a comparison between maximum likelihood and generalized least squares in a heteroscedastic linear model abstract we consider a linear model with normally distributed but heteroscedastic errors when the error variances are functionally related to the regression parameter one can use either maximum likelihood or generalized least squares to estimate the regression parameter we show that likelihood is more sensitive to small misspecifications in the functional relationship between the error variances and the regression parameter

fast structured direct spectral methods for differential equations with variable coefficients i the one dimensional case we study the rank structures of the matrices in fourier and chebyshev spectral methods for differential equations with variable coefficients in one dimension we show analytically that these matrices have a so called low rank property not only for constant or smooth variable coefficients but also for coefficients with steep gradients and or high variations large ratios in their maximum minimum function values we develop a matrix free direct spectral solver which uses only a small number of matrix vector products to construct a structured approximation to the original discretized matrix a without the need to explicitly form a this is followed by fast structured matrix factorizations and solutions the overall direct spectral solver has o n log 2 n complexity and o n memory requirement numerical tests for several important but notoriously difficult problems show the superior efficiency and accuracy of our direct spectral solver especially when iterative methods have severe difficulties

corrections to fast multilevel computation of low rank representation of mathcal h matrix blocks several asymptotic complexity expressions in 1 were overestimated and require correction all the errors resulted from the same incorrect assumption at the end of section ii the computational complexity of calculating the singular value decomposition svd of an m times n matrix mathbf a was presented as mathcal o m 2 n n 3 the cost of computing the full svd mathbf a mathbf u sigma mathbf v dagger using the r svd algorithm 2 assuming m ge n otherwise m and n are interchanged the algorithm in 1 however uses the svd for computing sigma mathbf v dagger and only the first n columns of mathbf u at an mathcal o mn 2 n 3 cost only 2 indeed the standard svd implementations ensure that this computation costs mathcal o mn min m n 3 accordingly 1 should have used svd costs of mathcal o m 2 n mathcal o mn 2 rather than mathcal o n 3 mathcal o m 2 n when m using these impacts several complexity expressions in 1 and its conclusion as detailed next

vers une decomposition de jordan des blocs des groupes reductifs finis abstract let g be a reductive algebraic group over an algebraic closure of a prime field f p defined over f q with frobenius endomorphism f let g f be the subgroup of rational points let l be a prime number assume that l is different from p if g f is in duality with g f then by a theorem of m broue and j michel m broue j michel blocs et series de lusztig dans un groupe reductif fini j reine angew math 395 1989 56 67 for any l bloc b of g f there exists a unique g f conjugacy class s of l semi simple elements such that some irreducible representation of b is in the rational lusztig s series associated to s if s 1 b is said to be unipotent if g is not connected with identity component g define the unipotent l blocs of g f as the l blocks that cover some unipotent l bloc of g f assuming l is good for g we construct from g f and s some reductive algebraic group g s f defined over f q and a one to one map from the set of l blocks of g f with associated class s onto the set of unipotent l blocks of g s f such that if b corresponds to b then there is a significant height preserving one to one map from the set of irreducible representations irr b onto the set irr b the respective defect groups of b and b are isomorphic the associated brauer categories are equivalent one do not assume that the center of g is connected the class of s may be isolated in g

randomized algorithms for the low rank approximation of matrices we describe two recently proposed randomized algorithms for the construction of low rank approximations to matrices and demonstrate their application inter alia to the evaluation of the singular value decompositions of numerically low rank matrices being probabilistic the schemes described here have a finite probability of failure in most cases this probability is rather negligible 10 17 is a typical value in many situations the new procedures are considerably more efficient and reliable than the classical deterministic ones they also parallelize naturally we present several numerical examples to illustrate the performance of the schemes

conjectures of alperin and broue for 2 blocks with elementary abelian defect groups of order 8 using the classification of finite simple groups we prove alperin s weight conjecture and the character theoretic version of broue s abelian defect group conjecture for 2 blocks of finite groups with an elementary abelian defect group of order 8

