a new l curve for ill posed problems the truncated singular value decomposition is a popular method for the solution of linear ill posed problems the method requires the choice of a truncation index which affects the quality of the computed approximate solution this paper proposes that an l curve which is determined by how well the given data right hand side can be approximated by a linear combination of the first few left singular vectors or functions be used as an aid for determining the truncation index

structural induction in institutions abstract we develop a general logic independent structural induction proof method at the level of abstract institutions this provides a solid and uniform mathematical foundation to induction proof methodologies for a wide variety of actual logic based formal specification frameworks our development is based technically upon an axiomatic approach to substitutions within institution theory

regularization parameter determination for discrete ill posed problems the straightforward solution of discrete ill posed linear systems of equations or least squares problems with error contaminated data does not in general give meaningful results because the propagated error destroys the computed solution the problems have to be modified to reduce their sensitivity to the error in the data the amount of modification is determined by a regularization parameter it can be difficult to determine a suitable value of the regularization parameter when no knowledge of the norm of error in the data is available this paper proposes a new simple technique for determining a value of the regularization parameter that can be applied in this situation it is based on comparing computed solutions determined by tikhonov regularization and truncated singular value decomposition analogous comparisons are proposed for large scale problems the technique for determining the regularization parameter implicity provides an estimate for the norm of the error in the data

robust adaptive beamforming based on conjugate gradient algorithms the mismatches of signal and array geometry will seriously degrade the performance of adaptive beamformer in this paper we propose two methods for robust adaptive beamforming based on the conjugate gradient cg algorithm the proposed beamformers offer a significant improvement in the computational complexity while providing the same performance of the best robust beamformers at present the first method belongs to the diagonal loading technique we derive a diagonal loading cgls algorithm cg applied to normal equations and propose a simple method to choose the loading level based on a coarse estimation of the desired signal power this parameter free method can effectively reduce the signal self cancellation at high signal to noise ratio the second method belongs to the regularization technique since the cg algorithm has a regularizing effect with iteration number being the regularization parameter the stopping criterion plays an important role on the robustness we develop three fast stopping criteria for cg iteration which reduce the stopping complexity from o n or o n 2 to o 1 the former two are the fast versions of existing methods and the later one is new moreover the new criterion based on fast ritz value estimation has better performance than others

exploiting randomness in acoustic impulse responses to achieve headphone compensation through deconvolution this paper presents a method of headphone earphone equalization based upon deconvolution of the headphone impulse response from other acoustic filters in the processing chain the methods presented are thus applicable to areas such as spatial audio where input signals are processed with binaural impulse responses the extraction of low order from higher order acoustic impulse responses is justified based upon an application of the theory pertaining to the clustering of the zeros of random coefficient polynomials about the unit circle an argument is presented supporting the applicability of this theory to acoustic impulse responses such an equalization method guarantees a considerable reduction in computational effort over traditional separate compensating filters a comparison is presented of the implementation of the equalization using the homomorphic method and least squares method with and without regularization to a traditional separate least squares inverse filter the proposed approach is shown to have a performance comparable to a separate least squares designed compensator often giving superior performance despite a large reduction in complexity

some fundamental algebraic tools for the semantics of computation part 3 indexed categories abstract this paper presents indexed categories which model uniformly defined families of categories and suggests that they are a useful tool for the working computer scientist an indexed category gives rise to a single flattened category as a disjoint union of its component categories plus some additional morphisms similarly an indexed functor which is a uniform family of functors between the components categories induces a flattened functor between the corresponding flattened categories under certain assumptions flattened categories are co complete if all their components are and flattened functors have left adjoints if all their components do several examples are given although this paper is part 3 of the series some fundamental algebraic tools for the semantics of computation it is entirely independent of parts 1 and 2

estimating solutions of first kind integral equations with nonnegative constraints and optimal smoothing a method is presented for estimating solutions of fredholm integral equations of the first kind given noisy data regularization is effected by a smoothing term which is the l 2 norm of the estimate we propose a scheme by which an approximately optimal amount of smoothing may be computed based only on the data and the assumed known noise variances numerical examples are given for estimating inverse laplace transforms

