procedure is studied through numerical experiments.

Discontinuous Galerkin Methods on hp-Anisotropic Meshes I: A Priori Error Analysis

We consider the a priori error analysis of hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form under weak assumptions on the mesh design and the local finite element spaces employed. In particular, we prove a priori hp-error bounds for linear target functionals of the solution, on (possibly) anisotropic computational meshes with anisotropic tensor-product polynomial basis functions. The theoretical results are illustrated by a numerical experiment.