Unbounded Error Research Articles

Representation of the exact solution and a stability analysis on the Fredholm integral equation of the first kind in reproducing kernel space

In this paper, a new method is given in order to solve an ill-posed problem on Fredholm integral equation of the first kind. The representation of the exact solution is given and the stability of the solution on Fredholm integral equation of the first kind is discussed in the reproducing kernel space. By the discussions, a conclusion is obtained the stability problem is a well-posed problem in the reproducing kernel space, namely, the measurement errors of the experimental data can not result in unbounded errors of the exact solution. The numerical experiment shows that the new method given in the paper is valid.

Quantum branching programs and space-bounded nonuniform quantum complexity

In this paper, the space complexity of non-uniform quantum algorithms is investigated using the model of quantum branching programs (QBPs). In order to clarify the relationship between QBPs and non-uniform quantum Turing machines, simulations between these two models are presented which allow to transfer upper and lower bound results. Exploiting additional insights about the connection between the running time and the precision of amplitudes, it is shown that non-uniform quantum Turing machines with algebraic amplitudes and QBPs with a suitable analogous set of amplitudes are equivalent in computational power if both models work with bounded or unbounded error. Furthermore, quantum ordered binary decision diagrams (QOBDDs) are considered, which are restricted QBPs that can be regarded as a non-uniform analog of one-way quantum finite automata. Upper and lower bounds are proved that allow a classification of the computational power of QOBDDs in comparison to usual deterministic and randomized variants of the model. Finally, an extension of QBPs is proposed where the performed unitary operation may depend on the result of a previous measurement. A simulation of randomized BPs by this generalized QBP model as well as exponential lower bounds for its ordered variant are presented.

On the complexity of simulating space-bounded quantum computations

This paper studies the space-complexity of predicting the long-term behavior of a class of stochastic processes based on evolutions and measurements of quantum mechanical systems. These processes generalize a wide range of both quantum and classical space-bounded computations, including unbounded error computations given by machines having algebraic number transition amplitudes or probabilities. It is proved that any space s quantum stochastic process from this class can be simulated probabilistically with unbounded error in space O(s), and therefore deterministically in space O(s2).

Semantical Counting Circuits

Counting functions can be defined syntactically or semantically depending on whether they count the number of witnesses in a non-deterministic or in a deterministic computation on the input. In the Turing-machine-based model these two ways of defining counting were proven to be equivalent for many important complexity classes. In the circuit-based model it was done for #P, but for low-level complexity classes such as #AC 0 and #NC 1 only the syntactical definitions were considered. We give appropriate semantical definitions for these two classes and prove them to be equivalent to the syntactical ones. We also consider semantically defined probabilistic complexity classes corresponding to AC 0 and NC ^{1} and prove that in the case of unbounded error, they are identical to their syntactical counterparts.

A linear lower bound on the unbounded error probabilistic communication complexity

The main mathematical result of this paper may be stated as follows: Given a matrix M∈{−1,1} n× n and any matrix M ̃ ∈ R n×n such that sign( M ̃ i,j)=M i,j for all i, j, then rank( M ̃ )⩾n/||M|| . Here || M|| denotes the spectral norm of the matrix M. This implies a general lower bound on the complexity of unbounded error probabilistic communication protocols. As a simple consequence, we obtain the first linear lower bound on the complexity of unbounded error probabilistic communication protocols for the functions defined by Hadamard matrices. This solves a long-standing open problem stated by Paturi and Simon (J. Comput. System Sci. 33 (1986) 106). We also give an upper bound on the margin of any embedding of a concept class in half spaces. Such bounds are of interest to problems in learning theory.

CLOSURE PROPERTY OF PROBABILISTIC TURING MACHINES AND ALTERNATING TURING MACHINES WITH SUBLOGARITHMIC SPACES

This paper first shows that (1) the class of languages recognized by o(log n) space-bounded two-way Monte Carlo Turing machines is not closed under concatenation, Kleene closure, and length-preserving homomorphism, and (2) the class of languages recognized by o(log log n) space-bounded two-way unbounded error probabilistic Turing machines is not closed under each of the above operations. We then show that the class of languages accepted by S(n) space-bounded two-way alternating Turing machines is not closed under each of the above operations for log log n ≤ S(n) ≤ o(log n).

