Showing papers on "Adaptive algorithm published in 2004"

Improved adaptive Gaussian mixture model for background subtraction

Abstract: Background subtraction is a common computer vision task. We analyze the usual pixel-level approach. We develop an efficient adaptive algorithm using Gaussian mixture probability density. Recursive equations are used to constantly update the parameters and but also to simultaneously select the appropriate number of components for each pixel.

Adaptive Finite Element Methods with convergence rates

Abstract: Adaptive Finite Element Methods for numerically solving elliptic equations are used often in practice. Only recently [12], [17] have these methods been shown to converge. However, this convergence analysis says nothing about the rates of convergence of these methods and therefore does, in principle, not guarantee yet any numerical advantages of adaptive strategies versus non-adaptive strategies. The present paper modifies the adaptive method of Morin, Nochetto, and Siebert [17] for solving the Laplace equation with piecewise linear elements on domains in ℝ2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm (which is equivalent to the H1 norm). Namely, it is shown that whenever s>0 and the solution u is such that for each n≥1, it can be approximated to accuracy O(n−s) in the energy norm by a continuous, piecewise linear function on a triangulation with n cells (using complete knowledge of u), then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process. Moreover, the number of arithmetic computations in the proposed method is also of order O(n) for each n≥1. The construction and analysis of this adaptive method relies on the theory of nonlinear approximation.

Audio Signal Processing for Next-Generation Multimedia Communication Systems

Abstract: Preface. Contributing Authors. 1: Introduction Yiteng (Arden) Huang, J. Benesty. 1. Multimedia Communications. 2. Challenges and Opportunities. 3. Organization of the Book. I: Speech Acquisition and Enhancement. 2: Differential Microphone Arrays G.W. Elko. 1. Introduction. 2. Differential Microphone Arrays. 3. Array Directional Gain. 4. Optimal Arrays for Isotropic Fields. 5. Design Examples. 6. Sensitivity to Microphone Mismatch and Noise. 7. Conclusions. 3: Spherical Microphone Arrays for 3D Sound Recording J. Meyer, G.W. Elko. 1. Introduction. 2. Fundamental Concept. 3. The Eigenbeamformer. 4. Modal-Beamformer. 5. Robustness Measure. 6. Beampattern Design. 7. Measurements. 8. Summary. 9. Appendix A. 4: Subband Noise Reduction Methods for Speech Enhancement E.J. Diethorn. 1. Introduction. 2. Wiener Filtering. 3. Speech Enhancement by Short-Time Spectral Modification. 4. Averaging Techniques for Envelope Estimation. 5. Example Implementation. 6. Conclusion. II: Acoustic Echo Cancellation. 5: Adaptive Algorithms for MIMO Acoustic Echo Cancellation J. Benesty, T. Gansler, Yiteng (Arden) Huang, M. Rupp. 1. Introduction. 2. Normal Equations and Identification of a MIMO System. 3. The Classical and Factorized Multichannel RLS. 4. The Multichannel Fast RLS. 5. TheMultichannel LMS Algorithm. 6. The Multichannel APA. 7. The Multichannel Exponentiated Gradient Algorithm. 8. The Multichannel Frequency-domain Adaptive Algorithm. 9. Conclusions. 6: Double-talk Detectors for Acoustic Echo Cancellers T. Gansler, J. Benesty. 1. Introduction. 2. Basics of AEC and DTD. 3. Double-talk Detection Algorithms. 4. Comparison of DTDs by Means of the ROC. 5. Discussion. 7: The WinEC: A Real-Time Hands-Free Stereo Communication System T. Gansler, V. Fischer, E.J. Diethorn, J. Benesty. 1. Introduction. 2. System Description. 3. Algorithms of the Echo Canceller Module. 4. Residual Echo and Noise Suppression. 5. Simulations. 6. Real-Time Tests with Different Modes of Operation. 7. Discussion. III: Sound Source Tracking and Separation. 8: Time Delay Estimation Jingdong Chen, Yiteng (Arden) Huang, J. Benesty. 1. Introduction. 2. Signal Models. 3. Generalized Cross-Correlation Method. 4. The Multichannel Cross-Correlation Algorithm. 5. Adaptive Eigenvalue Decomposition Algorithm. 6. Adaptive Multichannel Time Delay Estimation. 7. Experiments. 8. Conclusions. 9: Source Localization Yiteng (Arden) Huang, J. Benesty, G.W. Elko. 1. Introduction. 2. Source Localization Problem. 3. Measurement Model and Cramer-Rao lower Bound for Source Localization. 4. Maximum Liklihood Estimator. 5. Least Squares Estimate. 6. Example

