Non-adaptive Method Research Articles

An inverse problem for the acoustics equations: A multilevel adaptive algorithm

A numerical solution to an inverse problem for the acoustic equations using an optimization method for a stratified medium is presented. With the distribution of an acoustic wave field on the medium’s surface, the 1D distributions of medium’s density, as well as the velocity and absorption coefficient of the acoustic wave, are determined. Absorption in a Voigt body model is considered. The conjugate gradients and the Newton method are used for minimization. To increase the efficiency of the numerical method, a multilevel adaptive algorithm is proposed. The algorithm is based on a division of the whole procedure of solving the inverse problem into a series of consecutive levels. Each level is characterized by the number of parameters to be determined at the level. In moving from one level to another, the number of parameters changes adaptively according to the functional minimized and the convergence rate. The minimization parameters are chosen as illustrated by results of solving the inverse problem in a spectral domain, where the desired quantities are presented as Chebyshev polynomial series and minimization is carried out with respect to the coefficients of these series. The method is compared in efficiency with a nonadaptive method. The optimal parameters of the multilevel method are chosen. It is shown that the multilevel algorithm offers several advantages over the one without partitioning into levels. The algorithm produces primarily a more accurate solution to the inverse problem.

Adapting grid applications to safety using fault-tolerant methods: Design, implementation and evaluations

Grid applications have been prone to encountering problems such as failures or malicious attacks during execution in recent years, due to their distributed and large-scale features. The application itself, however, has limited power to address these problems. This paper presents the design, implementation, and evaluation of an adaptive framework— Dynasa, which strives to handle security problems using adaptive fault-tolerance (i.e., checkpointing and replication) during the execution of applications according to the status of the Grid environments. We evaluate our adaptive framework experimentally using the Grid5000 testbed and the experimental results have demonstrated that Dynasa enables the application itself to handle the security problems efficiently. The starting of the adaptive component is less than 1 s and the adaptive action is less than 0.1 s with the checkpoint interval of 20 s. Compared with non-adaptive method, experimental results demonstrate that Dynasa achieves better performance in terms of execution time, network bandwidth consumed, and CPU load, resulting in up to a 50% lower overhead.

PCR NONADAPTIVE GROUP TESTING OF DNA LIBRARIES FOR BIOMOLECULAR COMPUTING AND TAGGANT APPLICATIONS

Using a universal and parallel battery of PCR reactions, we give a nonadaptive group testing method for identifying the individual strands in a pooled sample of several different DNA sequences taken from a DNA library. The method discussed here has potential applications to DNA taggants, DNA memory and DNA computing.

On Monte Carlo algorithms applied to Dirichlet problems for parabolic operators in the setting of time-dependent domains

Dirichlet problems for second order parabolic operators in space-time domains Ω ⊂ ℝn+1 are of paramount importance in analysis, partial differential equations and applied mathematics. These problems can be approached in many different ways using techniques from partial differential equations, potential theory, stochastic differential equations, stopped diffusions and Monte Carlo methods. The performance of any technique depends on the structural assumptions on the operator, the geometry and smoothness properties of the space-time domain Ω, the smoothness of the Dirichlet data and the smoothness of the coefficients of the operator under consideration. In this paper, which mainly is of numerical nature, we attempt to further understand how Monte Carlo methods based on the numerical integration of stochastic differential equations perform when applied to Dirichlet problems for uniformly elliptic second order parabolic operators and how their performance vary as the smoothness of the boundary, Dirichlet data and coefficients change from smooth to non-smooth. Our analysis is set in the genuinely parabolic setting of time-dependent domains, which in itself adds interesting features previously only modestly discussed in the literature. The methods evaluated and discussed include elaborations on the non-adaptive method proposed by Gobet [ESAIM: Probability and Statistics 5: 261–297, 2001] based on approximation by half spaces and exit probabilities and the adaptive method proposed in [Lecture Notes in Computational Science and Engineering 44: 59–88, 2005] for weak approximation of stochastic differential equations.

Adaptivity and computational complexity in the numerical solution of ODEs

In this paper we analyze the problem of adaptivity for one-step numerical methods for solving ODEs, both IVPs and BVPs, with a view to generating grids of minimal computational cost for which the local error is below a prescribed tolerance (optimal grids). The grids are generated by introducing an auxiliary independent variable τ and finding a grid deformation map, t=Θ(τ), that maps an equidistant grid {τj} to a non-equidistant grid in the original independent variable, {tj}. An optimal deformation map Θ is determined by a variational approach. Finally, we investigate the cost of the solution procedure and compare it to the cost of using equidistant grids. We show that if the principal error function is non-constant, an adaptive method is always more efficient than a non-adaptive method.

