Wavelet boundary element methods – Adaptivity and goal-oriented error estimation

This article is dedicated to the adaptive wavelet boundary element method. It computes an approximation to the unknown solution of the boundary integral equation under consideration with a rate $N^{−s}_{dof}$, whenever the solution can be approximated with this rate in the setting determined by the underlying wavelet basis. The computational cost scale linearly in the number $N_{dof}$ of degrees of freedom. Goal-oriented error estimation for evaluating linear output functionals of the solution is also considered. An algorithm is proposed that approximately evaluates a linear output functional with a rate $N^{−(s+t)}_{dof}$, whenever the primal solution can be approximated with a rate $N^{-s}_{dof}$ and the dual solution can be approximated with a rate $N^{−t}_{dof}$, while the cost still scale linearly in $N_{dof}$. Numerical results for an acoustic scattering problem and for the point evaluation of the potential in case of the Laplace equation are reported to validate and quantify the approach

Development of a Bead-Based Multiplex Genotyping Method for Diagnostic Characterization of HPV Infection

The accurate genotyping of human papillomavirus (HPV) is clinically important because the oncogenic potential of HPV is dependent on specific genotypes. Here, we described the development of a bead-based multiplex HPV genotyping (MPG) method which is able to detect 20 types of HPV (15 high-risk HPV types 16, 18, 31, 33, 35, 39, 45, 51, 52, 53, 56, 58, 59, 66, 68 and 5 low-risk HPV types 6, 11, 40, 55, 70) and evaluated its accuracy with sequencing. A total of 890 clinical samples were studied. Among these samples, 484 were HPV positive and 406 were HPV negative by consensus primer (PGMY09/11) directed PCR. The genotyping of 484 HPV positive samples was carried out by the bead-based MPG method. The accuracy was 93.5% (95% CI, 91.0–96.0), 80.1% (95% CI, 72.3–87.9) for single and multiple infections, respectively, while a complete type mismatch was observed only in one sample. The MPG method indiscriminately detected dysplasia of several cytological grades including 71.8% (95% CI, 61.5–82.3) of ASCUS (atypical squamous cells of undetermined significance) and more specific for high grade lesions. For women with HSIL (high grade squamous intraepithelial lesion) and SCC diagnosis, 32 women showed a PPV (positive predictive value) of 77.3% (95% CI, 64.8–89.8). Among women >40 years of age, 22 women with histological cervical cancer lesions showed a PPV of 88% (95% CI, 75.3–100). Of the highest risk HPV types including HPV-16, 18 and 31 positive women of the same age groups, 34 women with histological cervical cancer lesions showed a PPV of 77.3% (95% CI, 65.0–89.6). Taken together, the bead-based MPG method could successfully detect high-grade lesions and high-risk HPV types with a high degree of accuracy in clinical samples

Adaptive wavelet algorithms for solving operator equations

This thesis treats various aspects of adaptive wavelet algorithms for solving operator equations. For a separable Hilbert space H, a linear functional f in H', and a boundedly invertible linear operator A:H->H', we consider the problem of finding u from H satisfying Au=f. Typically A is given by a variational formulation of a boundary value problem or integral equation, and H is a Sobolev space formulated on some domain or manifold, possibly incorporating essential boundary conditions. Often we will assume that A is self-adjoint and H-elliptic. General operators can be treated, e.g., by forming normal equations, although in particular situations quantitatively more attractive alternatives exist. In their pioneering works [Math. Comp., 70:27-75, 2001] and [Found. Comput. Math., 2(3):203-245, 2002] , Cohen, Dahmen and DeVore introduced adaptive wavelet paradigms for solving the problem numerically. Utilizing a Riesz basis W for H, the idea is to transform the original problem into a problem involving the coefficients of u with respect to the basis W. Writing the collection of these coefficients of u as U, U has to satisfy MU=F, where M is an infinitely sized stiffness matrix, and F is an infinitely sized load vector. Under certain assumptions concerning the cost of evaluating the entries of the stiffness matrix, the methods from the aforementioned works of Cohen, Dahmen, and DeVore for solving this infinite matrix-vector problem were shown to be of optimal computational complexity. In this thesis, we will verify those assumptions, extend the scope of problems for which the adaptive wavelet algorithms can be applied directly, and most importantly, develop and analyze modified adaptive algorithms with improved quantitative properties. Chapter 1 (Introduction) contains a general introduction to the thesis. Chapter 2 (Basic principles) contains a short introduction to the theory of adaptive wavelet algorithms. We start with recalling essential properties of wavelet bases, and briefly present basic results on best N-term approximation. Then we describe how an optimally convergent algorithm can be constructed using any linearly convergent iteration in the energy space. We include proofs of the most fundamental results, along with references to relevant literature. In Chapter 3 (Adaptive Galerkin methods), an adaptive wavelet method for solving linear operator equations is constructed that is a modification of the method from [Math. Comp., 70:27-75, 2001], in the sense that there is no recurrent coarsening of the iterands. In spite of this, it will be shown that the method has optimal computational complexity. Numerical results for a simple model problem indicate that the new method is more efficient than the existing method. In Chapter 4 (Using polynomial preconditioners), we investigate the possibility of using polynomial preconditioners in the context of adaptive wavelet methods. We propose a version of a preconditioned adaptive wavelet algorithm and show that it has optimal computational complexity. In Chapter 5 (Adaptive algorithm for nonsymmetric and indefinite elliptic problems), we modify the adaptive wavelet algorithm from Chapter 3 so that it applies directly, i.e., without forming the normal equation, not only to self-adjoint elliptic operators but also to operators of the form L=A+B, where A is self-adjoint elliptic and B is compact, assuming that the resulting operator equation is well-posed. We show that the algorithm has optimal computational complexity. Aiming at a further improvement of quantitative properties, in Chapter 6 (Adaptive algorithm with truncated residuals), a new class of adaptive wavelet algorithms for solving elliptic operator equations is introduced, which are proven to have optimal complexity assuming a certain property of the stiffness matrix. This assumption is confirmed for elliptic differential operators. In Chapter 7 (Computability of differential operators), restricting us to differential operators, we develop a numerical integration scheme that computes the entries of the stiffness matrix at the expense of an error that is consistent with the approximation error, whereas in each column the average computational cost per entry is O(1). As a consequence, we can conclude that the “fully discrete” adaptive wavelet algorithm has optimal computational complexity. In Chapter 8 (Computability of singular integral operators), we prove an analogous result for singular integral operators, by carefully distributing computational costs over the matrix entries in combination with choosing efficient quadrature schemes. Chapter 9 (Conclusion) finishes with a summary and discussion of the presented research topics, as well as with some suggestions for future research

An optimal adaptive wavelet method without coarsening of the iterands

In this paper, an adaptive wavelet method for solving linear operator equations is constructed that is a modification of the method from [Math. Comp, 70 (2001), pp. 27-75] by Cohen, Dahmen and DeVore, in the sense that there is no recurrent coarsening of the iterands. Despite this, it will be shown that the method has optimal computational complexity. Numerical results for a simple model problem indicate that the new method is more efficient than an existing alternative adaptive wavelet method