A comparative study of Markov chain Monte Carlo methods for conceptual rainfall‐runoff modeling

Abstract: [1] One challenge that faces hydrologists in water resources planning is to predict the catchment's response to a given rainfall. Estimation of parameter uncertainty (and model uncertainty) allows assessment of the risk in likely applications of hydrological models. Bayesian statistical inference provides an ideal means of assessing parameter uncertainty, whereby prior knowledge about the parameter is combined with information from the available data to produce a probability distribution (the posterior distribution) that describes uncertainty about the parameter and serves as a basis for selecting appropriate values for use in modeling applications. Widespread use of Bayesian techniques in hydrology has been hindered by difficulties in summarizing and exploring the posterior distribution. These difficulties have been largely overcome by recent advances in Markov chain Monte Carlo (MCMC) methods that involve Monte Carlo sampling of the posterior distribution. This study compares four MCMC sampling algorithms in the context of rainfall-runoff modeling. The algorithms compared include a conventional Metropolis-Hastings algorithm used previously in hydrological applications which uses a combination of block and single-site updating and an adaptive Metropolis algorithm that has characteristics that are well suited to model parameters with a high degree of correlation and interdependence, as is often evident in hydrological models. In addition to these, two other algorithms are evaluated to clarify the relative importance of updating all model parameters as a block versus updating each parameter one at a time. The MCMC techniques are compared for simplicity, ease of use, statistical efficiency in exploration of the parameter space, and speed of implementation, using 11 years of daily rainfall-runoff data from the Bass river catchment in Australia. The results show that the adaptive Metropolis algorithm is superior in many respects and can offer a relatively simple basis for assessing parameter uncertainty in hydrological modeling studies and that the efficiency of the adaptive algorithm is not solely attributed to the block-updating element of the algorithm.

Recursive least squares constant modulus algorithm for blind adaptive array

Abstract: We consider the problem of blind adaptive signal separation with an antenna array, based on the constant modulus (CM) criterion. An approximation to the CM cost function is proposed, which allows the use of the recursive least squares (RLS) optimization technique. A novel RLS constant modulus algorithm (RLS-CMA) is derived, where the modulus power of the array output can take on arbitrary positive real values (i.e., fractional values allowed). Simulations are performed to compare the performance of the proposed RLS-CMA to other well-known algorithms for blind adaptive beamforming. Results indicate that the RLS-CMA has a significantly faster convergence rate and better tracking ability.

An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems

Abstract: An efficient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator. An adaptive algorithm with variable time-step sizes and space meshes is proposed and studied which, at each time step, delays the mesh coarsening until the final iteration of the adaptive procedure, allowing only mesh and time-step size refinements before. It is proved that at each time step the adaptive algorithm is able to reduce the error indicators (and thus the error) below any given tolerance within a finite number of iteration steps. The key ingredient in the analysis is a new coarsening strategy. Numerical results are presented to show the competitive behavior of the proposed adaptive algorithm.

Adaptive mode decision for H.264 encoder

Abstract: An adaptive mode decision algorithm is presented, with rate-distortion optimisation that reduces complexity of the H.264 encoder without loss of image quality and compression ratio. The proposed algorithm uses the property of an all-zero coefficients block that is produced by quantisation and coefficient thresholding to effectively skip unnecessary modes. Experimental results show that the speed of the adaptive mode decision algorithm is two times faster than the full-mode decision algorithm of the JM72 reference encoder, without any coding loss.

