In linear inverse problems, we have data derived from a noisy linear transformation of some unknown parameters, and we wish to estimate these unknowns from the data. Separable inverse problems are a powerful generalization in which the transformation itself depends on additional unknown parameters and we wish to determine both sets of parameters simultaneously. When separable problems are solved by optimization, convergence can often be accelerated by elimination of the linear variables, a strategy which appears most prominently in the variable projection methods due to Golub and Pereyra. Existing variable elimination methods require an explicit formula for the optimal value of the linear variables, so they cannot be used in problems with Poisson likelihoods, bound constraints, or other important departures from least squares. To address this limitation, we propose a generalization of variable elimination in which standard optimization methods are modified to behave as though a variable has been eliminated. We verify that this approach is a proper generalization by using it to re-derive several existing variable elimination techniques. We then extend the approach to bound-constrained and Poissonian problems, showing in the process that many of the best features of variable elimination methods can be duplicated in our framework. Tests on difficult exponential sum fitting and blind deconvolution problems indicate that the proposed approach can have significant speed and robustness advantages over standard methods.
Structured linear algebra techniques such as block Gaussian elimination are known to be useful @cite_18 @cite_10 , but they are underutilized in practice. This is apparent from the fact that most optimization codes employ a limited set of broadly applicable linear algebra techniques @cite_7 , and very few are designed to accommodate user-defined linear solvers such as the ones we propose in . We contend that since significant speed gains are attainable with special linear solvers, optimization algorithm implementations should accommodate user-customized linear algebra by adding appropriate callback and reverse communication protocols.
@cite_18 @cite_10 @cite_7
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With the emergence of thread-level parallelism as the primary means for continued performance improvement, the programmability issue has reemerged as an obstacle to the use of architectural advances. We argue that evolving legacy libraries for dense and banded linear algebra is not a viable solution due to constraints imposed by early design decisions. We propose a philosophy of abstraction and separation of concerns that provides a promising solution in this problem domain. The first abstraction, FLASH, allows algorithms to express computation with matrices consisting of contiguous blocks, facilitating algorithms-by-blocks. Operand descriptions are registered for a particular operation a priori by the library implementor. A runtime system, SuperMatrix, uses this information to identify data dependencies between suboperations, allowing them to be scheduled to threads out-of-order and executed in parallel. But not all classical algorithms in linear algebra lend themselves to conversion to algorithms-by-blocks. We show how our recently proposed LU factorization with incremental pivoting and a closely related algorithm-by-blocks for the QR factorization, both originally designed for out-of-core computation, overcome this difficulty. Anecdotal evidence regarding the development of routines with a core functionality demonstrates how the methodology supports high productivity while experimental results suggest that high performance is abundantly achievable. This paper presents a fast direct solver for 3D discretized linear systems using the supernodal multifrontal method together with low-rank approximations. For linear systems arising from certain partial differential equations (PDEs) such as elliptic equations, during the Gaussian elimination of the matrices with Nested Dissection ordering, the fill-in of L and U factors loses its sparsity and contains dense blocks with low-rank property. Off-diagonal blocks can be efficiently approximated with low-rank matrices; diagonal blocks approximated with semiseparable structures called hierarchically semiseparable (HSS) representations. Matrix operations in the multifrontal method are performed in low-rank arithmetic. We present efficient way to organize the HSS structured operations along the elimination. To compress dense blocks into low-rank or HSS structures, we use effective cross approximation (CA) approach. We also use idea of adaptive balancing between robust arithmetic for computing the small dense blocks and low-rank matrix operations for handling with compressed ones while performing the Gaussian elimination. This new proposed solver can be essentially parallelized both on architecture with shared and distributed memory and can be used as effective preconditioner. To check efficient of our solver we compare it with Intel MKL PARDISO - the high performance direct solver. Memory and performance tests demonstrate up to 3 times performance and memory gain for the 3D problems with more than (10^6 ) unknowns. Therefore, proposed multifrontal HSS solver can solve large problems, which cannot be resolved by direct solvers because of large memory consumptions. In recent years it has been shown that for many linear algebra operations it is possible to create families of algorithms following a very systematic procedure. We do not refer to the fine tuning of a known algorithm, but to a methodology for the actual generation of both algorithms and routines to solve a given target matrix equation. Although systematic, the methodology relies on complex algebraic manipulations and non-obvious pattern matching, making the procedure challenging to be performed by hand, our goal is the development of a fully automated system that from the sole description of a target equation creates multiple algorithms and routines. We present CL1ck, a symbolic system written in Mathematica, that starts with an equation, decomposes it into multiple equations, and returns a set of loop-invariants for the algorithms -- yet to be generated -- that will solve the equation. In a successive step each loop-invariant is then mapped to its corresponding algorithm and routine. For a large class of equations, the methodology generates known algorithms as well as many previously unknown ones. Most interestingly, the methodology unifies algorithms traditionally developed in isolation. As an example, the five well known algorithms for the LU factorization are for the first time unified under a common root. We investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various fields such as machine learning, operations research and pattern recognition. In the first class (Min ULR) one wishes, given a possibly infeasible system of linear relations, to find a solution that violates as few relations as possible while satisfying all the others. In the second class (Min RVLS) the linear system is supposed to be feasible and one looks for a solution with as few nonzero variables as possible. For both Min ULR and Min RVLS the four basic types of relational operators =, ⩾, > and ≠ are considered. While Min RVLS with equations was mentioned to be NP-hard in (Garey and Johnson, 1979), we established in (Amaldi; 1992; Amaldi and Kann, 1995) that min ULR with equalities and inequalities are NP-hard even when restricted to homogeneous systems with bipolar coefficients. The latter problems have been shown hard to approximate in (, 1993). In this paper we determine strong bounds on the approximability of various variants of Min RVLS and min ULR, including constrained ones where the variables are restricted to take binary values or where some relations are mandatory while others are optional. The various NP-hard versions turn out to have different approximability properties depending on the type of relations and the additional constraints, but none of them can be approximated within any constant factor, unless P = NP. Particular attention is devoted to two interesting special cases that occur in discriminant analysis and machine learning. In particular, we disprove a conjecture of van Horn and Martinez (1992) regarding the existence of a polynomial-time algorithm to design linear classifiers (or perceptrons) that involve a close-to-minimum number of features.