some fundamental algebraic tools for the semantics of computation part 2 signed and abstract theories les theories algebriques dans le sens de lawvere sont abstraites en ce qui concerne les presentations operation donnee ni equations particulieres ne sont determinees par la classe d isomorphisme d une theorie de ce fait pas meme les rangs ou le nombre total d operations ne sont determines ce degre d abstraction n est pas approprie a la specification des concepts de programmation on a donc donne une signature particuliere

on an improved levin oscillatory quadrature method abstract performance of an improved levin quadrature method for oscillatory integrals is studied in the study the behavior of the target system of linear equations is analyzed and an error reduction factor is proposed to measure the behavior s impact on the integral result numerical investigations show that the error reduction factor is extremely small for ill conditioned case and the ill conditioning has little impact on the final integral result therefore the concerned quadrature method is numerically very stable and it has addressed the levin method s problem of being susceptible to the ill conditioning

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

sar image regularization with fast approximate discrete minimization synthetic aperture radar sar images like other coherent imaging modalities suffer from speckle noise the presence of this noise makes the automatic interpretation of images a challenging task and noise reduction is often a prerequisite for successful use of classical image processing algorithms numerous approaches have been proposed to filter speckle noise markov random field mrf modelization provides a convenient way to express both data fidelity constraints and desirable properties of the filtered image in this context total variation minimization has been extensively used to constrain the oscillations in the regularized image while preserving its edges speckle noise follows heavy tailed distributions and the mrf formulation leads to a minimization problem involving nonconvex log likelihood terms such a minimization can be performed efficiently by computing minimum cuts on weighted graphs due to memory constraints exact minimization although theoretically possible is not achievable on large images required by remote sensing applications the computational burden of the state of the art algorithm for approximate minimization namely the alpha expansion is too heavy specially when considering joint regularization of several images we show that a satisfying solution can be reached in few iterations by performing a graph cut based combinatorial exploration of large trial moves this algorithm is applied to joint regularization of the amplitude and interferometric phase in urban area sar images

on fuzzification of topological categories this paper shows that l m fuzzy topology of u hohle t kubiak and a sostak is an instance of a general fuzzification procedure for topological categories which amounts to the construction of a new topological category from a given one this fuzzification procedure motivates a partial dualization of the machinery of tower extension of topological constructs of d zhang thereby providing the procedure of tower extension of topological categories with the help of this dualization we arrive at the meta mathematical result that the concept of l m fuzzy topology and the notion of approach space of r lowen are dual to each other

interpolation in grothendieck institutions it is well known that interpolation properties of logics underlying specification formalisms play an important rule in the study of structured specifications they have also many other useful logical consequences in this paper we solve the interpolation problem for grothendieck institutions which have recently emerged as an important mathematical structure underlying heterogenous multilogic specification our main result can be used in the applications in several different ways it can be used to establish interpolation properties for multi logic grothendieck institutions but also to lift interpolation properties from unsorted logics to their many sorted variants the importance of the latter resides in the fact that unlike other structural properties of logics many sorted interpolation is a non trivial generalisation of unsorted interpolation the concepts results and the applications discussed in this paper are illustrated with several examples from conventional logic and algebraic specification theory

functorial theory of parameterized specifications in a general specification framework abstract a general specification framework based on the notion of indexed categories is introduced in order to study the structural aspects of specifications independent of the underlying logics similar to institutions this concept of specification frames allows to formulate a unified structural theory of various kinds of algebraic specifications which have been studied separately in the literature before in contrast to institutions we do not require to have satisfaction relations and conditions which allows to handle also behavioural specifications and semantics and various concepts of constraints in this framework in this framework we generalize the well known theory of parameterized algebraic specifications with initial semantics from the equational case to specification frames satisfying mainly three basic axioms the existence of pushouts free constructions and amalgamation moreover an axiomatic treatment of restriction is presented which allows to study in addition to refinement also implementations of parameterized specifications including restrictions finally we present an axiomatic framework for functorial semantics which opens the way to apply the theory not only to initial semantics but also to other kinds of functorial semantics including final and specific kinds of loose semantics