Cognitive radio testbed: further details and testing of a distributed genetic algorithm based cognitive engine for programmable radios

Abstract: This paper provides details of a distributed genetic algorithm (GA) based cognitive radio engine model for disaster communications and its implementation in a cognitive radio test bed using programmable radios. Future applications include tactical and covert communications. The cognitive system monitor (CSM) module presented here permits cross layer cognition and adaptation of a programmable radio by classifying the observed channel, matching channel behavior with operational goals, and passing these goals to a wireless system genetic algorithm (WSGA) adaptive controller module to evolve and optimize radio operation. The CSM module algorithm provides for parallel distributed operation and includes a learning classifier and meta-GA functions that work from a knowledge base (which may be distributed) in long term memory to synthesize matched channels and operational goals that are retained in short term memory. Experimental results show that the cognitive engine finds the best tradeoff between a host radio's operational parameters in changing wireless conditions, while the baseline adaptive controller only increases or decreases its data rate based on a threshold, often wasting usable bandwidth or excess power when it is not needed due its inability to learn.

Signal processing of sensor node data for vehicle detection

Abstract: We describe an algorithm and experimental work for vehicle detection using sensor node data. Both acoustic and magnetic signals are processed for vehicle detection. We propose a real-time vehicle detection algorithm called the adaptive threshold algorithm (ATA). The algorithm first computes the time-domain energy distribution curve and then slices the energy curve using a threshold updated adaptively by some decision states. Finally, the hard decision results from threshold slicing are passed to a finite-state machine, which makes the final vehicle detection decision. Real-time tests and offline simulations both demonstrate that the proposed algorithm is effective.

Adaptive Control Design for Nonaffine Models Arising in Flight Control

Abstract: Adaptive tracking control algorithms are developed for a class of models encountered in flight control that are nonaffine in the control input. The essence of the approach is to differentiate the function that is nonlinear in the control input and obtain an increased-order system that is linear in the derivative of the control signal and can be used as a new control variable. A systematic procedure is developed and related theoretical and practical issues are discussed. The proposed procedure, referred to as the controller for nonaffine plants, is developed for several cases of nonaffine models with unknown parameters. It is shown that the key aspect in the adaptive control design is the definition of the estimate of the derivative of system's state, which results in a convenient error model from which the adaptive laws can be written in a straightforward manner. The proposed approach is tested using a three-degree-of-freedom simulation of a typical fighter aircraft and is shown to result in a substantially improved system response.

Direct adaptive iterative learning control of nonlinear systems using an output-recurrent fuzzy neural network

Abstract: In this paper, a direct adaptive iterative learning control (DAILC) based on a new output-recurrent fuzzy neural network (ORFNN) is presented for a class of repeatable nonlinear systems with unknown nonlinearities and variable initial resetting errors. In order to overcome the design difficulty due to initial state errors at the beginning of each iteration, a concept of time-varying boundary layer is employed to construct an error equation. The learning controller is then designed by using the given ORFNN to approximate an optimal equivalent controller. Some auxiliary control components are applied to eliminate approximation error and ensure learning convergence. Since the optimal ORFNN parameters for a best approximation are generally unavailable, an adaptive algorithm with projection mechanism is derived to update all the consequent, premise, and recurrent parameters during iteration processes. Only one network is required to design the ORFNN-based DAILC and the plant nonlinearities, especially the nonlinear input gain, are allowed to be totally unknown. Based on a Lyapunov-like analysis, we show that all adjustable parameters and internal signals remain bounded for all iterations. Furthermore, the norm of state tracking error vector will asymptotically converge to a tunable residual set as iteration goes to infinity. Finally, iterative learning control of two nonlinear systems, inverted pendulum system and Chua's chaotic circuit, are performed to verify the tracking performance of the proposed learning scheme.

Data-aided and blind stochastic gradient algorithms for widely linear MMSE MAI suppression for DS-CDMA

Abstract: In this paper, three novel stochastic gradient algorithms for adjustment of the widely linear (WL) minimum mean-squared error (MMSE) filter for multiple access interference (MAI) suppression for direct-sequence code-division multiple access (DS-CDMA) are introduced and analyzed. In particular, we derive a data-aided WL least-mean-square (LMS) algorithm, a blind WL minimum-output-energy (MOE) algorithm, and a WL blind LMS (BLMS) algorithm. We give analytical expressions for the steady-state signal-to-interference-plus-noise ratios (SINRs) of the proposed WL algorithms, and we also investigate their speed of convergence. Wherever possible, comparisons with the corresponding linear adaptive algorithms are made. Both analytical considerations and simulations show, in good agreement, the superiority of the novel WL adaptive algorithms. Nevertheless, all proposed WL algorithms require a slightly lower computational complexity than their linear counterparts.