domination of blocks fusion systems and hyperfocal subgroups abstract we aim in this paper to investigate the connections between some properties of blocks and of their dominating blocks we find conditions to verify that a block is inertial if and only if its dominating block is inertial in some situations the equality of the factor fusion systems associated with a block and with its brauer correspondent block give information about the hyperfocal subgroups

the method of polarized traces for the 2d helmholtz equation we present a solver for the 2d high frequency helmholtz equation in heterogeneous acoustic media with online parallel complexity that scales optimally as o n l where n is the number of volume unknowns and l is the number of processors as long as l grows at most like a small fractional power of n the solver decomposes the domain into layers and uses transmission conditions in boundary integral form to explicitly define polarized traces i e up and down going waves sampled at interfaces local direct solvers are used in each layer to precompute traces of local green s functions in an embarrassingly parallel way the offline part and incomplete green s formulas are used to propagate interface data in a sweeping fashion as a preconditioner inside a gmres loop the online part adaptive low rank partitioning of the integral kernels is used to speed up their application to interface data the method uses second order finite differences the complexity scalings are empirical but motivated by an analysis of ranks of off diagonal blocks of oscillatory integrals they continue to hold in the context of standard geophysical community models such as bp and marmousi 2 where convergence occurs in 5 to 10 gmres iterations while the parallelism in this paper stems from decomposing the domain we do not explore the alternative of parallelizing the systems solves with distributed linear algebra routines

a theory of pseudoskeleton approximations abstract let an m n matrix a be approximated by a rank r matrix with an accuracy e we prove that it is possible to choose r columns and r rows of a forming a so called pseudoskeleton component which approximates a with o e r m n accuracy in the sense of the 2 norm on the way to this estimate we study the interconnection between the volume i e the determinant in the absolute value and the minimal singular value r of r r submatrices of an n r matrix with orthogonal columns we propose a lower bound better than one given by chandrasekaran and ipsen and by hong and pan for the maximum of r over all these submatrices and formulate a hypothesis on a tighter bound

fast monte carlo algorithms for matrices iii computing a compressed approximate matrix decomposition in many applications the data consist of or may be naturally formulated as an m times n matrix a which may be stored on disk but which is too large to be read into random access memory ram or to practically perform superlinear polynomial time computations on it two algorithms are presented which when given an m times n matrix a compute approximations to a which are the product of three smaller matrices c u and r each of which may be computed rapidly let a cur be the computed approximate decomposition both algorithms have provable bounds for the error matrix a a in the first algorithm c columns of a and r rows of a are randomly chosen if the m times c matrix c consists of those c columns of a after appropriate rescaling and the r times n matrix r consists of those r rows of a also after appropriate rescaling then the c times r matrix u may be calculated from c and r for any matrix x let x f and x 2 denote its frobenius norm and its spectral norm respectively it is proven that left a a right xi le min d mathrm rank d le k left a d right xi poly k 1 c left a right f holds in expectation and with high probability for both xi 2 f and for all k 1 ldots mbox rank a thus by appropriate choice of k left a a right 2 le epsilon left a right f also holds in expectation and with high probability this algorithm may be implemented without storing the matrix a in ram provided it can make two passes over the matrix stored in external memory and use o m n additional ram assuming that c and r are constants independent of the size of the input the second algorithm is similar except that it approximates the matrix c by randomly sampling a constant number of rows of c thus it has additional error but it can be implemented in three passes over the matrix using only constant additional ram to achieve an additional error beyond the best rank k approximation that is at most epsilon a f both algorithms take time which is a low degree polynomial in k 1 epsilon and 1 delta where delta 0 is a failure probability the first takes time linear in mbox max m n and the second takes time independent of m and n the proofs for the error bounds make important use of matrix perturbation theory and previous work on approximating matrix multiplication and computing low rank approximations to a matrix the probability distribution over columns and rows and the rescaling are crucial features of the algorithms and must be chosen judiciously