efficient maximum entropy reconstruction of nuclear magnetic resonance t1 t2 spectra this paper deals with the reconstruction of t1 t2 correlation spectra in nuclear magnetic resonance relaxometry the ill posed character and the large size of this inverse problem are the main difficulties to tackle while maximum entropy is retained as an adequate regularization approach the choice of an efficient optimization algorithm remains a challenging task our proposal is to apply a truncated newton algorithm with two original features first a theoretically sound line search strategy suitable for the entropy function is applied to ensure the convergence of the algorithm second an appropriate preconditioning structure based on a singular value decomposition of the forward model matrix is used to speed up the algorithm convergence furthermore we exploit the specific structures of the observation model and the hessian of the criterion to reduce the computation cost of the algorithm the performances of the proposed strategy are illustrated by means of synthetic and real data processing

parameter determination for tikhonov regularization problems in general form abstract tikhonov regularization is one of the most popular methods for computing an approximate solution of linear discrete ill posed problems with error contaminated data a regularization parameter 0 balances the influence of a fidelity term which measures how well the data are approximated and of a regularization term which dampens the propagation of the data error into the computed approximate solution the value of the regularization parameter is important for the quality of the computed solution a too large value of 0 gives an over smoothed solution that lacks details that the desired solution may have while a too small value yields a computed solution that is unnecessarily and possibly severely contaminated by propagated error when a fairly accurate estimate of the norm of the error in the data is known a suitable value of often can be determined with the aid of the discrepancy principle this paper is concerned with the situation when the discrepancy principle cannot be applied it then can be quite difficult to determine a suitable value of we consider the situation when the tikhonov regularization problem is in general form i e when the regularization term is determined by a regularization matrix different from the identity and describe an extension of the cose method for determining the regularization parameter in this situation this method has previously been discussed for tikhonov regularization in standard form i e for the situation when the regularization matrix is the identity it is well known that tikhonov regularization in general form with a suitably chosen regularization matrix can give a computed solution of higher quality than tikhonov regularization in standard form

inversion of nuclear magnetic resonance echo data based on maximum entropy abstractnuclear magnetic resonance nmr t2 inversion is an ill posed problem in which regularization techniques are usually adopted to suppress the oscillations caused by noise in the solutions the maximum entropy concept provides an unbiased way to obtain information from incomplete data and it implicitly imposes a positive constraint on probability distribution so we used the maximum entropy method to invert nmr echo data we have developed a simple and effective method for solving the objective function of the maximum entropy method first the solution was replaced by a positive function to achieve the positive constraint of the solution the objective function was converted to an unconstrained one and then the levenberg marquardt method was used to solve the newly obtained unconstrained objective function to suppress the highly tilted tail at the short relaxation time of the t2 distribution a modified or normalized shannon entropy function was used to replace the standard shannon entropy funct

schultz matrix iteration based method for stable solution of discrete ill posed problems abstract in this paper we propose an iterative method for solving discrete ill posed problems based on matrix iterations generated by schultz method and known to converge to the moore penrose pseudoinverse of a matrix practically letting the schultz matrix iteration be x k we construct the vector x k x k b where b is a data vector hence by construction the iterates converge to the minimum 2 norm solution of a least squares problem with coefficient matrix a and data vector b we derive theoretical properties of the sequence x k and show that it is quadratically convergent in the case of corrupted data we analyze the semi convergence behavior of the iterates and conclude that the iteration must be truncated to control the propagation of the noise error as a result we derive an error estimate for the case where the truncation parameter is chosen by the discrepancy principle in addition combining a projected approach with the new method we propose variants of the method that are well suited for large scale problems several numerical results are presented to illustrate the effectiveness of the method on well known test problems

on tori without conjugate points in this paper we consider riemannian metrics without conjugate points on an n torus recent work of j heber established that the gradient vector fields of busemann functions on the universal cover of such a manifold induce a natural foliation akin to the weak stable foliation for a riemannian manifold with negative sectional curvature on the unit tangent bundle the main result in the paper is that the metric is flat if this foliation is lipschitz we also prove that this foliation is lipschitz if and only if the metric has bounded asymptotes this confirms a conjecture of e hopf in this case