A locally-adaptive algorithm for measuring blocking artifacts in images and videos

Abstract: Block transform coding is the most popular approach for image and video compression. The objective measurement of blocking artifacts plays an important role in the design, optimization, and assessment of image and video coding systems. This paper presents a new algorithm for measuring blocking artifacts in images and videos. It exhibits unique and useful features: 1) it examines the blocks individually so that it can measure the severity of blocking artifacts locally; 2) it is a one-pass algorithm in the sense that the image needs to be accessed only once; 3) it takes into account the blocking artifacts for high bit rate images and the flatness for the very low bit rate images; 4) the blocking artifacts measure is well-defined in the range of 0-10. Experiments on various still images and videos show that this blockiness measure is very efficient in terms of computational complexity and memory usage, and can produce consistent blocking artifacts measurement.

Sliding window adaptive SVD algorithms

Abstract: The singular value decomposition (SVD) is an important tool for subspace estimation. In adaptive signal processing, we are especially interested in tracking the SVD of a recursively updated data matrix. This paper introduces a new tracking technique that is designed for rectangular sliding window data matrices. This approach, which is derived from the classical bi-orthogonal iteration SVD algorithm, shows excellent performance in the context of frequency estimation. It proves to be very robust to abrupt signal changes, due to the use of a sliding window. Finally, an ultra-fast tracking algorithm with comparable performance is proposed.

Distributed adaptive task allocation in heterogeneous computing environments to maximize throughput

Abstract: Summary form only given. We consider the task allocation problem for computing a large set of equal-sized independent tasks on heterogeneous computing systems. This problem represents the computation paradigm for a wide range of applications such as SETl@home and Monte Carlo simulations. We consider a general problem in which the interconnection between the nodes is modeled using a graph. We maximize the throughput of the system by using an extended network flow representation. We then develop a decentralized adaptive algorithm. This algorithm leads to a simple decentralized protocol that coordinates the resources in the system. The effectiveness of the proposed task allocation approach is verified through simulations.

Performance evaluation of adaptive ramp-metering algorithms using microscopic traffic simulation model

Abstract: Adaptive ramp metering has undergone significant theoretical developments in recent years. However, the applicability and potential effectiveness of such algorithms depend on a number of complex factors that are best investigated during a planning phase prior to any decision on their implementation. The use of traffic simulation models can provide a quick and cost-effective way to evaluate the performance of such algorithms prior to implementation on the target freeway network. In this paper, a capability-enhanced PARAMICS simulation model has been used in an evaluation study of three well-known adaptive ramp-metering algorithms: ALINEA, BOTTLE- NECK, and ZONE. ALINEA is a local feedback-control algorithm, and the other two are areawide coordinated algorithms. The evaluation has been conducted in a simulation environment over a stretch of the I-405 freeway in California, under both recurrent congestion and incident scenarios. Simulation results show that adaptive ramp-metering algorithms can reduce freeway congestion effectively compared to the fixed-time control. ALINEA shows good performance under both recurrent and nonrecurrent congestion scenarios. BOTTLENECK and ZONE can be improved by replacing their native local occupancy control algorithms with ALINEA. Compared to ALINEA, the revised BOTTLENECK and ZONE algorithms using ALINEA as the local control algorithm are found to be more efficient in reducing traffic congestion than ALINEA alone. The revised BOTTLENECK algorithm performs robustly under all scenarios. The results also indicate that ramp metering becomes less effective when traffic experiences severe congestion under incident scenarios.

A hybrid adaptive blind equalization algorithm for QAM signals in wireless communications

Abstract: A hybrid adaptive channel equalization technique for quadrature amplitude modulation (QAM) signals is proposed. The proposed algorithm, which is referred to as the modified constant modulus algorithm (MCMA), minimizes an error cost function that includes both amplitude and phase of the equalizer output. In addition to the amplitude-dependent term that is provided by the conventional constant modulus algorithm (CMA), the cost function includes an additive signal constellation matched error (CME) term. This term can be designed to satisfy a set of desirable properties. The MCMA is compared with the CMA for blind equalization. The performance is measured for wireless channels using both transient and steady-state behavior of the mean square error (MSE). It is shown that MCMA is superior and more robust in low signal-to-noise ratio (SNR) environments. Simulation results demonstrate that using MCMA improves adaptive channel equalization by increasing the convergence rate and decreasing the steady-state mean square error.

