From 305934dfbd2b37deb50cf93732e442c51bb1603b Mon Sep 17 00:00:00 2001 From: Luc Maisonobe Date: Wed, 6 Jan 2016 14:18:39 +0100 Subject: [PATCH] Field-based Adams-Bashforth integrator. --- .../AdamsBashforthFieldIntegrator.java | 374 ++++++++++++++++++ .../AdamsBashforthFieldIntegratorTest.java | 78 ++++ 2 files changed, 452 insertions(+) create mode 100644 src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegrator.java create mode 100644 src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegratorTest.java diff --git a/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegrator.java b/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegrator.java new file mode 100644 index 000000000..db6bf4f04 --- /dev/null +++ b/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegrator.java @@ -0,0 +1,374 @@ +/* + * Licensed to the Apache Software Foundation (ASF) under one or more + * contributor license agreements. See the NOTICE file distributed with + * this work for additional information regarding copyright ownership. + * The ASF licenses this file to You under the Apache License, Version 2.0 + * (the "License"); you may not use this file except in compliance with + * the License. You may obtain a copy of the License at + * + * http://www.apache.org/licenses/LICENSE-2.0 + * + * Unless required by applicable law or agreed to in writing, software + * distributed under the License is distributed on an "AS IS" BASIS, + * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + * See the License for the specific language governing permissions and + * limitations under the License. + */ + +package org.apache.commons.math4.ode.nonstiff; + +import org.apache.commons.math4.Field; +import org.apache.commons.math4.RealFieldElement; +import org.apache.commons.math4.exception.DimensionMismatchException; +import org.apache.commons.math4.exception.MaxCountExceededException; +import org.apache.commons.math4.exception.NoBracketingException; +import org.apache.commons.math4.exception.NumberIsTooSmallException; +import org.apache.commons.math4.linear.Array2DRowFieldMatrix; +import org.apache.commons.math4.linear.FieldMatrix; +import org.apache.commons.math4.ode.FieldExpandableODE; +import org.apache.commons.math4.ode.FieldODEState; +import org.apache.commons.math4.ode.FieldODEStateAndDerivative; +import org.apache.commons.math4.util.MathArrays; + + +/** + * This class implements explicit Adams-Bashforth integrators for Ordinary + * Differential Equations. + * + *

Adams-Bashforth methods (in fact due to Adams alone) are explicit + * multistep ODE solvers. This implementation is a variation of the classical + * one: it uses adaptive stepsize to implement error control, whereas + * classical implementations are fixed step size. The value of state vector + * at step n+1 is a simple combination of the value at step n and of the + * derivatives at steps n, n-1, n-2 ... Depending on the number k of previous + * steps one wants to use for computing the next value, different formulas + * are available:

+ * + * + *

A k-steps Adams-Bashforth method is of order k.

+ * + *

Implementation details

+ * + *

We define scaled derivatives si(n) at step n as: + * 

+ * s1(n) = h y'n for first derivative
+ * s2(n) = h2/2 y''n for second derivative
+ * s3(n) = h3/6 y'''n for third derivative
+ * ...
+ * sk(n) = hk/k! y(k)n for kth derivative
+ *

+ * + *

The definitions above use the classical representation with several previous first + * derivatives. Lets define + * 

+ *   qn = [ s1(n-1) s1(n-2) ... s1(n-(k-1)) ]T
+ *
+ * (we omit the k index in the notation for clarity). With these definitions, + * Adams-Bashforth methods can be written: + *

+ * + *

Instead of using the classical representation with first derivatives only (yn, + * s1(n) and qn), our implementation uses the Nordsieck vector with + * higher degrees scaled derivatives all taken at the same step (yn, s1(n) + * and rn) where rn is defined as: + * 

+ * rn = [ s2(n), s3(n) ... sk(n) ]T
+ *
+ * (here again we omit the k index in the notation for clarity) + *

+ * + *

Taylor series formulas show that for any index offset i, s1(n-i) can be + * computed from s1(n), s2(n) ... sk(n), the formula being exact + * for degree k polynomials. + * 

