Nth-order Activity of Continuous Systems a,b,c Rodrigo Castro and c Ernesto Kofman a ETH Zürich, Switzerland b University of Buenos Aires & c CIFASIS-CONICET,

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Presentation transcript:

nth-order Activity of Continuous Systems a,b,c Rodrigo Castro and c Ernesto Kofman a ETH Zürich, Switzerland b University of Buenos Aires & c CIFASIS-CONICET, Argentina

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Introduction – Activity: Original definition – Need for an nth-order extension nth-Order Quantization – Zero (static), First and Second Order – nth-Order Quantization – Quantized State Systems (QSS) nth-Order Activity – The error perspective – nth-Order error dynamics – Definition of nth-Order Activity Examples – Example I: 1st. Order Non-stiff system – Example II: 2nd. order Stiff system Conclusions & Future Work Agenda

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Activity: Original definitionIntroduction The original definition of activity takes into account changes only in the signal values. x 1 (t) x 2 (t) x 3 (t) x 1 (t 0 )=x 2 (t 0 )=x 3 (t 0 ) x 1 (t f )=x 2 (t f )=x 3 (t f ) t0t0 tftf

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Activity: Original definitionIntroduction The original definition of activity takes into account changes only in the signal values. As a consequence, for a monotonically increasing or decreasing signal – the activity can be fully determined only by the distance between the final and the initial value, x 1 (t) x 2 (t) x 3 (t) x 1 (t 0 )=x 2 (t 0 )=x 3 (t 0 ) x 1 (t f )=x 2 (t f )=x 3 (t f ) t0t0 tftf

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Activity: Original definitionIntroduction The original definition of activity takes into account changes only in the signal values. As a consequence, for a monotonically increasing or decreasing signal – the activity can be fully determined only by the distance between the final and the initial value, – without using at all the information about how it goes from the initial to the final value. x 1 (t) x 2 (t) x 3 (t) x 1 (t 0 )=x 2 (t 0 )=x 3 (t 0 ) x 1 (t f )=x 2 (t f )=x 3 (t f ) t0t0 tftf

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Activity: Original definitionIntroduction The original definition of activity takes into account changes only in the signal values. As a consequence, for a monotonically increasing or decreasing signal – the activity can be fully determined only by the distance between the final and the initial value, – without using at all the information about how it goes from the initial to the final value. x 1 (t) x 2 (t) x 3 (t) x 1 (t 0 )=x 2 (t 0 )=x 3 (t 0 ) x 1 (t f )=x 2 (t f )=x 3 (t f )  A 1 =A 2 =A 3 t0t0 tftf

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Activity: Original definitionIntroduction The original definition of activity takes into account changes only in the signal values. As a consequence, for a monotonically increasing or decreasing signal – the activity can be fully determined only by the distance between the final and the initial value, – without using at all the information about how it goes from the initial to the final value. x 1 (t) x 2 (t) x 3 (t) x 1 (t 0 )=x 2 (t 0 )=x 3 (t 0 ) x 1 (t f )=x 2 (t f )=x 3 (t f )  A 1 =A 2 =A 3 t0t0 tftf A 1 =A 2 =A 3

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan x 2 (t) Activity: Original definitionIntroduction When a continuous signal is quantized with a zero-order quantization function – we obtain the well-known piecewise constant trajectory t f1 t f2 t f3 t0t0 m1m1 m2m2 m3m3 x i (t)q i (t)

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan x 2 (t) Activity: Original definitionIntroduction When a continuous signal is quantized with a zero-order quantization function – we obtain the well-known piecewise constant trajectory t f1 t f2 t f3 t0t0 m1m1 m2m2 m3m3 – For each interval of time at which x(t) is monotonic, the number of signal quantum crossings is: x i (t)q i (t)