A sparse grid space-time discretization scheme for parabolic problems

In this paper, we consider the discretization in space and time of parabolic differential equations where we use the so-called space-time sparse grid technique. It employs the tensor product of a one-dimensional multilevel basis in time and a proper multilevel basis in space. This way, the additional order of complexity of a direct space-time discretization can be avoided, provided that the solution fulfills a certain smoothness assumption in space-time, namely that its mixed space-time derivatives are bounded. This holds in many applications due to the smoothing properties of the propagator of the parabolic PDE (heat kernel). In the more general case, the space-time sparse grid approach can be employed together with adaptive refinement in space and time and then leads to similar approximation rates as the non-adaptive method for smooth functions. We analyze the properties of different space-time sparse grid discretizations for parabolic differential equations from both, the theoretical and practical point of view, discuss their implementational aspects and report on the results of numerical experiments.

An adaptive approach to offline handwritten word recognition

An adaptive handwritten word recognition method is presented. A recursive architecture based on interaction between flexible character classification and deductive decision making is developed. The recognition process starts from the initial coarse level using a minimum number of features, then increases the discrimination power by adding other features adaptively and recursively until the result is accepted by the decision maker. For the computational aspect of a feasible solution, a unified decision metric, recognition confidence; is derived from two measurements: pattern confidence, evaluation of absolute confidence using shape features, and lexical confidence, evaluation of the relative string dissimilarity in the lexicon. Practical implementation and experimental results in reading the handwritten words of the address components of US mail pieces are provided. Up to a 4 percent improvement in recognition performance is achieved compared to a nonadaptive method. The experimental result shows that the proposed method has advantages in producing valid answers using the same number of features as conventional methods.

Digital mammography: Comparison of adaptive and nonadaptive CAD methods for mass detection

The authors compared the performance of adaptive and nonadaptive computer-aided diagnostic (CAD) methods for breast mass detection with digital mammography. Both adaptive and nonadaptive modular CAD methods employed recent advances in multiresolution and mutiorientation wavelet transforms for improved feature extraction. The nonadaptive method uses fixed parameters for the image preprocessing modules. The adaptive method, a new class of algorithms, adapts to image content by selecting parameters for the image preprocessing modules within a parameter range. Comparison of the two methods was performed for each individual CAD module with a region-of-interest (ROI) database containing all mass types and normal tissue. Receiver operating characteristic (ROC) analysis clearly demonstrated an improvement in performance for the three adaptive modules and a significant overall difference between the two methods. The average ROC area index (Az) values were 0.86 and 0.95 for the nonadaptive and adaptive methods, respectively. The corresponding P value is .0145. For a previously reported database of full mammographic images containing 50 abnormal cases with all mass types and 50 normal images, the adaptive CAD method had a sensitivity of 96% (1.71 false-positive results per image) compared with 89% (1.91 false-positive results per image) for the nonadaptive CAD method. The adaptive CAD method demonstrated better performance. A study is in progress to determine the generalizability of the adaptive CAD method by applying it to larger retrospective image databases with different film digitizers.

The adaptive modified rhyme test

This paper discusses an application of the adaptive maximum-likelihood procedure (MLP) [Watson and Pelli, Percept. Psychophys. 33, 113–120 (1983)] to speech intelligibility measurements conducted with the modified rhyme test (MRT) [House et al., J. Acoust. Soc. Am. 37, 158–166 (1965)]. The MLP procedure allows the experimenter to preset the target intelligibility at any value and determine the corresponding speech-to-noise (S/N) ratio. This makes it an efficient tool for estimating the change in the S/N ratio required to maintain identical intelligibility in different noise conditions of the communication channel. The MLP-MRT was tested on a group of 12 listeners. The S/N ratio corresponding to intelligibilities of 50%, 70%, and 88% was determined in five test runs of 30 items each. Subsequently, the S/N ratios estimated by the adaptive procedure were verified by measuring the intelligibility scores by a nonadaptive method. Results showed that intelligibility scores obtained by the nonadaptive method differed by less than 5% from the intelligibility preset in the adaptive procedure. [Part of the work was done as a collaborative project between NRC and the Defence and Civil Institute of Environmental Medicine, during the author’s stay with the Acoustics and Signal Processing Group, NRC, Ottawa, Canada.]

An Adaptive High-Speed Automatic Binarizing Method by Analog Processing.

A binaraized image processing is often used by many measurement systems. It is important to select the suitable threshold level corresponding to measuring surroundings and measuring objects. The digital adaptive binarizing methods are difficult to binarize the image data at high-speed and to measure successively for a long time. In addition it is difficult to construct the simple binarizing system. This paper describes an adaptive high-speed automatic binarizing method by analog processing. A principle of the method is to detect the peak value and valley value of the CCD sensor outputs, and is to set the threshold level automatically by using their values. The features of the method are: (1) to distinguish between opaque and transparent objects; (2) to set threshold level automatically according to measuring surroundings. We constructed the prototype system by using analog electric circuit and applied the system to dimension measurement of both transparent and opaque objects. The binaraized outputs to opaque object by the system are accurate within 5 pixel and that of transparent object are accurate within 9 pixel. This results are sufficient precision compared with non-adaptive method. When the proposed method is applied to one dimensional CCD, the sampling time is twice as long as the scanning time of CCD. And the sampling time in the case of two dimensional CCD is equal to video frame time (about 33 ms). As a result, it is shown that the proposed method is effective for an adaptive high-speed automatic binarizing method.

