75,870 research outputs found
Improving the numerical stability of fast matrix multiplication
Fast algorithms for matrix multiplication, namely those that perform
asymptotically fewer scalar operations than the classical algorithm, have been
considered primarily of theoretical interest. Apart from Strassen's original
algorithm, few fast algorithms have been efficiently implemented or used in
practical applications. However, there exist many practical alternatives to
Strassen's algorithm with varying performance and numerical properties. Fast
algorithms are known to be numerically stable, but because their error bounds
are slightly weaker than the classical algorithm, they are not used even in
cases where they provide a performance benefit.
We argue in this paper that the numerical sacrifice of fast algorithms,
particularly for the typical use cases of practical algorithms, is not
prohibitive, and we explore ways to improve the accuracy both theoretically and
empirically. The numerical accuracy of fast matrix multiplication depends on
properties of the algorithm and of the input matrices, and we consider both
contributions independently. We generalize and tighten previous error analyses
of fast algorithms and compare their properties. We discuss algorithmic
techniques for improving the error guarantees from two perspectives:
manipulating the algorithms, and reducing input anomalies by various forms of
diagonal scaling. Finally, we benchmark performance and demonstrate our
improved numerical accuracy
Fast linear algebra is stable
In an earlier paper, we showed that a large class of fast recursive matrix
multiplication algorithms is stable in a normwise sense, and that in fact if
multiplication of -by- matrices can be done by any algorithm in
operations for any , then it can be done
stably in operations for any . Here we extend
this result to show that essentially all standard linear algebra operations,
including LU decomposition, QR decomposition, linear equation solving, matrix
inversion, solving least squares problems, (generalized) eigenvalue problems
and the singular value decomposition can also be done stably (in a normwise
sense) in operations.Comment: 26 pages; final version; to appear in Numerische Mathemati
Error control of the vectorial nondirective stable plane wave multilevel fast multipole algorithm
Novel formulas are presented that allow the rapid estimation of the number of terms L that needs to be taken into account in the translation operator of the vectorial Nondirective Stable Plane Wave Multilevel Fast Multipole Algorithm (NSPWMLFMA). This is especially important for low frequencies, since the L needed for error-controllability can be substantially higher than the L required in the scalar case. Although these formulas were originally derived for use in the NSPWMLFMA, they are equally useful in at least three other fast matrix multiplication methods
Generalized companion matrix for approximate GCD
We study a variant of the univariate approximate GCD problem, where the
coefficients of one polynomial f(x)are known exactly, whereas the coefficients
of the second polynomial g(x)may be perturbed. Our approach relies on the
properties of the matrix which describes the operator of multiplication by gin
the quotient ring C[x]=(f). In particular, the structure of the null space of
the multiplication matrix contains all the essential information about GCD(f;
g). Moreover, the multiplication matrix exhibits a displacement structure that
allows us to design a fast algorithm for approximate GCD computation with
quadratic complexity w.r.t. polynomial degrees.Comment: Submitted to MEGA 201
Optimal approximate matrix product in terms of stable rank
We prove, using the subspace embedding guarantee in a black box way, that one
can achieve the spectral norm guarantee for approximate matrix multiplication
with a dimensionality-reducing map having
rows. Here is the maximum stable rank, i.e. squared ratio of
Frobenius and operator norms, of the two matrices being multiplied. This is a
quantitative improvement over previous work of [MZ11, KVZ14], and is also
optimal for any oblivious dimensionality-reducing map. Furthermore, due to the
black box reliance on the subspace embedding property in our proofs, our
theorem can be applied to a much more general class of sketching matrices than
what was known before, in addition to achieving better bounds. For example, one
can apply our theorem to efficient subspace embeddings such as the Subsampled
Randomized Hadamard Transform or sparse subspace embeddings, or even with
subspace embedding constructions that may be developed in the future.
Our main theorem, via connections with spectral error matrix multiplication
shown in prior work, implies quantitative improvements for approximate least
squares regression and low rank approximation. Our main result has also already
been applied to improve dimensionality reduction guarantees for -means
clustering [CEMMP14], and implies new results for nonparametric regression
[YPW15].
We also separately point out that the proof of the "BSS" deterministic
row-sampling result of [BSS12] can be modified to show that for any matrices
of stable rank at most , one can achieve the spectral norm
guarantee for approximate matrix multiplication of by deterministically
sampling rows that can be found in polynomial
time. The original result of [BSS12] was for rank instead of stable rank. Our
observation leads to a stronger version of a main theorem of [KMST10].Comment: v3: minor edits; v2: fixed one step in proof of Theorem 9 which was
wrong by a constant factor (see the new Lemma 5 and its use; final theorem
unaffected
Accurate and efficient algorithms for boundary element methods in electromagnetic scattering: a tribute to the work of F. Olyslager
Boundary element methods (BEMs) are an increasingly popular approach to model electromagnetic scattering both by perfect conductors and dielectric objects. Several mathematical, numerical, and computational techniques pullulated from the research into BEMs, enhancing its efficiency and applicability. In designing a viable implementation of the BEM, both theoretical and practical aspects need to be taken into account. Theoretical aspects include the choice of an integral equation for the sought after current densities on the geometry's boundaries and the choice of a discretization strategy (i.e. a finite element space) for this equation. Practical aspects include efficient algorithms to execute the multiplication of the system matrix by a test vector (such as a fast multipole method) and the parallelization of this multiplication algorithm that allows the distribution of the computation and communication requirements between multiple computational nodes. In honor of our former colleague and mentor, F. Olyslager, an overview of the BEMs for large and complex EM problems developed within the Electromagnetics Group at Ghent University is presented. Recent results that ramified from F. Olyslager's scientific endeavors are included in the survey
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