We present the algebraic foundations of the symmetric Zassenhaus algorithm and some of its varian... more We present the algebraic foundations of the symmetric Zassenhaus algorithm and some of its variants. These algorithms have proven effective in devising higher-order methods for solving the time-dependent {S}chr\"{o}dinger equation in the semiclassical regime. We find that the favourable properties of these methods derive directly from the structural properties of a Z2-graded Lie algebra. Commutators in this Lie algebra can be simplified explicitly, leading to commutator-free methods. Their other structural properties are crucial in proving unitarity, stability, convergence, error bounds and quadratic costs of Zassenhaus based methods. These algebraic structures have also found applications in Magnus expansion based methods for time-varying potentials where they allow significantly milder constraints for convergence and lead to highly effective schemes. The algebraic foundations laid out in this work pave the way for extending higher-order Zassenhaus and Magnus schemes to other equations of quantum mechanics.
The computation of the semiclassical Schrödinger equation featuring time-dependent potentials is ... more The computation of the semiclassical Schrödinger equation featuring time-dependent potentials is of great importance in quantum control of atomic and molecular processes. It presents major challenges because of the presence of a small parameter. Assuming periodic boundary conditions, the standard approach in tackling this problem consists of semi-discretisation with a spectral method, followed by a Magnus expansion. It is typical to discretise in time, thereafter replacing all integrals occurring in the Magnus expansion with quadratures. Following this, an exponential splitting is usually prescribed.
In this paper we sketch an alternative strategy where semi-discretisation and approximation of integrals is done at the very end, following an exponential splitting. This approach allows us to consider significantly larger time steps and gives us the flexibility to handle a variety of potentials, inclusive of highly oscillatory potentials. Our analysis commences from the investigation of the free Lie algebra generated by differentiation and by multiplication with the interaction potential. It turns out that this algebra possesses structure that renders it amenable to a very effective form of asymptotic splitting: an exponential splitting where consecutive terms are scaled by increasing powers of the small parameter. This leads to methods that attain high spatial and temporal accuracy and whose cost scales like O(M log M), where M is the number of degrees of freedom in the discretisation.
The computation of the semiclassical Schrödinger equation presents major challenges because of th... more The computation of the semiclassical Schrödinger equation presents major challenges because of the presence of a small parameter. Assuming periodic boundary conditions, the standard approach consists of semi-discretisation with a spectral method, followed by an exponential splitting. In this paper we sketch an alternative strategy. Our analysis commences from the investigation of the free Lie algebra generated by differentiation and by multiplication with the inter- action potential: it turns out that this algebra possesses structure that renders it amenable to a very effective form of asymptotic splitting: exponential splitting where consecutive terms are scaled by increasing powers of the small parameter. This leads to methods that attain high spatial and temporal accuracy and whose cost scales like O(M log M), where M is the number of degrees of freedom in the discretisation.
The question of when two nondeterministic concurrent systems are behaviourally related has occupi... more The question of when two nondeterministic concurrent systems are behaviourally related has occupied a large part of the literature on process algebra and has yielded a variety of equivalences (and congruences) and preorders (and precongruences) all based on the notion of bisimulations. Recently one of the authors has tried to unify a class of these bisimulation based relations by a parametrised notion of bisimu- lation and shown that the properties of the bisimilarity relations are often inherited from those of the underlying relationships between the observables. In addition to the usual strong and weak bisimilarity relations, it is possible to capture some other bisimilarity relations – those sensitive to costs, performance, dis- tribution or locations etc – by parametrised bisimulations. In this paper we present an equational axiomatization of all equivalence relations that fall in the class of parametrised bisimilarities without empty observables. Our axiomatization has been inspired by the axiomatization of observational congruence by Bergstra and Klop and attempts to extend it for parametrised bisimilarities. The axiomatization has been proven to be complete for finite process graphs relative to a complete axiomatization for the relations on observables. In the process, we also show that in the absence of empty observables, all preorders and equivalence relations are also precongruences and congruences, respectively.
The question of when two systems are behaviourally equal has occupied a large part of the literat... more The question of when two systems are behaviourally equal has occupied a large part of the literature on verification and has yielded various equivalences (and congruences). These equivalence relations are most useful in comparing systems whose executions are not necessarily finite. An axiomatization of these equivalences gives us both, a nice algebraic handle on processes, and a proof system for checking the equality of two processes. Comparison of efficiency of non-terminating processes like an operating system has been largely untackled.
We have presented here, an axiomatization for a certain subset of ordering induced bisimilarities. This axiomatization yields the axiomatization for equivalences like observational equivalence and inefficiency bisimulation as special cases. The axiomatization has been proven to be complete for finite state processes, and can be used as a proof system for checking the equality of systems.
We present the algebraic foundations of the symmetric Zassenhaus algorithm and some of its varian... more We present the algebraic foundations of the symmetric Zassenhaus algorithm and some of its variants. These algorithms have proven effective in devising higher-order methods for solving the time-dependent {S}chr\"{o}dinger equation in the semiclassical regime. We find that the favourable properties of these methods derive directly from the structural properties of a Z2-graded Lie algebra. Commutators in this Lie algebra can be simplified explicitly, leading to commutator-free methods. Their other structural properties are crucial in proving unitarity, stability, convergence, error bounds and quadratic costs of Zassenhaus based methods. These algebraic structures have also found applications in Magnus expansion based methods for time-varying potentials where they allow significantly milder constraints for convergence and lead to highly effective schemes. The algebraic foundations laid out in this work pave the way for extending higher-order Zassenhaus and Magnus schemes to other equations of quantum mechanics.