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan x 2 (t) Activity: Original definitionIntroduction When a continuous signal is quantized with a zero-order quantization function – we obtain the well-known piecewise constant trajectory t f1 t f2 t f3 t0t0 m1m1 m2m2 m3m3 – For each interval of time at which x(t) is monotonic, the number of signal quantum crossings is: x i (t)q i (t)

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan x 2 (t) Activity: Original definitionIntroduction When a continuous signal is quantized with a zero-order quantization function – we obtain the well-known piecewise constant trajectory t f1 t f2 t f3 t0t0 x 1 (t) m1m1 m2m2 m3m3 – For each interval of time at which x(t) is monotonic, the number of signal quantum crossings is: x i (t)q i (t)

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan x 2 (t) Activity: Original definitionIntroduction When a continuous signal is quantized with a zero-order quantization function – we obtain the well-known piecewise constant trajectory t f1 t f2 t f3 t0t0 x 1 (t)x 3 (t) m1m1 m2m2 m3m3 – For each interval of time at which x(t) is monotonic, the number of signal quantum crossings is: x i (t)q i (t)

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan x 2 (t) Activity: Original definitionIntroduction When a continuous signal is quantized with a zero-order quantization function – we obtain the well-known piecewise constant trajectory x 1 (t 0 )=x 2 (t 0 )=x 3 (t 0 ) x 1 (t f1 )=x 2 (t f2 )=x 3 (t f3 )  A 1 =A 2 =A 3 t f1 t f2 t f3 t0t0 x 1 (t)x 3 (t) m1m1 m2m2 m3m3 – For each interval of time at which x(t) is monotonic, the number of signal quantum crossings is: x i (t)q i (t)

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Zero-order quantization functions are those used in first-order accurate QSS numerical integration methods – QSS1, LIQSS1, CQSS, BQSS Activity: Original definitionIntroduction – For each interval of time at which x(t) is monotonic, the number of signal quantum crossings is: x i (t)q i (t)

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Zero-order quantization functions are those used in first-order accurate QSS numerical integration methods – QSS1, LIQSS1, CQSS, BQSS – For these methods, the number of signal quantum crossings can establish a lower bound for the number of integration steps required to approximate the analytical solution with an accuracy (maximum error) bounded by the quantum size Activity: Original definitionIntroduction – For each interval of time at which x(t) is monotonic, the number of signal quantum crossings is: x i (t)q i (t)

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan “Classical”, “First order” Activity – Offers then a link between integration accuracy (Quantum size) and computational effort (# integration steps) Need for an nth-order extension Introduction

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan “Classical”, “First order” Activity – Offers then a link between integration accuracy (Quantum size) and computational effort (# integration steps) Nice features: – Convenient and intuitive visual relation between the solution x(t) and its quantized version q(t) – Can be easily expressed in terms of maxs and mins: Need for an nth-order extension Introduction

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan “Classical”, “First order” Activity – Offers then a link between integration accuracy (Quantum size) and computational effort (# integration steps) Nice features: – Convenient and intuitive visual relation between the solution x(t) and its quantized version q(t) – Can be easily expressed in terms of maxs and mins: Disadvantages: – It works only for first order accurate methods. – Not valid for higher order accurate methods Existing QSS methods: QSS 1 to 4, LIQSS 1 to 4, DQSS 1 to 3 Need for an nth-order extension Introduction

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan “Classical”, “First order” Activity – Offers then a link between integration accuracy (Quantum size) and computational effort (# integration steps) Nice features: – Convenient and intuitive visual relation between the solution x(t) and its quantized version q(t) – Can be easily expressed in terms of maxs and mins: Disadvantages: – It works only for first order accurate methods. – Not valid for higher order accurate methods Existing QSS methods: QSS 1 to 4, LIQSS 1 to 4, DQSS 1 to 3 – Intuition: QSS1 vs. QSS2 For a given ∆Q, 1 st. Order Activity is the same # of Steps is NOT the same Need for an nth-order extension IntroductionQSS1QSS2 zero order quantization q(t) piecewise constant first order quantization q(t) piecewise linear