Space-Bounded Quantum Complexity

This paper investigates the computational power of space-bounded quantum Turing machines. The following facts are proved for space-constructible space bounds s satisfying s(n)=Ω(logn): 1.Any quantum Turing machine (QTM) running in space s can be simulated by an unbounded error probabilistic Turing machine (PTM) run- ning in space O(s). No assumptions on the probability of error or running time for the QTM are required, although it is assumed that all transition amplitudes of the QTM are rational. 2.Any PTM that runs in space s and halts absolutely (i.e., has finite worst-case running time) can be simulated by a QTM running in space O(s). If the PTM operates with bounded error, then the QTM may be taken to operate with bounded error as well, although the QTM may not halt absolutely in this case. In the case of unbounded error, the QTM may be taken to halt absolutely. We therefore have that unbounded error, space O(s) bounded quantum Turing machines and probabilistic Turing machines are equivalent in power and, furthermore, that any QTM running in space s can be simulated deterministically in NC2(2s)⊆DSPACE(s2)∩DTIME(2O(s)). We also consider quantum analogues of nondeterministic and one-sided error probabilistic space-bounded classes and prove some simple facts regarding these classes.

A Dynamical Estimation of Forecast Error Covariances in an Assimilation System

Abstract The author uses an approximation of the extended Kalman filter to estimate the forecast and analysis error covariances of an operational assimilation system. The fundamental differences with Kalman filtering are the basic-state trajectory, which is kept close to the operational analyses, and the analysis weights, which follow the usual approximations of optimal interpolation with local data selection. The estimation error covariances for the model state are updated during the analysis and prediction cycles. Although no model error term is specified, the estimation error variances grow according to the dynamics on poorly observed areas. The unbounded error growth found in the Southern Hemisphere has to be limited by a representation of error saturation to account for nonlinearities in the atmosphere. A weak relaxation to climatology is introduced in order to improve the independence on the initial covariances. The behavior of the error variances and correlations is shown to be particularly interes...

Sequential data assimilation with a nonlinear quasi‐geostrophic model using Monte Carlo methods to forecast error statistics

A new sequential data assimilation method is discussed. It is based on forecasting the error statistics using Monte Carlo methods, a better alternative than solving the traditional and computationally extremely demanding approximate error covariance equation used in the extended Kalman filter. The unbounded error growth found in the extended Kalman filter, which is caused by an overly simplified closure in the error covariance equation, is completely eliminated. Open boundaries can be handled as long as the ocean model is well posed. Well‐known numerical instabilities associated with the error covariance equation are avoided because storage and evolution of the error covariance matrix itself are not needed. The results are also better than what is provided by the extended Kalman filter since there is no closure problem and the quality of the forecast error statistics therefore improves. The method should be feasible also for more sophisticated primitive equation models. The computational load for reasonable accuracy is only a fraction of what is required for the extended Kalman filter and is given by the storage of, say, 100 model states for an ensemble size of 100 and thus CPU requirements of the order of the cost of 100 model integrations. The proposed method can therefore be used with realistic nonlinear ocean models on large domains on existing computers, and it is also well suited for parallel computers and clusters of workstations where each processor integrates a few members of the ensemble.

Exact lower time bounds for computing Boolean functions on CREW PRAMs

The time complexity of Boolean functions on abstract concurrent-read exclusive-write parallel random access machines (CREW PRAMs) is considered. We improve results of Cook, Dwork, and Reischuk ( SIAM J. Comput. 15 (1986), 87–97) , and extend work of Kutyłowski ( SIAM J. Comput. 20 (1991), 824–833) , who proved a lower time bound for the OR function on such machines that equals the upper bound. We provide a general means for obtaining exact (i.e., correct up to an additive constant) lower bounds, which works for many Boolean functions, in particular all symmetric functions. The new approach is based on the fact that Boolean functions can be represented as polynomials with integer coefficients and that the degree of such a polynomial can be taken as a complexity measure. For some functions, e.g., AND and PARITY, the exact time bound also holds for nondeterministic machines. For probabilistic machines, we obtain exact lower time bounds for PARITY in the unbounded error model and, utilizing results by Szegedy (Ph.D. dissertation, University of Chicago, 1989) , prove a general lower bound valid for all Boolean functions in the bounded error model. We further show that the (bounded error) probabilistic time complexity of Boolean functions on CREW PRAMs differs at most by a constant factor from the deterministic time complexity. We also obtain exact bounds for machines that allow a few processors to try to write to the same cell simultaneously. These bounds are stronger than those which follow automatically from the exclusive-write bounds. No tight bounds for this model were known before.