+ * s1(n-i) = s1(n) + ∑j>0 (j+1) (-i)j sj+1(n)
+ *
+ * The previous formula can be used with several values for i to compute the transform between + * classical representation and Nordsieck vector. The transform between rn + * and qn resulting from the Taylor series formulas above is: + * 
+ * qn = s1(n) u + P rn
+ *
+ * where u is the [ 1 1 ... 1 ]T vector and P is the (k-1)×(k-1) matrix built + * with the (j+1) (-i)j terms with i being the row number starting from 1 and j being + * the column number starting from 1: + * 
+ *        [  -2   3   -4    5  ... ]
+ *        [  -4  12  -32   80  ... ]
+ *   P =  [  -6  27 -108  405  ... ]
+ *        [  -8  48 -256 1280  ... ]
+ *        [          ...           ]
+ *

+ * + *

Using the Nordsieck vector has several advantages: + *

+ * + *

The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows: + *

+ * where A is a rows shifting matrix (the lower left part is an identity matrix): + * 
+ *        [ 0 0   ...  0 0 | 0 ]
+ *        [ ---------------+---]
+ *        [ 1 0   ...  0 0 | 0 ]
+ *    A = [ 0 1   ...  0 0 | 0 ]
+ *        [       ...      | 0 ]
+ *        [ 0 0   ...  1 0 | 0 ]
+ *        [ 0 0   ...  0 1 | 0 ]
+ *

+ * + *

The P-1u vector and the P-1 A P matrix do not depend on the state, + * they only depend on k and therefore are precomputed once for all.

+ * + * @param the type of the field elements + * @since 3.6 + */ +public class AdamsBashforthFieldIntegrator> extends AdamsFieldIntegrator { + + /** Integrator method name. */ + private static final String METHOD_NAME = "Adams-Bashforth"; + + /** + * Build an Adams-Bashforth integrator with the given order and step control parameters. + * @param field field to which the time and state vector elements belong + * @param nSteps number of steps of the method excluding the one being computed + * @param minStep minimal step (sign is irrelevant, regardless of + * integration direction, forward or backward), the last step can + * be smaller than this + * @param maxStep maximal step (sign is irrelevant, regardless of + * integration direction, forward or backward), the last step can + * be smaller than this + * @param scalAbsoluteTolerance allowed absolute error + * @param scalRelativeTolerance allowed relative error + * @exception NumberIsTooSmallException if order is 1 or less + */ + public AdamsBashforthFieldIntegrator(final Field field, final int nSteps, + final double minStep, final double maxStep, + final double scalAbsoluteTolerance, + final double scalRelativeTolerance) + throws NumberIsTooSmallException { + super(field, METHOD_NAME, nSteps, nSteps, minStep, maxStep, + scalAbsoluteTolerance, scalRelativeTolerance); + } + + /** + * Build an Adams-Bashforth integrator with the given order and step control parameters. + * @param field field to which the time and state vector elements belong + * @param nSteps number of steps of the method excluding the one being computed + * @param minStep minimal step (sign is irrelevant, regardless of + * integration direction, forward or backward), the last step can + * be smaller than this + * @param maxStep maximal step (sign is irrelevant, regardless of + * integration direction, forward or backward), the last step can + * be smaller than this + * @param vecAbsoluteTolerance allowed absolute error + * @param vecRelativeTolerance allowed relative error + * @exception IllegalArgumentException if order is 1 or less + */ + public AdamsBashforthFieldIntegrator(final Field field, final int nSteps, + final double minStep, final double maxStep, + final double[] vecAbsoluteTolerance, + final double[] vecRelativeTolerance) + throws IllegalArgumentException { + super(field, METHOD_NAME, nSteps, nSteps, minStep, maxStep, + vecAbsoluteTolerance, vecRelativeTolerance); + } + + /** Estimate error. + *

+ * Error is estimated by interpolating back to previous state using + * the state Taylor expansion and comparing to real previous state. + *