sparsity and its applications this volume is based on a series of lectures prepared by a group of internationally renowned experts in the field for a meeting on the general theme of sparsity and its applications at the university of technology loughborough in april 1983 an introductory section briefly surveys the field of sparsity research and summarises the main ideas discussed in each contribution the contributions themselves examine the design analysis implementation and application of computer algorithms with a view to producing mathematical software which can be used as an efficent tool in different areas of science and engineering technology

on stable equivalences and blocks with one simple module using a stable equivalence due to rouquier we show that alperin s weight conjecture holds for any p block of a finite group with defect 2 whose brauer correspondent has a unique isomorphism class of simple modules

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

on the fast matrix multiplication in the boundary element method by panel clustering the boundary element method bem leads to a system of linear equations with a full matrix while fem yields sparse matrices this fact seems to require much computational work for the definition of the matrix for the solution of the system and in particular for the matrix vector multiplication which always occurs as an elementary in this paper a method for the approximate matrix vector multiplication is described which requires much less arithmetical work in addition the storage requirements are strongly reduced

blocks with defect group z2n z2n z2m abstract in this paper we prove that a block algebra with defect group z 2 n z 2 n z 2 m where n 2 and m is arbitrary is morita equivalent to its brauer correspondent

2 blocks whose defect group is homocyclic and whose inertial quotient contains a singer cycle abstract we consider a block b of a finite group with defect group d c 2 m n and inertial quotient e containing a singer cycle an element of order 2 n 1 this implies e e f where e c 2 n 1 f c n and e acts transitively on the elements in d of order 2 we classify the basic morita equivalence classes of b over a complete discrete valuation ring o when m 1 b is basic morita equivalent to the principal block of one of s l 2 2 n f d e or j 1 where j 1 occurs only when n 3 when m 1 b is basic morita equivalent to d e

a sparse approximate inverse preconditioner for the conjugate gradient method a method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix a is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations it is proved that in exact arithmetic the preconditioner is well defined if a is an h matrix the results of numerical experiments are presented

a heterogeneous stochastic fem framework for elliptic pdes we introduce a new concept of sparsity for the stochastic elliptic operator div a x which reflects the compactness of its inverse operator in the stochastic direction and allows for spatially heterogeneous stochastic structure this new concept of sparsity motivates a heterogeneous stochastic finite element method hsfem framework for linear elliptic equations which discretizes the equations using the heterogeneous coupling of spatial basis with local stochastic basis to exploit the local stochastic structure of the solution space we also provide a sampling method to construct the local stochastic basis for this framework using the randomized range finding techniques the resulting hsfem involves two stages and suits the multi query setting in the offline stage the local stochastic structure of the solution space is identified in the online stage the equation can be efficiently solved for multiple forcing functions an online error estimation and correction procedure through monte carlo sampling is given numerical results for several problems with high dimensional stochastic input are presented to demonstrate the efficiency of the hsfem in the online stage

on donovan s conjecture abstract this article contains reduction theorems for some weaker variants of donovan s conjecture which supposes that for every finite group d of prime order p there are only finitely many morita equivalence classes of p blocks of finite groups having defect groups isomorphic to d i when restricting the conjecture to the case of abelian defect groups it can be reduced to only considering p blocks of a class of groups that are close to being direct products of simple groups ii when restricting the conjecture to the case of principal blocks of finite groups having an abelian sylow p subgroup it can be reduced to only considering p blocks of simple groups a weaker version of donovan s conjecture would suppose that there are only finitely many cartan matrices of p blocks of finite groups with defect group isomorphic to a given one iii this conjecture can be reduced to only considering p blocks of quasisimple groups with centers having orders prime to p

quasi isolated blocks and brauer s height zero conjecture this paper has two main results firstly we complete the parametrisation of all p blocks of nite quasi simple groups by nding the so called quasi isolated blocks of exceptional groups of lie type for bad primes this relies on the explicit decomposition of lusztig induction from suitable levi subgroups our second major result is the proof of one direction of brauer s long standing height zero conjecture on blocks of nite groups using the