A normalized robust mixed-norm adaptive algorithm for system identification

Abstract: A normalized robust mixed-norm (NRMN) algorithm for system identification in the presence of impulsive noise is introduced. The standard robust mixed-norm (RMN) algorithm exhibits slow convergence, requires a stationary operating environment, and employs a constant step-size that needs to be determined a priori. To overcome these limitations, the proposed NRMN algorithm introduces a time-varying learning rate and, thus, no longer requires a stationary environment, a major drawback of the RMN algorithm. The proposed NRMN exhibits increased convergence rate and substantially reduces the steady-state coefficient error, as compared to the least mean square (LMS), normalized LMS (NLMS), least absolute deviation (LAD), and RMN algorithm.

A variable leaky LMS adaptive algorithm

Abstract: The LMS algorithm has found wide application in many areas of adaptive signal processing and control. We introduce a variable leaky LMS algorithm, designed to overcome the slow convergence of standard LMS in cases of high input eigenvalue spread. The algorithm uses a greedy punish/reward heuristic together with a quantized leak adjustment function to vary the leak. Simulation results confirm that the new algorithm can significantly outperform standard LMS when the input eigenvalue spread is high.

Numerical algorithm for adaptive autoreclosure and protection of medium-voltage overhead lines

Abstract: In this paper, a new numerical algorithm for medium-voltage overhead lines, autoreclosure, is described. The subfunction of the autoreclosure scheme that would inhibit the first shot after detecting a solid fault (as compared with an arc fault) is evaluated and presented. It is based on one terminal data processing and it is derived in the time domain. In the algorithm the fault nature (arcing or arcless fault) is estimated using linear least error squares estimation technique. The arc, occurring on the fault point during arcing faults on overhead lines, is included in the problem consideration. In addition, by introducing the prefault load current in the existing model, better algorithm performances and a more reliable adaptive algorithm for autoreclosure are achieved. The algorithm is derived for the case of three-phase symmetrical fault. The results of the algorithm testing through computer simulation are presented. Particularly the algorithm sensitivity to arc elongation effects, supplying network parameters, and processing of the signals in the presence of harmonics are tested and analyzed.

A locally adaptive algorithm for measuring blocking artifacts in images and videos

Abstract: Block transform coding is the most popular approach for image and video compression. The objective measurement of blocking artifacts plays an important role in the design, optimization, and assessment of image and video coding systems. This paper presents a new algorithm for measuring image quality of a BDCT coded images or videos. It exhibits unique and useful features: (1) it examines the blocks individually so that it can measure the severity of blocking artifacts locally; (2) it is a one-pass algorithm in the sense that the image needs to be accessed only once; (3) it takes into account the blocking artifacts for high bit rate images and the flatness for the very low bit rate images; (4) the quality measure is well defined in the range of 0–10. Experiments on various still images and videos show that the new quality measure is very efficient in terms of computational complexity and memory usage, and can produce consistent blocking artifacts measurement.

Spreading code optimization and adaptation in CDMA via discrete stochastic approximation

Abstract: The aim of this paper is to develop discrete stochastic approximation algorithms that adaptively optimize the spreading codes of users in a code-division multiple-access (CDMA) system employing linear minimum mean-square error (MMSE) receivers. The proposed algorithms are able to adapt to slowly time-varying channel conditions. One of the most important properties of the algorithms is their self-learning capability-they spend most of the computational effort at the global optimizer of the objective function. Tracking analysis of the adaptive algorithms is presented together with mean-square convergence. An adaptive-step-size algorithm is also presented for optimally adjusting the step size based on the observations. Numerical examples, illustrating the performance of the algorithms in multipath fading channels, show substantial improvement over heuristic algorithms.

