Ab-initio Functional Decomposition of Kalman Filter: A Feasibility Analysis on Constrained Least Squares Problems

D’Amore, Luisa; Cacciapuoti, Rosalba; Mele, Valeria

doi:10.1007/978-3-030-43222-5_7

Cited by 2 publications

(11 citation statements)

References 21 publications

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“…DD‐KF involves decomposition of the whole computational problem, partitioning of the solution and a slight modification of KF algorithm allowing a correction at run‐time of local solutions, through its exchange between adjacent subdomains. The analysis carried out in Reference 6 highlighted the reliability of the proposed framework mainly in terms of the accuracy of local solutions with respect the global solution, instead, here we highlight main features of DD‐KF algorithm demonstrating that it allows us to overcome the inherent bottlenecks of KF algorithm. In order to analyze the benefits arising from decomposition we present scalability analysis of DD‐KF algorithm measuring the scale‐up factor which expresses the performance gain of the algorithm in terms of reduction of its time complexity 7 .…”

Section: Motivations and Related Workmentioning

confidence: 86%

“…In this regards, in Reference 6 we presented an innovative DD framework for using KF in large‐scale applications. We called it DD‐KF approach.…”

Section: Motivations and Related Workmentioning

confidence: 99%

“…Definition (DD‐CLS model 6) Let S be the overdetermined linear system in () and

A \in ℝ^{(m_{0} + m_{1}) \times n}

b \in ℝ^{m_{0} + m_{1}}

the matrix and the vector defined in () and

R_{0} \in ℝ^{m_{0} \times m_{0}}

R_{1} \in ℝ^{m_{1} \times m_{1}}

R = diag (R_{0}, R_{1}) \in ℝ^{(m_{0} + m_{1}) \times (m_{0} + m_{1})}

be the weight matrices with m 0 > n and m 1 > 0. Let us consider the index set of columns of A , I = {1, … , n }.…”

Section: The Backgroundmentioning

confidence: 99%

“…To apply KF algorithm to

P_{1}^{n + 1}

and

P_{2}^{n + 1}

in () and (), we need

{\hat{x}}_{1, 0} \in ℝ^{n_{1}}

and

{\hat{x}}_{2, 0} \in ℝ^{n_{2}}

(i.e., reduction of

{\hat{x}}_{0} = {(H_{0}^{T} {RH}_{0})}^{- 1} H_{0}^{T} R y_{0}

). These vectors are calculated as follows: DD of I without overlap (i.e., s = 0).Theorem 6 Let us consider

H_{0} \in ℝ^{m_{0} \times n}

y_{0} \in ℝ^{m_{0}}

, and

R_{0} \in ℝ^{m_{0} \times m_{0}}

, where m 0 > n , m 1 > 0 , system H 0 x 0 = y 0 and the set I. Let us consider the following steps: Then

{\hat{x}}_{0} = {(H_{0}^{T} R_{0} H_{0})}^{- 1} H_{0}^{T} R_{0} y_{0} \in ℝ^{}

…”

Section: The Backgroundmentioning

confidence: 99%

“…Theorem 6 Let us consider

H_{0} \in ℝ^{m_{0} \times n}

y_{0} \in ℝ^{m_{0}}

, and

R_{0} \in ℝ^{m_{0} \times m_{0}}

, where m 0 > n , m 1 > 0 , system H 0 x 0 = y 0 and the set I. Let us consider the following steps: Then

{\hat{x}}_{0} = {(H_{0}^{T} R_{0} H_{0})}^{- 1} H_{0}^{T} R_{0} y_{0} \in ℝ^{n}

, which is the solution in least squares sense of H 0 x 0 = y 0 , is obtained as follows :

{\hat{x}}_{0} |_{I_{1}} = x_{1}, {\hat{x}}_{0} |_{I_{2}} = x_{2},

where

{\hat{x}}_{0} |_{I_{i}}

is the restriction of

{\hat{x}}_{0}

to I i , for i = 1, 2.…”

Section: The Backgroundmentioning

confidence: 99%

See 4 more Smart Citations

A scalable Kalman filter algorithm: Trustworthy analysis on constrained least square model

D’Amore

Cacciapuoti

Mele

2020

Concurrency and Computation

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Summary Kalman filter (KF) is one of the most important and common estimation algorithms. We introduce an innovative designing of Kalman filter algorithm based on domain decomposition (we call it DD‐KF). DD‐KF involves decomposition of the whole computational problem, partitioning of the solution and a slight modification of KF algorithm allowing a correction at run‐time of local solutions. The resulted parallel algorithm consists of concurrent copies of KF algorithm, each one requiring the same amount of computations on each subdomain and an exchange of boundary conditions between adjacent subdomains. Main advantage of this approach is that it can be potentially applied in a moderately nonintrusive manner to existing codes for tracking and controlling systems in location, navigation, in computer graphics and in much more state estimation problems. To highlight the capability of DD‐KF of exploiting the computing power provided by future designs of microprocessors based on multi/many‐cores CPU/GPU technologies, we consider DD both at physical core level and at microprocessor level and we discuss scalability of DD‐KF algorithm at coarse and fine grained level. Throughout the present work, we derive and discuss DD‐KF algorithm for solving constrained least square model, which underlies any data sampling and estimation problem.