The computation of the semiclassical Schrödinger equation featuring time-dependent potentials is ... more The computation of the semiclassical Schrödinger equation featuring time-dependent potentials is of great importance in quantum control of atomic and molecular processes. It presents major challenges because of the presence of a small parameter. Assuming periodic boundary conditions, the standard approach in tackling this problem consists of semi-discretisation with a spectral method, followed by a Magnus expansion. It is typical to discretise in time, thereafter replacing all integrals occurring in the Magnus expansion with quadratures. Following this, an exponential splitting is usually prescribed.
In this paper we sketch an alternative strategy where semi-discretisation and approximation of integrals is done at the very end, following an exponential splitting. This approach allows us to consider significantly larger time steps and gives us the flexibility to handle a variety of potentials, inclusive of highly oscillatory potentials. Our analysis commences from the investigation of the free Lie algebra generated by differentiation and by multiplication with the interaction potential. It turns out that this algebra possesses structure that renders it amenable to a very effective form of asymptotic splitting: an exponential splitting where consecutive terms are scaled by increasing powers of the small parameter. This leads to methods that attain high spatial and temporal accuracy and whose cost scales like O(M log M), where M is the number of degrees of freedom in the discretisation.
The computation of the semiclassical Schrödinger equation presents major challenges because of th... more The computation of the semiclassical Schrödinger equation presents major challenges because of the presence of a small parameter. Assuming periodic boundary conditions, the standard approach consists of semi-discretisation with a spectral method, followed by an exponential splitting. In this paper we sketch an alternative strategy. Our analysis commences from the investigation of the free Lie algebra generated by differentiation and by multiplication with the inter- action potential: it turns out that this algebra possesses structure that renders it amenable to a very effective form of asymptotic splitting: exponential splitting where consecutive terms are scaled by increasing powers of the small parameter. This leads to methods that attain high spatial and temporal accuracy and whose cost scales like O(M log M), where M is the number of degrees of freedom in the discretisation.
The question of when two nondeterministic concurrent systems are behaviourally related has occupi... more The question of when two nondeterministic concurrent systems are behaviourally related has occupied a large part of the literature on process algebra and has yielded a variety of equivalences (and congruences) and preorders (and precongruences) all based on the notion of bisimulations. Recently one of the authors has tried to unify a class of these bisimulation based relations by a parametrised notion of bisimu- lation and shown that the properties of the bisimilarity relations are often inherited from those of the underlying relationships between the observables. In addition to the usual strong and weak bisimilarity relations, it is possible to capture some other bisimilarity relations – those sensitive to costs, performance, dis- tribution or locations etc – by parametrised bisimulations. In this paper we present an equational axiomatization of all equivalence relations that fall in the class of parametrised bisimilarities without empty observables. Our axiomatization has been inspired by the axiomatization of observational congruence by Bergstra and Klop and attempts to extend it for parametrised bisimilarities. The axiomatization has been proven to be complete for finite process graphs relative to a complete axiomatization for the relations on observables. In the process, we also show that in the absence of empty observables, all preorders and equivalence relations are also precongruences and congruences, respectively.
The question of when two systems are behaviourally equal has occupied a large part of the literat... more The question of when two systems are behaviourally equal has occupied a large part of the literature on verification and has yielded various equivalences (and congruences). These equivalence relations are most useful in comparing systems whose executions are not necessarily finite. An axiomatization of these equivalences gives us both, a nice algebraic handle on processes, and a proof system for checking the equality of two processes. Comparison of efficiency of non-terminating processes like an operating system has been largely untackled.
We have presented here, an axiomatization for a certain subset of ordering induced bisimilarities. This axiomatization yields the axiomatization for equivalences like observational equivalence and inefficiency bisimulation as special cases. The axiomatization has been proven to be complete for finite state processes, and can be used as a proof system for checking the equality of systems.
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Papers by Pranav Singh
In this paper we sketch an alternative strategy where semi-discretisation and approximation of integrals is done at the very end, following an exponential splitting. This approach allows us to consider significantly larger time steps and gives us the flexibility to handle a variety of potentials, inclusive of highly oscillatory potentials. Our analysis commences from the investigation of the free Lie algebra generated by differentiation and by multiplication with the interaction potential. It turns out that this algebra possesses structure that renders it amenable to a very effective form of asymptotic splitting: an exponential splitting where consecutive terms are scaled by increasing powers of the small parameter. This leads to methods that attain high spatial and temporal accuracy and whose cost scales like O(M log M), where M is the number of degrees of freedom in the discretisation.
We have presented here, an axiomatization for a certain subset of ordering induced bisimilarities. This axiomatization yields the axiomatization for equivalences like observational equivalence and inefficiency bisimulation as special cases. The axiomatization has been proven to be complete for finite state processes, and can be used as a proof system for checking the equality of systems.
In this paper we sketch an alternative strategy where semi-discretisation and approximation of integrals is done at the very end, following an exponential splitting. This approach allows us to consider significantly larger time steps and gives us the flexibility to handle a variety of potentials, inclusive of highly oscillatory potentials. Our analysis commences from the investigation of the free Lie algebra generated by differentiation and by multiplication with the interaction potential. It turns out that this algebra possesses structure that renders it amenable to a very effective form of asymptotic splitting: an exponential splitting where consecutive terms are scaled by increasing powers of the small parameter. This leads to methods that attain high spatial and temporal accuracy and whose cost scales like O(M log M), where M is the number of degrees of freedom in the discretisation.
We have presented here, an axiomatization for a certain subset of ordering induced bisimilarities. This axiomatization yields the axiomatization for equivalences like observational equivalence and inefficiency bisimulation as special cases. The axiomatization has been proven to be complete for finite state processes, and can be used as a proof system for checking the equality of systems.