the structure of blocks with a klein four defect group we prove erdmann s conjecture stating that every block with a klein four defect group has a simple module with trivial source and deduce from this that puig s finiteness conjecture holds for source algebras of blocks with a klein four defect group the proof uses the classification of finite simple groups

diagnostics for heteroscedasticity in regression summary for the usual regression model without replication we provide a diagnostic test for heteroscedasticity based on the score statistic a graphical procedure to complement the score test is also presented some key u ords influence linear model residual score test diagnostic methods in linear regression are used to examine the appropriateness of assumptions underlying the modelling process and to locate unusual characteristics of the data that may influence conclusions the recent literature on diagnostics is dominated by studies of methods for the detection of influential observations cook weisberg 1982 provide a review diagnostics for the relevance of specific assumptions however have not received the same degree of attention even though these may be of equal importance our purpose here is to provide appropriate diagnostic techniques to aid in an assessment of the validity of the usual assumption of homoscedasticity when little or no replication is present available methods for studying this assumption include both graphical and nongraphical procedures the usual graphical procedure consists of plotting the ordinary least squares residuals against fitted values or an explanatory variable a megaphone shaped pattern is taken as evidence that the variance depends on the quantity plotted on tlle abscissa weisberg 1980 chapter 6 in 3 we suggest several ways in which this

perfect isometries for blocks with abelian defect groups and cyclic inertial quotients of order 4 abstract let b be a p block of a finite group g with an abelian defect group p and e a root of b in c g p if the inertial quotient e n g p e p c g p is a cyclic group of order 4 there is a perfect isometry from the group of generalized characters of the group algebra of the semidirect product of e and p onto the group of generalized characters of g in b

2 blocks with abelian defect groups we give a classification up to morita equivalence of 2 blocks of quasi simple groups with abelian defect groups as a consequence we show that donovan s conjecture holds for elementary abelian 2 groups and that the entries of the cartan matrices are bounded in terms of the defect for arbitrary abelian 2 groups we also show that a block with defect groups of the form c2m c2m for m 2 has one of two morita equivalence types and hence is morita equivalent to the brauer correspondent block of the normaliser of a defect group this completes the analysis of the morita equivalence types of 2 blocks with abelian defect groups of rank 2 from which we conclude that donovan s conjecture holds for such 2 groups a further application is the completion of the determination of the number of irreducible characters in a block with abelian defect groups of order 16 the proof uses the classification of finite simple groups

existence of ℋ matrix approximants to the inverse fe matrix of elliptic operators with l coefficients this article deals with the existence of blockwise low rank approximants so called matrices to inverses of fem matrices in the case of uniformly elliptic operators with l coefficients unlike operators arising from boundary element methods for which the matrix theory has been extensively developed the inverses of these operators do not benefit from the smoothness of the kernel function however it will be shown that the corresponding green functions can be approximated by degenerate functions giving rise to the existence of blockwise low rank approximants of fem inverses numerical examples confirm the correctness of our estimates as a side product we analyse the matrix property of the inverse of the fe mass matrix

adaptive finite element method for fractional differential equations using hierarchical matrices abstract a robust and fast solver for the fractional differential equation fdes involving the riesz fractional derivative is developed using an adaptive finite element method it is based on the utilization of hierarchical matrices h matrices for the representation of the stiffness matrix resulting from the finite element discretization of the fdes we employ a geometric multigrid method for the solution of the algebraic system of equations we combine it with an adaptive algorithm based on a posteriori error estimation a posteriori error estimation based adaptive algorithm is used to deal with general type singularities arising in the solution of the fdes through various test examples we demonstrate the efficiency of the method and the high accuracy of the numerical solution even in the presence of singularities the proposed technique has been verified effectively through fundamental examples including riesz left right riemann liouville fractional derivative and furthermore it can be readily extended to more general fractional differential equations with different boundary conditions and low order terms