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Section: Motivations and Related Workmentioning

confidence: 86%

“…In this regards, in Reference 6 we presented an innovative DD framework for using KF in large‐scale applications. We called it DD‐KF approach.…”

Section: Motivations and Related Workmentioning

confidence: 99%

“…Definition (DD‐CLS model 6) Let S be the overdetermined linear system in () and

A \in ℝ^{(m_{0} + m_{1}) \times n}

b \in ℝ^{m_{0} + m_{1}}

the matrix and the vector defined in () and

R_{0} \in ℝ^{m_{0} \times m_{0}}

R_{1} \in ℝ^{m_{1} \times m_{1}}

R = diag (R_{0}, R_{1}) \in ℝ^{(m_{0} + m_{1}) \times (m_{0} + m_{1})}

be the weight matrices with m 0 > n and m 1 > 0. Let us consider the index set of columns of A , I = {1, … , n }.…”

Section: The Backgroundmentioning

confidence: 99%

“…To apply KF algorithm to

P_{1}^{n + 1}

and

P_{2}^{n + 1}

in () and (), we need

{\hat{x}}_{1, 0} \in ℝ^{n_{1}}

and

{\hat{x}}_{2, 0} \in ℝ^{n_{2}}

(i.e., reduction of

{\hat{x}}_{0} = {(H_{0}^{T} {RH}_{0})}^{- 1} H_{0}^{T} R y_{0}

). These vectors are calculated as follows: DD of I without overlap (i.e., s = 0).Theorem 6 Let us consider

H_{0} \in ℝ^{m_{0} \times n}

y_{0} \in ℝ^{m_{0}}

, and

R_{0} \in ℝ^{m_{0} \times m_{0}}

, where m 0 > n , m 1 > 0 , system H 0 x 0 = y 0 and the set I. Let us consider the following steps: Then

{\hat{x}}_{0} = {(H_{0}^{T} R_{0} H_{0})}^{- 1} H_{0}^{T} R_{0} y_{0} \in ℝ^{}

…”

Section: The Backgroundmentioning

confidence: 99%

“…Theorem 6 Let us consider

H_{0} \in ℝ^{m_{0} \times n}

y_{0} \in ℝ^{m_{0}}

, and

R_{0} \in ℝ^{m_{0} \times m_{0}}

, where m 0 > n , m 1 > 0 , system H 0 x 0 = y 0 and the set I. Let us consider the following steps: Then

{\hat{x}}_{0} = {(H_{0}^{T} R_{0} H_{0})}^{- 1} H_{0}^{T} R_{0} y_{0} \in ℝ^{n}

, which is the solution in least squares sense of H 0 x 0 = y 0 , is obtained as follows :

{\hat{x}}_{0} |_{I_{1}} = x_{1}, {\hat{x}}_{0} |_{I_{2}} = x_{2},

where

{\hat{x}}_{0} |_{I_{i}}

is the restriction of

{\hat{x}}_{0}

to I i , for i = 1, 2.…”

Section: The Backgroundmentioning

confidence: 99%

See 3 more Smart Citations

A scalable Kalman filter algorithm: Trustworthy analysis on constrained least square model

D’Amore

Cacciapuoti

Mele

2020

Concurrency and Computation

Self Cite

View full text Add to dashboard Cite

show abstract

Parallel framework for dynamic domain decomposition of data assimilation problems: a case study on Kalman Filter algorithm

D’Amore

Cacciapuoti

2021

Comp and Math Methods

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We focus on Partial Differential Equation (PDE)-based Data Assimilation problems (DA) solved by means of variational approaches and Kalman filter algorithm. Recently, we presented a Domain Decomposition framework (we call it DD-DA, for short) performing a decomposition of the whole physical domain along space and time directions, and joining the idea of Schwarz's methods and parallel in time approaches. For effective parallelization of DD-DA algorithms, the computational load assigned to subdomains must be equally distributed. Usually computational cost is proportional to the amount of data entities assigned to partitions. Good quality partitioning also requires the volume of communication during calculation to be kept at its minimum. In order to deal with DD-DA problems where the observations are nonuniformly distributed and general sparse, in the present work we employ a parallel load balancing algorithm based on adaptive and dynamic defining of boundaries of DD-which is aimed to balance workload according to data location. We call it DyDD. As the numerical model underlying DA problems arising from the so-called discretize-then-optimize approach is the constrained least square model (CLS), we will use CLS as a reference state estimation problem and we validate DyDD on different scenarios.

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Ab-initio Functional Decomposition of Kalman Filter: A Feasibility Analysis on Constrained Least Squares Problems

Cited by 2 publications

References 21 publications

A scalable Kalman filter algorithm: Trustworthy analysis on constrained least square model

A scalable Kalman filter algorithm: Trustworthy analysis on constrained least square model

Parallel framework for dynamic domain decomposition of data assimilation problems: a case study on Kalman Filter algorithm

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