+ * @param previousState state vector at step start + * @param predictedState predicted state vector at step end + * @param predictedScaled predicted value of the scaled derivatives at step end + * @param predictedNordsieck predicted value of the Nordsieck vector at step end + * @return estimated normalized local discretization error + */ + private T errorEstimation(final T[] previousState, + final T[] predictedState, + final T[] predictedScaled, + final FieldMatrix predictedNordsieck) { + + T error = getField().getZero(); + for (int i = 0; i < mainSetDimension; ++i) { + final T yScale = predictedState[i].abs(); + final T tol = (vecAbsoluteTolerance == null) ? + yScale.multiply(scalRelativeTolerance).add(scalAbsoluteTolerance) : + yScale.multiply(vecRelativeTolerance[i]).add(vecAbsoluteTolerance[i]); + + // apply Taylor formula from high order to low order, + // for the sake of numerical accuracy + T variation = getField().getZero(); + int sign = predictedNordsieck.getRowDimension() % 2 == 0 ? -1 : 1; + for (int k = predictedNordsieck.getRowDimension() - 1; k >= 0; --k) { + variation = variation.add(predictedNordsieck.getEntry(k, i).multiply(sign)); + sign = -sign; + } + variation = variation.subtract(predictedScaled[i]); + + final T ratio = predictedState[i].subtract(previousState[i]).add(variation).divide(tol); + error = error.add(ratio.multiply(ratio)); + + } + + return error.divide(mainSetDimension).sqrt(); + + } + + /** {@inheritDoc} */ + @Override + public FieldODEStateAndDerivative integrate(final FieldExpandableODE equations, + final FieldODEState initialState, + final T finalTime) + throws NumberIsTooSmallException, DimensionMismatchException, + MaxCountExceededException, NoBracketingException { + + sanityChecks(initialState, finalTime); + final T t0 = initialState.getTime(); + final T[] y = equations.getMapper().mapState(initialState); + setStepStart(initIntegration(equations, t0, y, finalTime)); + final boolean forward = finalTime.subtract(initialState.getTime()).getReal() > 0; + + // compute the initial Nordsieck vector using the configured starter integrator + start(equations, getStepStart(), finalTime); + + // reuse the step that was chosen by the starter integrator + AdamsFieldStepInterpolator interpolator = + new AdamsFieldStepInterpolator(getStepSize(), getStepStart(), scaled, nordsieck, + forward, equations.getMapper()); + + // main integration loop + setIsLastStep(false); + do { + + T[] predictedY = null; + final T[] predictedScaled = MathArrays.buildArray(getField(), y.length); + Array2DRowFieldMatrix predictedNordsieck = null; + T error = getField().getZero().add(10); + while (error.subtract(1.0).getReal() >= 0.0) { + + // predict a first estimate of the state at step end + final FieldODEStateAndDerivative stepEnd = interpolator.getCurrentState(); + predictedY = stepEnd.getState(); + + // evaluate the derivative + final T[] yDot = computeDerivatives(stepEnd.getTime(), predictedY); + + // predict Nordsieck vector at step end + for (int j = 0; j < predictedScaled.length; ++j) { + predictedScaled[j] = getStepSize().multiply(yDot[j]); + } + predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck); + updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck); + + // evaluate error + error = errorEstimation(y, predictedY, predictedScaled, predictedNordsieck); + + if (error.subtract(1.0).getReal() >= 0.0) { + // reject the step and attempt to reduce error by stepsize control + final T factor = computeStepGrowShrinkFactor(error); + rescale(filterStep(getStepSize().multiply(factor), forward, false)); + interpolator = new AdamsFieldStepInterpolator(getStepSize(), getStepStart(), scaled, nordsieck, + forward, equations.getMapper()); + + } + } + + // discrete events handling + System.arraycopy(predictedY, 0, y, 0, y.length); + setStepStart(acceptStep(interpolator, finalTime)); + scaled = predictedScaled; + nordsieck = predictedNordsieck; + + if (!isLastStep()) { + + if (resetOccurred()) { + // some events handler has triggered changes that + // invalidate the derivatives, we need to restart from scratch + start(equations, getStepStart(), finalTime); + interpolator = new AdamsFieldStepInterpolator(getStepSize(), getStepStart(), scaled, nordsieck, + forward, equations.getMapper()); + } + + // stepsize control for next step + final T factor = computeStepGrowShrinkFactor(error); + final T scaledH = getStepSize().multiply(factor); + final T nextT = getStepStart().getTime().add(scaledH); + final boolean nextIsLast = forward ? + nextT.subtract(finalTime).getReal() >= 0 : + nextT.subtract(finalTime).getReal() <= 0; + T hNew = filterStep(scaledH, forward, nextIsLast); + + final T filteredNextT = getStepStart().getTime().add(hNew); + final boolean filteredNextIsLast = forward ? + filteredNextT.subtract(finalTime).getReal() >= 0 : + filteredNextT.subtract(finalTime).getReal() <= 0; + if (filteredNextIsLast) { + hNew = finalTime.subtract(getStepStart().getTime()); + } + + rescale(hNew); + interpolator = new AdamsFieldStepInterpolator(getStepSize(), getStepStart(), scaled, nordsieck, + forward, equations.getMapper()); + + } + + } while (!isLastStep()); + + final FieldODEStateAndDerivative finalState = getStepStart(); + setStepStart(null); + setStepSize(null); + return finalState; + + } + + /** Rescale the instance. + *

Since the scaled and Nordsieck arrays are shared with the caller, + * this method has the side effect of rescaling this arrays in the caller too.