an adaptive pruning algorithm for the discrete l curve criterion we describe a robust and adaptive implementation of the l curve criterion the algorithm locates the corner of a discrete l curve which is a log log plot of corresponding residual norms and solution norms of regularized solutions from a method with a discrete regularization parameter such as truncated svd or regularizing cg iterations our algorithm needs no predefined parameters and in order to capture the global features of the curve in an adaptive fashion we use a sequence of pruned l curves that correspond to considering the curves at different scales we compare our new algorithm to existing algorithms and demonstrate its robustness by numerical examples

relations between svd and gsvd of discrete regularization problems in standard and general form abstract we consider an algorithm due to elden that transforms general discrete regularization problems into standard form we derive a simple relationship between the svd associated with the standard form problem and the gsvd associated with the general problem and we discuss the accuracy of the gsvd computed via this transformation

on the existence of free models in abstract algebraic institutions abstract to provide a formal framework for discussing specifications of abstract data types we restrict the notion of institution due to goguen and burstall 1984 which formalises the concept of a logical system for writing specifications and deal with abstract algebraic institutions these are institutions equipped with a notion of submodel which satisfy a number of technical conditions our main results concern the problem of the existence of free constructions in abstract algebraic institutions we generalise a characterization of algebraic specification languages that guarantee the existence of reachable initial models for any consistent set of axioms given by mahr and makowsky 1984 then the more general problem of the existence of free functors left adjoints to forgetful functors for any theory morphism is analysed we give a construction of a free model of a theory over a model of a subtheory with respect to an arbitrary theory morphism which requires only the existence of initial models this yields a characterization of strongly liberal abstract algebraic institutions we also show how to specialise there characterisation results for the partial algebras

solving fredholm integrals of the first kind with tensor product structure in 2 and 2 5 dimensions we present an efficient algorithm to solve a class of two and 2 5 dimensional 2 d and 2 5 d fredholm integrals of the first kind with a tensor product structure and nonnegativity constraint on the estimated parameters of interest in an optimization framework a zeroth order regularization functional is used to incorporate a priori information about the smoothness of the parameters into the problem formulation we adapt the butler reeds dawson 1981 algorithm to solve this optimization problem in three steps in the first step the data are compressed using singular value decomposition svd of the kernels the tensor product structure of the kernel is exploited so that the compressed data is typically a thousand fold smaller than the original data this size reduction is crucial for fast optimization in the second step the constrained optimization problem is transformed to an unconstrained optimization problem in the compressed data space in the third step a suboptimal value of the smoothing parameter is chosen by the brd method steps 2 and 3 are iterated until convergence of the algorithm we demonstrate the performance of the algorithm on simulated data

a new method for nmr data inversion based on double parameter regularization nuclear magnetic resonance nmr technology plays a significant role in petroleum exploration nmr data can be processed using inversion methods to reflect the relaxation information of all the components a new double parameter regularization dpr method was proposed for the inversion of nmr data whose regularization terms consist of tikhonov regularization and maximum entropy regularization the objective function for the dpr method was solved using the levenberg marquardt method the proportional coefficient of the regularization parameter was obtained using an iteration procedure the optimum regularization parameter of the dpr method was selected using an s curve the relationship between the optimum regularization parameter and the signal to noise ratio snr of the data was evaluated moreover we compared the results of the nmr inversion obtained from the norm smooth method the maximum entropy method and the dpr method for simulated data we evaluated how the proportional coefficient of the re

an axiomatic approach to structuring specifications in this paper we develop an axiomatic approach to structured specifications in which both the underlying logical system and corresponding institution of the structured specifications are treated as abstract institutions which means two levels of institution independence this abstract axiomatic approach provides a uniform framework for the study of structured specifications independently from any actual choice of specification building operators and moreover it unifies the theory and the model oriented approaches within this framework we develop concepts and results about abstract structured specifications such as co limits model amalgamation compactness interpolation sound and complete proof theory and pushout style parameterization with sharing all of them in a top down manner dictated by the upper level of institution independence