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan “Classical”, “First order” Activity – Offers then a link between integration accuracy (Quantum size) and computational effort (# integration steps) Nice features: – Convenient and intuitive visual relation between the solution x(t) and its quantized version q(t) – Can be easily expressed in terms of maxs and mins: Disadvantages: – It works only for first order accurate methods. – Not valid for higher order accurate methods Existing QSS methods: QSS 1 to 4, LIQSS 1 to 4, DQSS 1 to 3 – Intuition: QSS1 vs. QSS2 For a given ∆Q, 1 st. Order Activity is the same # of Steps is NOT the same  A formal extension for Activity of nth-order is required. Need for an nth-order extension IntroductionQSS1QSS2 zero order quantization q(t) piecewise constant first order quantization q(t) piecewise linear

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – Zero-order (static) Zero (static) and First Ordernth-Order Quantization k 1 =A1/∆Q

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – Zero-order (static) Zero (static) and First Ordernth-Order Quantization k 1 =A1/∆Q polynomial segments j=0,1,2,…

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – Zero-order (static) Zero (static) and First Ordernth-Order Quantization err(t) k 1 =A1/∆Q polynomial segments j=0,1,2,…

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – Zero-order (static) Zero (static) and First Ordernth-Order Quantization err(t) q(t) piecewise constant k 1 =A1/∆Q polynomial segments j=0,1,2,…

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – Zero-order (static) – First-order Zero (static) and First Ordernth-Order Quantization err(t) q(t) piecewise constant k 1 =A1/∆Q polynomial segments j=0,1,2,…

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – Zero-order (static) – First-order Zero (static) and First Ordernth-Order Quantization err(t) q(t) piecewise constant k 1 =A1/∆Q k 2 <k 1 1 polynomial segments j=0,1,2,…

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – Zero-order (static) – First-order Zero (static) and First Ordernth-Order Quantization err(t) q(t) piecewise constant k 1 =A1/∆Q k 2 <k 1 1 polynomial segments j=0,1,2,…

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – Zero-order (static) – First-order Zero (static) and First Ordernth-Order Quantization err(t) q(t) piecewise constant k 1 =A1/∆Q k 2 <k 1 1 polynomial segments j=0,1,2,…

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – Zero-order (static) – First-order Zero (static) and First Ordernth-Order Quantization err(t) q(t) piecewise constant q(t) piecewise linear k 1 =A1/∆Q k 2 <k 1 1 polynomial segments j=0,1,2,…

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – Zero-order (static) – First-order Zero (static) and First Ordernth-Order Quantization err(t) q(t) piecewise constant q(t) piecewise linear No visual “quantization grid” available anymore Now also “how” the signal grows matters (e.g. only one event needed to quantize x(t)=k.t) k 1 =A1/∆Q k 2 <k 1 1 polynomial segments j=0,1,2,…

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – Second-order Second Ordernth-Order Quantization k 3 <k 2

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – Second-order Second Ordernth-Order Quantization err(t) q(t) piecewise parabolic k 3 <k 2

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – Second-order – The quantization scheme directly determines the number of “polynomial segments” (steps) required Second Ordernth-Order Quantization err(t) q(t) piecewise parabolic k 3 <k 2

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – Second-order – The quantization scheme directly determines the number of “polynomial segments” (steps) required – We will start with a definition of quantization of order n which will lead us to a definition of activity of order n Second Ordernth-Order Quantization err(t) q(t) piecewise parabolic k 3 <k 2

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – nth-order (with n>0) nth-Ordernth-Order Quantization

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – nth-order (with n>0) nth-Ordernth-Order Quantization err(t) q(t) piecewise nth-order

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Quantization: the key “error-driven” process – nth-order (with n>0) m=1,2,…,n nth-Ordernth-Order Quantization err(t) q(t) piecewise nth-order