Skeletons from the Treecode Closet

We consider treecodes (N-body programs which use a tree data structure) from the standpoint of their worst-case behavior. That is, we derive upper bounds on the largest possible errors that are introduced into a calculation by use of various multipole acceptability criteria (MAC). We find that the conventional Barnes-Hut MAC can introduce potentially unbounded errors unless θ < 1/3, and that this behavior while rare, is demonstrable in astrophysically reasonable examples. We consider two other MACs closely related to the BH MAC. While they do not admit the same unbounded errors, they nevertheless require extraordinary amounts of CPU time to guarantee modest levels of accuracy. We derive new error bounds based on some additional, easily computed moments of the mass distribution. These error bounds form the basis for four new MACs which can be used to limit the absolute or relative error introduced by each multipole evaluation, or, with the introduction of some additional data structures, the absolute or rms error in the final acceleration of each particle. Using the Sum Squares MAC to analytically place a 1% bound on the rms error in a series of test models, we find that it significantly outperforms the θ = 0.65 BH MAC in terms of both accuracy (mean, rms, and maximum error) and performance (floating point operation count).

Restricted relativizations of probabilistic polynomial time

We consider several questions on the computational power of PP, the class of sets accepted by polynomial time-bounded probabilistic Turing machines with two-sided unbounded error probability. In particular, we consider the questions whether PP is as powerful as PSPACE and whether PP is more powerful than PH (the polynomial-time hierarchy). These questions have remained open. In fact, only a few results have been shown even in the relativized case. In the present paper we deal with the above questions in a manner similar to Long (1985), Balcázar (1986), and Long and Selman (1986). We introduce a quantitative restriction on access to a given oracle for probabilistic oracle machines. The restriction, denoted as PR R(·), allows polynomial time-bounded probabilistic oracle machines to query only polynomially many positive strings but exponentially many negative strings in its computation tree. We show that for every oracle A that PP R(NP)( A) is included in PP( A). This strengthens the result shown recently by Biegel (1989). In fact, we have their result as an immediate consequence. We next show some positive relativization results on the above questions. We show that seprating PP from PSPACE with some sparse oracle implies the separation of PP from PSPACE in the nonrelativized case. We also show that separating PP from PH with some sparse oracle implies the separation in the nonrelativized case. We also compare D # P and with PH. In particular, we show that D # P = PH iff D # ( S) = PH ( S) for some sparse oracle set S. Thus, the inequality D # P ≠ PH can also be relativized for all sparse oracle sets.

Seasonal sources and sinks of atmospheric CO&lt;sub&gt;2&lt;/sub&gt; Direct inversion of filtered data

A two-dimensional atmospheric transport model with transport fields derived from a GCM is used to perform the inverse calculation of determining surface CO 2 sources from surface concentration data. The concentration data are filtered spatially and temporally with a minimum period of 6 months and a maximum spatial wave number of 5 in order to suppress the unbounded error amplification that occurs in such inverse problems. The distribution of sources and sinks is consistent with recent results concerning oceanic uptake of CO 2 . The results also show an additional northern hemisphere sink that is interpreted as additional biotic growth and a seasonal tropical source that may represent seasonal burning of cleared biomass. DOI: 10.1111/j.1600-0889.1989.tb00129.x

Probabilistic communication complexity

Communication is a bottleneck in many distributed computations. In VLSI, communication constraints dictate lower bounds on the performance of chips. The two-processor information transfer model measures the communication requirements to compute functions. We study the unbounded error probabilistic version of this model. Because of its weak notion of correct output, we believe that this model measures the “intrinsic” communication complexity of functions. We present exact characterizations of the unbounded error communication complexity in terms of arrangements of hyperplanes and approximations of matrices. These characterizations establish the connection with certain classical problems in combinatorial geometry which are concerned with the configurations of points in d-dimensional real space. With the help of these characterizations, we obtain some upper and lower bounds on communication complexity. The upper bounds which we obtained for the functions—equality and verification of Hamming distance—are considerably better than their counterparts in the deterministic, the nondeterministic, and the bounded error probabilistic models. We also exhibit a function which has log n complexity. We present a counting argument to show that most functions have linear complexity. Further, we apply the logarithmic lower bound on communication complexity to obtain an Ω(n log n) bound on the time of 1-tape unbounded error probabilistic Turing machines. We believe that this is the first nontrivial lower bound obtained for such machines.