a fast summation tree code for matern kernel the matern family of functions is a widely used covariance kernel in spatial statistics for gaussian process modeling which in many instances requires calculations with a covariance matrix in this paper we design a fast summation algorithm for the matern kernel in order to efficiently perform matrix vector multiplications this algorithm is based on the barnes hut tree code framework and addresses several practical issues the anisotropy of the kernel the nonuniform distribution of the point set and a tight error estimate of the approximation even though the algorithmic details differ from the standard tree code in several aspects empirically the computational cost of our algorithm scales as o n log n f orn points comprehensive numerical experiments are shown to demonstrate the practicality of the design

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

fast construction of hierarchical matrix representation from matrix vector multiplication we develop a hierarchical matrix construction algorithm using matrix vector multiplications based on the randomized singular value decomposition of low rank matrices the algorithm uses o logn applications of the matrix on structured random test vectors and o nlogn extra computational cost where n is the dimension of the unknown matrix numerical examples on constructing green s functions for elliptic operators in two dimensions show efficiency and accuracy of the proposed algorithm

metric based upscaling we consider divergence form elliptic operators in dimension n ge 2 with l coefficients although solutions of these operators are only holder continuous we show that they are differentiable c1 with respect to harmonic coordinates it follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales this new numerical homogenization method is based on the transfer of a new metric in addition to traditional averaged homogenized quantities from subgrid scales into computational scales error bounds can be given and this method can also be used as a compression tool for differential operators 2006 wiley periodicals inc

butterfly factorization via randomized matrix vector multiplications this paper presents an adaptive randomized algorithm for computing the butterfly factorization of an m times n matrix with m approx n provided that both the matrix and its transpose can be rapi

fast monte carlo algorithms for matrices i approximating matrix multiplication motivated by applications in which the data may be formulated as a matrix we consider algorithms for several common linear algebra problems these algorithms make more efficient use of computational resources such as the computation time random access memory ram and the number of passes over the data than do previously known algorithms for these problems in this paper we devise two algorithms for the matrix multiplication problem suppose a and b which are m times n and n times p respectively are the two input matrices in our main algorithm we perform c independent trials where in each trial we randomly sample an element of 1 2 ldots n with an appropriate probability distribution cal p on 1 2 ldots n we form an m times c matrix c consisting of the sampled columns of a each scaled appropriately and we form a c times n matrix r using the corresponding rows of b again scaled appropriately the choice of cal p and the column and row scaling are crucial features of the algorithm when these are chosen judiciously we show that cr is a good approximation to ab more precisely we show that left ab cr right f o left a right f left b right f sqrt c where cdot f denotes the frobenius norm i e a 2 f sum i j a ij 2 this algorithm can be implemented without storing the matrices a and b in ram provided it can make two passes over the matrices stored in external memory and use o c m n p additional ram to construct c and r we then present a second matrix multiplication algorithm which is similar in spirit to our main algorithm in addition we present a model the pass efficient model in which the efficiency of these and other approximate matrix algorithms may be studied and which we argue is well suited to many applications involving massive data sets in this model the scarce computational resources are the number of passes over the data and the additional space and time required by the algorithm the input matrices may be presented in any order of the entries and not just row or column order as is the case in many applications where e g the data has been written in by multiple agents in addition the input matrices may be presented in a sparse representation where only the nonzero entries are written

fast multilevel computation of low rank representation of mathcal h matrix blocks a physics based algorithm for accelerating the computation of method of moments matrix blocks low rank approximation is presented the algorithm relies on efficient sampling of phase and amplitude compensated interactions using nonuniform grids rank revealing analysis is applied in a multilevel fashion to matrices of reduced column and row dimensions that describe subdomains interactions with these coarse grids rather than to the original matrix blocks as a result significant savings are achieved especially for the inherently more compressible dynamic quasi planar and quasi static cases the algorithm s reduced storage and computation time requirements are estimated analytically and verified numerically for representative examples