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan The quantization process keeps track of the dynamics of the error between an input signal and its quantized version – It is the key mechanism used by QSS integrators for error control – QSS: Already discussed in previous presentations. Quick recap: Quantized State Systems (QSS) nth-Order Quantization

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan The quantization process keeps track of the dynamics of the error between an input signal and its quantized version – It is the key mechanism used by QSS integrators for error control – QSS: Already discussed in previous presentations. Quick recap: Quantized State Systems (QSS) nth-Order Quantization quantized integrator pure integrator

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan The quantization process keeps track of the dynamics of the error between an input signal and its quantized version – It is the key mechanism used by QSS integrators for error control – QSS: Already discussed in previous presentations. Quick recap: Quantized State Systems (QSS) nth-Order Quantization quantized integrator pure integrator Event={c 0 }QSS1

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan The quantization process keeps track of the dynamics of the error between an input signal and its quantized version – It is the key mechanism used by QSS integrators for error control – QSS: Already discussed in previous presentations. Quick recap: Quantized State Systems (QSS) nth-Order Quantization quantized integrator pure integrator Event={c 0 }QSS1 QSS2Event={c 0,c 1 }

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan The quantization process keeps track of the dynamics of the error between an input signal and its quantized version – It is the key mechanism used by QSS integrators for error control – QSS: Already discussed in previous presentations. Quick recap: Quantized State Systems (QSS) nth-Order Quantization quantized integrator pure integrator Event={c 0 }QSS1 QSS2Event={c 0,c 1 } QSS3Event={c 0,c 1,c 2 }

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan The original definition of Activity integrates the rate of change of the signal x(t): The error perspectiventh-Order Activity

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan The original definition of Activity integrates the rate of change of the signal x(t): When q(t) is the result of a zero-order quantization, the rate of change of the signal x(t) coincides with the rate of growth of the error |q(t)-x(t)| The error perspectiventh-Order Activity

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan The original definition of Activity integrates the rate of change of the signal x(t): When q(t) is the result of a zero-order quantization, the rate of change of the signal x(t) coincides with the rate of growth of the error |q(t)-x(t)| – Consequently this formula works: The error perspectiventh-Order Activity

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan The original definition of Activity integrates the rate of change of the signal x(t): When q(t) is the result of a zero-order quantization, the rate of change of the signal x(t) coincides with the rate of growth of the error |q(t)-x(t)| – Consequently this formula works: The error perspectiventh-Order Activity But …

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan When q(t) is the result of a high order quantization, the rate of growth of the error |q(t)-x(t)| looks different: nth-Order error dynamicsnth-Order Activity

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan When q(t) is the result of a high order quantization, the rate of growth of the error |q(t)-x(t)| looks different: nth-Order error dynamicsnth-Order Activity

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan When q(t) is the result of a high order quantization, the rate of growth of the error |q(t)-x(t)| looks different: nth-Order error dynamicsnth-Order Activity =

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan When q(t) is the result of a high order quantization, the rate of growth of the error |q(t)-x(t)| looks different: Let us replace x(t) by its Taylor series expansion: nth-Order error dynamicsnth-Order Activity =

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan When q(t) is the result of a high order quantization, the rate of growth of the error |q(t)-x(t)| looks different: Let us replace x(t) by its Taylor series expansion: nth-Order error dynamicsnth-Order Activity = =

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan When q(t) is the result of a high order quantization, the rate of growth of the error |q(t)-x(t)| looks different: Let us replace x(t) by its Taylor series expansion: Then, the dynamics of the error can be expressed as: nth-Order error dynamicsnth-Order Activity = =

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan When q(t) is the result of a high order quantization, the rate of growth of the error |q(t)-x(t)| looks different: Let us replace x(t) by its Taylor series expansion: Then, the dynamics of the error can be expressed as: nth-Order error dynamicsnth-Order Activity = = =0 in all *QSSn methods

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Definition of nth-Order Activity nth-Order Activity So, we have error