Cellular multiadaptive analogic architecture: a computational framework for UAV applications

Abstract: An efficient adaptive algorithm in real-time applications should make optimal use of the available computing power for reaching some specific design goals. Relying on appropriate strategies, the spatial resolution/temporal rate can be traded against computational complexity; and sensitivity traded against robustness, in an adaptive process. In this paper, we present an algorithmic framework where a spatial multigrid computing is placed within a temporal multirate structure, and at each spatial grid point, the computation is based on an adaptive multiscale approach. The algorithms utilize an analogic (analog and logic) architecture consisting of a high-resolution optical sensor, a low-resolution cellular sensor-processor and a digital signal processor. The proposed framework makes the acquisition of a spatio-temporally consistent image flow possible even in case of extreme variations (relative motion) in the environment. It ideally supports the handling of various difficult problems on a moving platform including terrain identification, navigation parameter estimation, and multitarget tracking. The proposed spatio-temporal adaptation relies on a feature-based optical-flow estimation that can be efficiently calculated on available cellular nonlinear network (CNN) chips. The quality of the adaptation is evaluated compared to nonadaptive spatio-temporal behavior where the input flow is oversampled, thus resulting in redundant data processing with an unnecessary waste of computing power. We also use a visual navigation example recovering the yaw-pitch-roll parameters from motion-field estimates in order to analyze the adaptive hierarchical algorithmic framework proposed and highlight the application potentials in the area of unmanned air vehicles.

An Adaptive Edge-detection Method Based on Canny Algorithm

Abstract: This paper is based on Canny algorithm.An adaptive Canny algorithm of edge-detection method is proposed.This algorithm not only keeps the Canny's excellent performance in good localization,only one response to a single edge and good detection,but also improves the performance in the detail edge-detection and good detection. Canny adaptive algorithm divides image into sub-images and detects them with adaptive threshold value according to the whole image edge information, that improves the automaticity of edge-detection.With the mathematic analysis and test result,it is demonstrated that the adaptive edge-detection method is an efficient improving approach on edge-detection.

Uniform circular arrays for smart antennas

Abstract: One of the most recent innovations to overcome the demands for increased capacity nowadays, is that of deploying smart antennas for wireless communications. A smart or adaptive antenna is defined as one that combines multiple antenna elements with a signal processing capability to optimize its radiation or reception pattern automatically, in response to the signal environment. The paper concentrates on the use of circular arrays for smart antennas. An overview of smart antennas is provided. The necessity for implementing uniform circular arrays (UCAs) for adaptive antennas and several adaptive algorithms applicable to UCAs are mentioned. For the simulation process, a specific UCA configuration has been examined, where adaptive beamforming was performed in both azimuth and elevation (for fixed elevation and azimuth angles, respectively) according to certain requirements in the design (directions of signals of interest and signals not of interest) imposed by the wireless environment.

An Optimal Adaptive Finite Element Method

Abstract: Although existing adaptive finite element methods for solving second order elliptic equations often perform better in practical computations than nonadaptive ones, usually they are not even proven to converge. Only recently in the work of Dorfler [SIAM J. Numer. Anal., 33 (1996), pp. 1106--1124] and that of Morin, Nochetto, and Siebert [SIAM J. Numer. Anal., 38 (2000), pp. 466--488], adaptive methods were constructed for which convergence could be demonstrated. However, convergence alone does not imply that the method is more efficient than its nonadaptive counterpart. In [ Numer. Math.}, 97 (2004), pp. 219--268], Binev, Dahmen, and DeVore added a coarsening step to the routine of Morin, Nochetto, and Siebert, and proved that the resulting method is quasi-optimal in the following sense: If the solution is such that for some s > 0, the error in energy norm of the best continuous piecewise linear approximations subordinate to any partition with n triangles is $\mathcal{O}(n^{-s})$, then given an $\eps>0$, the adaptive method produces an approximation with an error less than $\eps$ subordinate to a partition with $\mathcal{O}(\eps^{-1/s})$ triangles, in only $\mathcal{O}(\eps^{-1/s})$ operations. In this paper, employing a different type of adaptive partition, we develop an adaptive method with properties similar to those of Binev, Dahmen, and DeVore's method, but unlike their method, our coarsening routine will be based on a transformation to a wavelet basis, and we expect it to have better quantitative properties. Furthermore, all our results are valid uniformly in the size of possible jumps of the diffusion coefficients. Since the algorithm uses solely approximations of the right-hand side, we can even allow right-hand sides in $H^{-1}(\Omega)$ that lie outside $L_2(\Omega)$, at least when they can be sufficiently well approximated by piecewise constants. In our final adaptive algorithm, all tolerances depend on an a posteriori estimate of the current error instead of an a priori one; this can be expected to provide quantitative advantages.