+ * @param newStepSize new step size to use in the scaled and Nordsieck arrays + */ + public void rescale(final T newStepSize) { + + final T ratio = newStepSize.divide(getStepSize()); + for (int i = 0; i < scaled.length; ++i) { + scaled[i] = scaled[i].multiply(ratio); + } + + final T[][] nData = nordsieck.getDataRef(); + T power = ratio; + for (int i = 0; i < nData.length; ++i) { + power = power.multiply(ratio); + final T[] nDataI = nData[i]; + for (int j = 0; j < nDataI.length; ++j) { + nDataI[j] = nDataI[j].multiply(power); + } + } + + setStepSize(newStepSize); + + } + + +} diff --git a/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegratorTest.java b/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegratorTest.java new file mode 100644 index 000000000..408e64692 --- /dev/null +++ b/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegratorTest.java @@ -0,0 +1,78 @@ +/* + * Licensed to the Apache Software Foundation (ASF) under one or more + * contributor license agreements. See the NOTICE file distributed with + * this work for additional information regarding copyright ownership. + * The ASF licenses this file to You under the Apache License, Version 2.0 + * (the "License"); you may not use this file except in compliance with + * the License. You may obtain a copy of the License at + * + * http://www.apache.org/licenses/LICENSE-2.0 + * + * Unless required by applicable law or agreed to in writing, software + * distributed under the License is distributed on an "AS IS" BASIS, + * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + * See the License for the specific language governing permissions and + * limitations under the License. + */ + +package org.apache.commons.math4.ode.nonstiff; + + +import org.apache.commons.math4.Field; +import org.apache.commons.math4.RealFieldElement; +import org.apache.commons.math4.exception.MathIllegalStateException; +import org.apache.commons.math4.exception.MaxCountExceededException; +import org.apache.commons.math4.exception.NumberIsTooSmallException; +import org.apache.commons.math4.util.Decimal64Field; +import org.junit.Test; + +public class AdamsBashforthFieldIntegratorTest extends AbstractAdamsFieldIntegratorTest { + + protected > AdamsFieldIntegrator + createIntegrator(Field field, final int nSteps, final double minStep, final double maxStep, + final double scalAbsoluteTolerance, final double scalRelativeTolerance) { + return new AdamsBashforthFieldIntegrator(field, nSteps, minStep, maxStep, + scalAbsoluteTolerance, scalRelativeTolerance); + } + + protected > AdamsFieldIntegrator + createIntegrator(Field field, final int nSteps, final double minStep, final double maxStep, + final double[] vecAbsoluteTolerance, final double[] vecRelativeTolerance) { + return new AdamsBashforthFieldIntegrator(field, nSteps, minStep, maxStep, + vecAbsoluteTolerance, vecRelativeTolerance); + } + + @Test(expected=NumberIsTooSmallException.class) + public void testMinStep() { + doDimensionCheck(Decimal64Field.getInstance()); + } + + @Test + public void testIncreasingTolerance() { + // the 7 and 121 factors are only valid for this test + // and has been obtained from trial and error + // there are no general relationship between local and global errors + doTestIncreasingTolerance(Decimal64Field.getInstance(), 7, 121); + } + + @Test(expected = MaxCountExceededException.class) + public void exceedMaxEvaluations() { + doExceedMaxEvaluations(Decimal64Field.getInstance()); + } + + @Test + public void backward() { + doBackward(Decimal64Field.getInstance(), 4.3e-8, 4.3e-8, 1.0e-16, "Adams-Bashforth"); + } + + @Test + public void polynomial() { + doPolynomial(Decimal64Field.getInstance(), 5, 0.004, 6.0e-10); + } + + @Test(expected=MathIllegalStateException.class) + public void testStartFailure() { + doTestStartFailure(Decimal64Field.getInstance()); + } + +}