fast monte carlo algorithms for matrices ii computing a low rank approximation to a matrix in many applications the data consist of or may be naturally formulated as an m times n matrix a it is often of interest to find a low rank approximation to a i e an approximation d to the matrix a of rank not greater than a specified rank k where k is much smaller than m and n methods such as the singular value decomposition svd may be used to find an approximation to a which is the best in a well defined sense these methods require memory and time which are superlinear in m and n for many applications in which the data sets are very large this is prohibitive two simple and intuitive algorithms are presented which when given an m times n matrix a compute a description of a low rank approximation d to a and which are qualitatively faster than the svd both algorithms have provable bounds for the error matrix a d for any matrix x let x f and x 2 denote its frobenius norm and its spectral norm respectively in the first algorithm c columns of a are randomly chosen if the m times c matrix c consists of those c columns of a after appropriate rescaling then it is shown that from c tc approximations to the top singular values and corresponding singular vectors may be computed from the computed singular vectors a description d of the matrix a may be computed such that mathrm rank d le k and such that left a d right xi 2 le min d mathrm rank d le k left a d right xi 2 poly k 1 c left a right 2 f holds with high probability for both xi 2 f this algorithm may be implemented without storing the matrix a in random access memory ram provided it can make two passes over the matrix stored in external memory and use o cm c 2 additional ram the second algorithm is similar except that it further approximates the matrix c by randomly sampling r rows of c to form a r times c matrix w thus it has additional error but it can be implemented in three passes over the matrix using only constant additional ram to achieve an additional error beyond the best rank k approximation that is at most epsilon a 2 f both algorithms take time which is polynomial in k 1 epsilon and log 1 delta where delta 0 is a failure probability the first takes time linear in mbox max m n and the second takes time independent of m and n our bounds improve previously published results with respect to the rank parameter k for both the frobenius and spectral norms in addition the proofs for the error bounds use a novel method that makes important use of matrix perturbation theory the probability distribution over columns of a and the rescaling are crucial features of the algorithms which must be chosen judiciously

effective matrix free preconditioning for the augmented immersed interface method we present effective and efficient matrix free preconditioning techniques for the augmented immersed interface method aiim aiim has been developed recently and is shown to be very effective for interface problems and problems on irregular domains gmres is often used to solve for the augmented variable s associated with a schur complement a in aiim that is defined along the interface or the irregular boundary the efficiency of aiim relies on how quickly the system for a can be solved for some applications there are substantial difficulties involved such as the slow convergence of gmres particularly for free boundary and moving interface problems and the inconvenience in finding a preconditioner due to the situation that only the products of a and vectors are available here we propose matrix free structured preconditioning techniques for aiim via adaptive randomized sampling using only the products of a and vectors to construct a hierarchically semiseparable matrix approximation to a several improvements over existing schemes are shown so as to enhance the efficiency and also avoid potential instability the significance of the preconditioners includes 1 they do not require the entries of a or the multiplication of a t with vectors 2 constructing the preconditioners needs only o log n matrix vector products and o n storage where n is the size of a 3 applying the preconditioners needs only o n flops 4 they are very flexible and do not require any a priori knowledge of the structure of a the preconditioners are observed to significantly accelerate the convergence of gmres with heuristical justifications of the effectiveness comprehensive tests on several important applications are provided such as navier stokes equations on irregular domains with traction boundary conditions interface problems in incompressible flows mixed boundary problems and free boundary problems the preconditioning techniques are also useful for several other problems and methods

scopes reduction and morita equivalence classes of blocks in finite classical groups ii abstract we extend our work of j algebra 230 2000 378 423 on unipotent blocks of classical groups to blocks of orthogonal groups containing characters labelled by degenerate symbols we also show that for classical groups in the linear prime case there are only finitely many morita types of unipotent blocks with a given defect group