On the Power of Adaption

Optimal error bounds for adaptive and nonadaptive numerical methods are compared. Since the class of adaptive methods is much larger, a well-chosen adaptive method might seem to be better than any nonadaptive method. Nevertheless there are several results saying that under natural assumptions adaptive methods are not better than nonadaptive ones. There are also other results, however, saying that adaptive methods can be significantly better than nonadaptive ones as well as bounds on how much better they can be. It turns out that the answer to the “adaption problem” depends very much on what is known a priori about the problem in question; even a seemingly small change of the assumptions can lead to a different answer.

An efficient three-dimensional demagnetizing field calculation scheme (abstract)

We present an efficient three-dimensional demagnetizing field calculation scheme for micromagnetic simulations of the domain walls in garnet-like materials. We take advantage of the fact that the domain wall is nearly planar and distribute computational power nonuniformly between the bulk of the material and the wall regions. We use fast Fourier convolutions along one dimension (approximately parallel to the domain wall) and a two-dimensional variant of Greengard’s adaptive fast multipole method in the other dimensions. Even a 2D nonadaptive multipole method has the time complexity of the order of N, for N points in contrast to full Fourier convolution (Nlog N) or direct methods (N2). The adaptive method is based on maintaining a point-to-point error bound which allows us to work at high resolution in the region of the domain wall and coarser resolution in the bulk material. In this case, the calculation scales proportional to the domain wall size rather than the sample volume. The error bound calculation is important for our method and it depends on the Gegenbauer’s addition theorem for the modified Bessel functions. Due to the complex nature of this addition theorem, we have relied on numerical methods, comparing our method to the direct calculations for small size samples. We have implemented this scheme on an SGI workstation and we are reporting the outcome of the 2D adaptive fast multipole method implementation. We have studied the relationship between error bounds, domain wall size, and actual time. We have also compared our results with that of M. Redjdal’s method.

Sequential Adaptation in Estimating Elastic Boundary-Condition Influence Matrices

A method is discussed for adaptive estimation of the boundary-condition influence matrix of a linear elastic system, based on modal measurements (eigenvalues and eigenvectors) and on knowledge of the influence matrix of the system with respect to reference boundary conditions. The estimation is terminated adaptively by an algorithm motivated by the concept of sequential analysis. The boundary-condition influence matrix is re-estimated with measurement of each additional mode, until a termination criterion indicates that adequate accuracy has been attained. The advantage of adaptive termination of the estimation is the enhanced computational (and possibly instrumental) efficiency of estimating with minimal modal data. An analytical technique for comparing the adaptive termination with a reasonable non-adaptive method has been developed and demonstrated by application to a uniform beam. When uncertainty in the boundary conditions is represented by convex models, it is shown that the adaptive estimation can terminate much earlier than the non-adaptive procedure.

The Minimal Cost of Approximating Linear Operators Using Perturbed Information-The Asymptotic Setting

We study the asymptotic behavior of the minimal cost of computing an ϵ-approximation to linear continuous operators, as ϵ → 0+. An approximation is computed based on perturbed values of linear and continuous functionals which can be chosen adaptively. Obtaining a value of a functional with given precision is connected with some cost determined by the cost function. Under some assumptions, we show that the minimal (information) cost of computing an ϵ-approximation grows essentially as fast as the minimal information cost in the worst case setting. It can grow significantly slower only on a boundary set of problem elements. A nonadaptive method with the information cost proportional to the minimal worst case cost is constructed.

Dynamics of liquid membranes. I: Non-adaptive finite difference methods

A non-adaptive method and a Lagrangian-Eulerian finite difference technique are used to analyse the dynamic response of liquid membrancs to imposed pressure variations. The non-adaptive method employs a fixed grid and upwind differences for the convection terms, whereas the Lagrangian-Eulerian technique uses operator splitting and decomposes the mixed convection-diffusion system of equations into a sequence of convection and diffusion operators. The convection operator is solved exactly by means of the method of characteristics, and its results are interpolated onto the fixed (Eulerian) grid used to solve the diffusion operator. It is shown that although the method of characteristics eliminates the numerical diffusion associated with upwinding the convection terms in a fixed Eulerian grid, the Lagrangian-Eulerian method may yield overshoots and undershoots near steep flow gradients or when rapid pressure gradients are imposed, owing to the interpolation of the results of the convection operator onto the fixed grid used to solve the diffusion operator. This interpolation should be monotonic and positivity-preserving and should satisfy conservation of mass and linear momentum. It is also shown that both the non-adaptive and Lagrangian-Eulerian finite difference methods produce almost identical (within 1%) results when liquid membranes are subjected to positive and negative step and ramp changes in the pressure coefficient. However, because of their non-adaptive character, these techniques require an estimate of the (unknown) length of the membrane and do not use all the grid points in the calculations. The liquid membrane dynamic response is also analysed as a function of the Froude number, convergence parameter and nozzle exit angle for positive and negative step and ramp changes in the pressure coefficient.