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Definition of nth-Order Activity nth-Order Activity So, we have error The discontinuities in the polynomial segments (i.e., the integration steps) occur when:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Definition of nth-Order Activity nth-Order Activity err(t j+1 )=∆Q So, we have error The discontinuities in the polynomial segments (i.e., the integration steps) occur when:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Definition of nth-Order Activity nth-Order Activity err(t j+1 )=∆Q holds, producing “one new step”: So, we have error The discontinuities in the polynomial segments (i.e., the integration steps) occur when: Therefore, at each new discontinuity instant t j+1

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Definition of nth-Order Activity nth-Order Activity err(t j+1 )=∆Q holds, producing “one new step”: So, we have error The discontinuities in the polynomial segments (i.e., the integration steps) occur when: Therefore, at each new discontinuity instant t j+1

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Definition of nth-Order Activity nth-Order Activity err(t j+1 )=∆Q holds, producing “one new step”: So, we have error The discontinuities in the polynomial segments (i.e., the integration steps) occur when: Therefore, at each new discontinuity instant t j+1 Adding up these steps throughout all segments:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Definition of nth-Order Activity nth-Order Activity err(t j+1 )=∆Q holds, producing “one new step”: So, we have error The discontinuities in the polynomial segments (i.e., the integration steps) occur when: Therefore, at each new discontinuity instant t j+1 Adding up these steps throughout all segments:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Definition of nth-Order Activity nth-Order Activity err(t j+1 )=∆Q define Activity of order n holds, producing “one new step”: So, we have error The discontinuities in the polynomial segments (i.e., the integration steps) occur when: Therefore, at each new discontinuity instant t j+1 Adding up these steps throughout all segments:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Definition of nth-Order Activity nth-Order Activity err(t j+1 )=∆Q define Activity of order n holds, producing “one new step”: So, we have error The discontinuities in the polynomial segments (i.e., the integration steps) occur when: Therefore, at each new discontinuity instant t j+1 Adding up these steps throughout all segments:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan The new definition retains the notions that Activity is – a property related to the inherent dynamics a signal – independent of the accuracy (quantum size) of choice Definition of nth-Order Activity nth-Order Activity

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan The new definition retains the notions that Activity is – a property related to the inherent dynamics a signal – independent of the accuracy (quantum size) of choice Definition of nth-Order Activity nth-Order Activity First Order Activitynth-Order Activity n=1

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan st order system Example I: Non-stiff systemExamples Solution: System:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan st order system Example I: Non-stiff systemExamples Solution: System: nth-order Activity:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan st order system Example I: Non-stiff systemExamples Solution: System: x a (t) “analytical” or “exact” x(0)=1   t f =5 nth-order Activity:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan st order system Example I: Non-stiff systemExamples Solution: System: nth-order Activity:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan st order system Example I: Non-stiff systemExamples Solution: System: nth-order Activity:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan st order system Example I: Non-stiff systemExamples Solution: System: nth-order Activity:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan st order system Example I: Non-stiff systemExamples Solution: System: nth-order Activity:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan st order system Example I: Non-stiff systemExamples Solution: System: nth-order Activity: Practical and theoretical number of steps match reasonably close

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan st order system Example I: Non-stiff systemExamples Solution: System: nth-order Activity: Practical and theoretical number of steps match reasonably close =∆Q b / ∆Q a ;  =k b / k a  =  =( ) 1/2  =( ) 1/3

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan st order system Example I: Non-stiff systemExamples Solution: System: nth-order Activity: QSS1 Zoom in: t=[1,4] x a (t) “analytical” or “exact” q(t) x(t)

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan st order system Example I: Non-stiff systemExamples Solution: System: nth-order Activity: x a (t) “analytical” or “exact” q(t) x(t) QSS2 Zoom in: t=[1,4]

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan nd order system Example II: Stiff systemExamples System:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan nd order system Example II: Stiff systemExamples System:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan nd order system Example II: Stiff systemExamples Analytical solution: System:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan nd order system Example II: Stiff systemExamples Analytical solution: System: nth-order Activity:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan nd order system Example II: Stiff systemExamples Analytical solution: System: nth-order Activity:

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan nd order system Example II: Stiff systemExamples Analytical solution: System: nth-order Activity: 

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan nd order system Example II: Stiff systemExamples Analytical solution: System: nth-order Activity: 

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan nd order system Example II: Stiff systemExamples Analytical solution: System: nth-order Activity: 

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan nd order system We simulate with non-stiff (QSS) and stiff (LIQSS) solvers Example II: Stiff systemExamples System: nth-order Activity: QSS1LIQSS1

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan nd order system We simulate with non-stiff (QSS) and stiff (LIQSS) solvers Example II: Stiff systemExamples System: nth-order Activity: High frequency spurious oscillations in q 2 (t) QSS1LIQSS1 spurious oscillations avoided

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan nd order system First Order Activity Example II: Stiff systemExamples System: nth-order Activity: Practical and theoretical # of Steps match closely for LIQSS

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan nd order system First Order Activity Example II: Stiff systemExamples System: nth-order Activity: Practical and theoretical # of Steps match closely for LIQSS

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan nd order system First Order Activity Example II: Stiff systemExamples System: nth-order Activity: Practical and theoretical # of Steps match closely for LIQSS For q 2 (t) with QSS, the practical # of Steps is obviously unacceptable

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan nd order system Higher Order Activity (2 nd, 3 rd ): Expected results verified. – Results are shown below only for the “conflictive variable” q 2 (t) Example II: Stiff systemExamples Practical and theoretical # of Steps match closely for LIQSS

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan nd order system Higher Order Activity (2 nd, 3 rd ): Expected results verified. – Results are shown below only for the “conflictive variable” q 2 (t) Example II: Stiff systemExamples Practical and theoretical # of Steps match closely for LIQSS For q 2 (t) with QSS, the practical # of Steps is obviously unacceptable

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan We have presented a generalization of the concept of activity for continuous time signals. Conclusions

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan We have presented a generalization of the concept of activity for continuous time signals. The classical definition of activity measures the rate of change of the signal Conclusions

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan We have presented a generalization of the concept of activity for continuous time signals. The classical definition of activity measures the rate of change of the signal The new definition of activity of nth-order takes into account the rate of change of the derivatives. – Now the “how” [a continuous signal evolves] matters. – Not only the signal’s maxima and minima – Less intuitive, less visualizable Conclusions

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan We have presented a generalization of the concept of activity for continuous time signals. The classical definition of activity measures the rate of change of the signal The new definition of activity of nth-order takes into account the rate of change of the derivatives. – Now the “how” [a continuous signal evolves] matters. – Not only the signal’s maxima and minima – Less intuitive, less visualizable Activity of order n provides a means by which establishing an ideal lower bound for the number of integration steps – against which comparing the performance of a (suitably selected) Quantization-based integration method of order n. Conclusions

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan We have presented a generalization of the concept of activity for continuous time signals. The classical definition of activity measures the rate of change of the signal The new definition of activity of nth-order takes into account the rate of change of the derivatives. – Now the “how” [a continuous signal evolves] matters. – Not only the signal’s maxima and minima – Less intuitive, less visualizable Activity of order n provides a means by which establishing an ideal lower bound for the number of integration steps – against which comparing the performance of a (suitably selected) Quantization-based integration method of order n. Activity of order n is, so far, almost exclusively of theoretical relevance – For calculating A (n) we need to know the analytical solution of a system… – … but that is exactly what we can’t know prior to simulation !!! (for most cases of practical interest) Conclusions

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Explore how the knowledge of activity measures for each variable in a given system can be exploited to: – derive optimal model partitions into multiple parallel processing nodes (cores, processors, servers) in order to maximize speedups as compared against a serial (single node) simulation. Ongoing work

Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich, Switzerland, Jan Q&A